Thermodynamics is the seines of the energy transformation

T.D. Karapetyan

PHYSICAL AND COLLOIDAL

CHEMISTRY

Textbook to students of III and IV semesters of pharmaceutical

faculty

This textbook is adopted by the

Methodical Council of Foreign

Students of the University

YEREVAN 2006

2

YEREVAN STATE MEDICAL UNIVERSITY

AFTER M.HERATSI

Chemistry Department of Pharmaceutical Faculty

Completed by

TAMARA DAVID KARAPETYAN

Candidate of biological sciences

Associated professor of the Chemistry Department of Pharma-

ceutical Faculty

The textbook “PHYSICAL AND COLLOIDAL CHEMISTRY” is intended

to be studied by YSMU students of pharmaceutics faculty as well as by those of

General medicine and Medical and Biological colleges.

The present textbook includes all the chapters of Physical and Colloidal Chemistry

according to the curriculum.

The teaching material is illustrated by tables and pictures.

Yerevan State Medical University . 2006. 224 page

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4

CONTENTS

PART ONE 9

PHISICAL CHEMISTRY 9

INTRODUCTION 9

CHAPTER 1 10

THERMODYNAMICS 10

BASIC DEFINITIONS 11

§ 1. Types of thermodynamic processes 12

§ 2. Internal energy, work, heat 14

Tasks examples with solutions: 16

LAWS OF THERMODYNAMICS 16

§ 3. The Zeroth Law of thermodynamics 16

§ 4. The First law of thermodynamics 17

§ 5. Heat capacity 19


§ 6. Thermochemistry. Hess’s Law 21


§ 7. The Second law of thermodynamics 23


§ 8. The Third Law of thermodynamics 29

§ 9. Thermodynamic potentials 30


§ 10. The chemical potential 34

§ 11. Application of thermodynamic laws 36

Review questions: 37

CHAPTER 2 40

CHEMICAL EQUILIBRIUM 40

§ 1. Equilibrium constant 40

Chemical equilibrium in heterogeneous reactions 42

5

§ 2. Equation of the isotherm of a chemical reaction 42

§ 3. Equations of the isobar and isochor of chemical reaction 45



CHAPTER 3 49

PHASE EQUILIBRIUM 49

§ 1. Basic definitions 49

§ 2. Gibbs phase rule 52

§ 3. The Clausius – Clapeyron equation 53


§ 4. ONE–COMPONENT SYSTEMS 55

TWO–COMPONENT SYSTEMS 57

§ 5. Liquid-solid phase diagram 58

Application of thermal analyses in pharmacy 60

TWO–COMPONENT SYSTEMS OF TWO LIQUIDS 61

§ 6. The systems of two completely miscible liquids 61

§ 7. Liquid-vapor diagrams 63

§ 8. Distillation 65

§ 9. The systems of two partially miscible liquids 67

The lever rule 68

§ 10. The systems of two immiscible liquids 69

§ 11. The partition law 70

§ 12. Extraction 71

§ 13. Application of phase diagrams in pharmacy 73


CHAPTER 4 77

ELECTROCHEMISTRY 77

6

§ 1. Solutions of strong electrolytes 77

§ 2. The Debye-Huckel theory 78

§ 3. Ion motion in the electric field 81

ELECTRICAL CONDUCTIVITY 82

§ 4. Specific conductivity 82

§ 5. Molar conductivity 83

§ 6. Conductometric determinations 84



§ 7. ELECTRODE PROCESSES 88

TYPES OF ELECTRODES 91

§ 8. The electrodes of the first kind 91

§ 9. The electrodes of the second kind 92

§ 10. Redox electrodes 93

§ 11. Ion-selective membrane electrodes 95

§ 12. The diffusion and membrane potentials 96

§ 13. Bioelectrochemistry 97

§ 14. Galvanic cell 99

§ 15. Potentiometric determination 102

§ 16. Applications of e.m.f. measurements 103



§ 17. Electrolysis 108

§ 18. Electrode polarization 109

§19. Polarography 110


CHAPTER 5 112

CHEMICAL KINETICS 112

§ 1. The rate of reaction 112

§ 2. The reaction rate dependence on the reactants

concentration 114

KINETIC EQUATIONS 116

7

§ 3. Zero-order reactions 116

§ 4. First-order reactions 116

§ 5. Second-order reactions 117

§ 6. Third-order reactions 119

§ 7. Determination of the reaction’s order 119


COMPLEX REACTIONS 120

§ 8. Simultaneous (parallel) reactions 121

§ 9. Consecutive reactions 122

§ 10. Coupled reactions 123

§ 11. Reversible reactionsb 123

§ 12. Chain reactions 125

§ 13. Photochemical reactions 127

§ 14. Kinetics of heterogeneous reactions 128

§ 15. The reaction rate dependence on temperature 128

§ 16. Calculation of activation energy 132


CATALYSIS 133

Basic principles 133

§ 17. Homogeneous catalysis 134

§ 18. Enzyme catalysis 136

§ 19. Heterogeneous catalysis 139


PART TWO 145

SURFACES AND COLLOIDS 145

INTRODUCTION 145

CHAPTER 6 146

SURFACE PHENOMENA 146

§ 1. Surface tension 147

§ 2. Sorption 147

§ 3. Adsorption on the liquid surface 149

8

§ 4. Application of surfactants in medicine and pharmacy 151

§ 5. Adsorption of gases on solids 151

§ 6. Adsorption on solids from solution 155

§ 7. Electrolytes adsorption on solids from solutions 157


§ 8. Chromatography 159

§ 9. Classification of chromatographic methods 160


§ 10. Applying of the surface phenomena in pharmacy 165

CHAPTER 7 166

COLLOIDAL CHEMISTRY 166

INTRODUCTION 166

§ 1. Classification of the disperse systems 168

§ 2. Preparation and purification of the disperse systems 169

MOLECULAR – KINETIC PROPERTIES OF COLLOIDS 171

§ 3. Brownian motion 171

§ 4. Diffusion 172

§ 5. Osmotic pressure 174

§ 6. Sedimentation 174

OPTICAL PROPERTIES OF COLLOIDS 175

§ 7. Light scattering 176

§ 8. Light absorption 176


ELECTRICAL PROPERTIES OF COLLOIDS 178

§ 9. The structure of a double layer 179

§ 10. The electrical potentials at the double layer 181

§ 11. Methods of zeta-potential determination 184

§ 12. Micellar theory of colloid particle structure 184

§ 13. Stability and coagulation of colloids 185

§ 14. Protection of colloids 189

§ 15. Peptization 190


9

OTHER TYPES OF DISPERSE SYSTEMS 192

§ 16. Suspensions 192

§ 17. Emulsions 193

§ 18. Foams 195

§ 19. Aerosols 196

§ 20. Powders 199

§ 21. Solutions of surfactants 200

SOLUTIONS OF HIGH MOLECULAR SUBSTANCES 203

§ 22. Structure of polymers 204

§ 23. Dissolution of polymers 205

§ 24. Stability of polymer solutions 207

§ 25. Osmotic pressure of polymer solutions 208

§ 26. Viscosity of polymer solutions 210

§ 27. Polyelectrolytes 213

10

PART ONE

PHYSICAL CHEMISTRY

INTRODUCTION

The Physical Chemistry is one of the important and necessary subject not

only for pharmaceutics, but also for other many siences as organic chemistry and

biochemistry, physiology, general medicine etc. Besides, it has great application

for analysis in different areas.

Physical chemistry studies chemical processes along with their physical ef-

fects. It establishes connection between chemical and physical properties of the

matter. Physical chemistry joins chemistry, physics and mathematics that allow

creating mathematical models of the biological systems and making quantitative

accounts to carry out of the process.

Physical chemistry studies all the systems, where chemical reactions can

take place. It tries to solve the problem of the direction and limit of the process, to

determine the rate of the process and factors, influencing the rate, optimal condi-

tions for the process realization. By physical chemistry laws one can work out sci-

entific foundation of the chemical and pharmaceutical technology, to find ways for

manage the process. It helps to find new ways for new drug synthesis.

Knowledge of the physical chemistry permits to predict structure of the re-

quired drug, to find ways for its administration into the organism, to know how a

drug can be absorbed in the blood, distributed in the organism between different

organs and what ways it leaves the organism.

Physical chemistry is necessary for all those sciences the pharmaceutics

deals whit come. Students are often astonished that there are many physical and

chemical subjects during their study at the Medical University. However, aren’t it

that the organism and its parts are not only biological, but physicochemical systems

too and the processes which take place in them, describes with physicochemical

rules. Of course, we must say that biological processes are more complicated then

we do them in vitro. But physical chemistry gives us abilities to understand and to

manage them.

11

CHAPTER 1

THERMODYNAMICS

Thermodynamics (from Greek words for “heat” and “power”) studies heat,

work, energy and their changes during a process, occurred in the system. In a

broader sense, thermodynamics studies the relationships between macroscopic

properties of a system.

Thermodynamics deals with:

the conversion of energy from one form to another;

the energy effect in physical and chemical processes and their de-

pendence on the conditions under which a process takes place;

the possibility, direction and limits of spontaneous process (process,

in which work and energy are not supplied from outside of system);

determination of the conditions of equilibrium;

amount of maximum useful work that could be obtained from the

given process.

One part of the thermodynamics is Chemical Thermodynamics. It deals with

the transformation between chemical energy and other forms of energy. It studies

direction and possibility of the chemical process and calculates heat effect of the

reaction.

As we can see, thermodynamics is very important in our life as well as for

all physical and chemical processes. Our life and all processes proceeding in hu-

man’s activity are impossible without energy and its transformation. It involves

biological (vegetative and animal organisms), inorganic, industrial ranges, etc. So-

lar energy is Earth’s primary energy source. Solar energy is responsible for heating

the atmosphere and Earth’s surface, for the growth of vegetation and for global

climate patterns. And all this is possible due to transformation of one kind of ener-

gy into another, which is the object of study of thermodynamics.

Thermodynamically methods and calculations have a great advantage, be-

cause they allow to know heat amounts absorbed or escaped during the process

(i.e., to define amount of wasted energy or energy consumption), possibility and

direction of the process without knowing its microscopic mechanisms and without

making experiments.

12

However, thermodynamic method has two demerits. It deals only with mac-

roscopic systems consisting of large quantity of particles. Besides, the second de-

merit is that time as a parameter is absent in thermodynamic equations. That means

that the thermodynamics can give no information concerning the rate of a process.

BASIC DEFINITIONS

Before studying thermodynamics, it is necessary to define some basic defini-

tions, associated with this topic.

In physical chemistry, the universe is divided into two parts – the system

and its surrounding.

The system is a body or a group of interacting bodies that we consider apart

from its surrounding. It may be a reaction vessel, an engine, a biological cell, and

so on.

The surrounding is all around the system and so much greater than a sys-

tem that the heat exchanges between them are not accompanied changing in the

temperature of a surrounding.

There are some types of systems. The type of system depends on the charac-

teristics of the boundary, which limits it from the surroundings.

An open system can exchange energy and matter with its surround-

ings through the boundary of the system. For example, a cup of tea,

or an animal organism, or an open flask with solution in it.

A closed system can exchange only energy with its surroundings, but

not matter. For example, if the flask is closed so that water vapor

cannot escape from it or is condensed in the flask.

An isolated system is one that cannot exchange energy or matter

with its surroundings and has a constant volume. All the processes

can occur only inside of such system. The conception of an isolated

system has an ideal meaning, because there is no ideal isolator for

system walls.

An adiabatic system cannot exchange the heat with its surround-

ings.

Systems can also be homogenous and heterogeneous. Homogenous system

does not contain different parts of the system with different properties, i.e. there is

no boundary surface (interface) between them. If the system has boundary surface

(interface) between different parts of the system, it’s called heterogeneous.

13

The state of the system is described by its properties, the so called parame-

ters of the system. There are two types of parameters – extensive and intensive.

Intensive parameters do not depend on the quantity of substance and are the

same as for the whole system and as for its parts (for example, temperature, densi-

ty, viscosity). Properties such as enthalpy, entropy, free energy that depend on the

quantity of substance are called extensive parameters of the system.

The system can also be characterized by its functions. Thermodynamic func-

tion, value of which depends only on the state of the system, is called state func-

tion. The change of such function in a process depends only on the initial and final

states of the system. For example, if we have water in the glass with initial 200C

and final 600 C temperature, no matter how the initial state is converted to the final

state. The changes in temperature will always have the same value: ΔT=T1-T2. The

symbol Δ always means the final value minus the initial value. In the other exam-

ple, changes in the potential energy always the same, if we are on the same point

on the top of a mountain, no matter how we get there. The function value of which

depends on the path, which the system is brought from one state to the other, is

called path function. As we will see, work and heat are those functions.

§ 1. Types of thermodynamic processes

When a state of a thermodynamic system is changed, we say a thermody-

namic process occurs in a system. There are a few types of thermodynamic pro-

cesses.

In a cyclic process the final state is the same as the initial state. The change

in any state function for a cyclic process is obviously zero: ΔU=ΔH=ΔV=ΔT=0.

Spontaneous process takes place without work being done on the system. In

other words, such a process occurs only at the expense of the own energy reserve

of the system (without energy entry from the surroundings). It happens naturally: a

gas expands to fill the available volume, a hot body cools to the temperature of its

surroundings, a body falls from a higher level to a lower one, etc. Non-

spontaneous process must be brought about by doing work to the system: compres-

sion a gas to a smaller volume, cools an object with a refrigerator, etc. In closed

system every spontaneous process finishes in equilibrium state. A state of a system

that remains constant in time without the intervention of any external process is

called an equilibrium state.

14

Fig 1.1. An example of

a reversible process

Reversible process in thermodynamics has some other meaning, than in

chemistry, where it means only that the process occurred in two opposite direc-

tions. Let us consider the system on Fig.1.1. The sys-

tem consists of a gas contained within the piston with

cylinder walls. The gas pressure is marked Pint and the

external pressure is Pext. If they are equal, the system is

in equilibrium. If the external pressure above the pis-

ton increased by an infinitesimal amount, the gas

compressed slightly. If the external pressure reduced

infinitesimally, then the gas expands slightly. In each

increasing of external pressure the volume of the sys-

tem decreases and the gas pressure increases until the

pressure of a system again balances the external pres-

sure. Suppose we carry out an infinite number of infin-

itesimal changes in the external and internal pressure.

The finite volume change consists of an infinite number of infinitesimal steps and

takes an infinite amount of time to carry out. In this process, the difference between

the pressures on the two sides of the piston is always infinitesimally small and the

system remains infinitesimally close to the state of equilibrium. It is an example of

a reversible process. So, a reversible process is the one where the system is always

infinitesimally close to equilibrium, and infinitesimally changing in external condi-

tions accompanies changing in a process direction, under conditions that direction

changing does not accompany changing of the energy in surroundings. For exam-

ple, reversibility of any exothermic reaction supposes that heat amount escaped

during a reaction could be gathered and expanded for reversible endothermic reac-

tion. It means that changing of energetically state of surroundings does not occur

when direction of a process occurs. It is clear that a reversible process obviously

has an ideal meaning because in any process outflow of energy occurs and in real

conditions there is no way to gather all the escaped energy in surrounding.

In reversible process, the work obtained from the system is called the maxi-

mum useful work. It means that all internal energy turns to the work without loss

of energy. In irreversible (one-way) process part of internal energy wasted on car-

ried out work and another part outflows to surroundings in the form of heat. The

example of it is the fast irreversible piston arising in a gas container due to gas ex-

pansion. Waste of energy in the form of heat in such a process occurs due to work

against frictional forces and work against external pressure. In case of reversible

15

process it occurs so slowly and movement of a piston is so infinitesimally that

waste of energy does not occur.

§ 2. Internal energy, work, heat

Energy is a fundamental property of a system and its capacity to do work. In

thermodynamics, the total energy of a system is called internal energy, U. The in-

ternal energy is total kinetic and potential energy of the particles (molecules, at-

oms etc.) composing the system. It does not include the kinetics of motion of the

whole system or its potential energy. The internal energy depends both on the

amount of the substance and on the conditions under which the process takes place.

It’s a state function and an extensive property of a system. It is impossible to de-

termine the absolute value of the internal energy of a system, but only changes in

internal energy due to process, ΔU. It is determined as U2 - U1, where U1 and U2 are

the internal energy values in the initial and final states of the system when any pro-

cess occurs in the system. Infinitesimal change of internal energy is marked as dU.

The quantity of the internal energy change is positive if internal energy of the sys-

tem increases, and negative, if it decreases.

For gases, an internal energy depends on the gas pressure and on the temper-

ature, for liquids – on the volume and the temperature. For ideal gases it depends

only on the temperature.

We may change the internal energy of a closed system only in two ways: by

transferring energy as work or as heat. Heat and work are forms of energy transfer.

Heat, q, is an energy transfer between the system and surroundings due to a tem-

perature difference and be realized by chaotic motion of molecules. Work, W, is an

energy transfer between the system and surroundings due to macroscopic force

acting through a distance and is realized by organized motion of molecules.

When bodies having different temperatures are in contact, collisions between

molecules of two bodies lead to energy transfer from the hotter body to the colder.

Heat is work done at the molecular level.

Heat and work are not state functions; it is senseless to ask how much heat a

system contains (or how much work it contains). Heat and work are defined only in

terms of the process. Before and after the process they do not exist.

The signs of W, q and ΔU depend on the energetic state of a system. When

heat flows into the system from the surroundings during a process, q is positive

(q>0); an outflow of heat from the system to the surroundings means q is negative.

16

When work is done on the system by the surroundings (in a compression), W is

positive; when the system does work on its surroundings (in an expansion), W is

negative. A positive q and positive W each increase the internal energy and in these

cases ΔU is positive.

Internal energy, heat and work are measured in the same units, the joule (J).

Changes in molar internal energy are expressed in kilojoules per mole (kJmol–1).

Depending on the conditions under which the process takes place, the work

is determined by different equations.

For an isobaric (constant pressure) process integration of the work equation:

W = pdV 1.1

gives W = p (V2-V1)

Using the equation of ideal gas state for one mol ideal gas:

pV = RT 1.2

allow to obtain the following expression: 1.2 If we consider the expansion of one mole of gas, then using of the equation pV=RT, we can write

W = R(T2-T1) 1.3

From this equation we are able to know the meaning of the universal gas constant,

R. If T2 - T1 = 10C, so R = W. That means, that R is equal to the work of the gas

expansion, when its temperature increases in 1oC.

In an isothermal process (T=const) substituting p = nRT/V (from the equa-

tion of gas state pV = nRT) in Eq.1.1 we obtain the expression W = nRTV

dV. After

integration (at constant temperature, it may be taken outside the integral together

with n and R as constant quantity) we get:

W = nRT V

dV and W = 2.303 nRT lg

1

2

V

V or W = 2.303 nRT lg

2

1

P

P 1.4

In an adiabatic process (q=const) a gas does not absorb heat from its sur-

roundings, expansion work occurred at the expense of the internal energy. The in-

ternal energy of an ideal gas depends only on the temperature: consequently, ΔU is

equal to the product of the heat capacity Cv and the change in temperature:

W = Cv (T1-T2) 1.5

For an isochoric (constant volume) process dV = 0, consequently W = 0. In

this case, a system isn’t able to do expansion work.

17

Tasks examples with solutions:

1. Calculate the work done when 50 g of iron reacts with hydrochloric acid

in (a) a closed vessel of fixed volume, (b) an open beaker at 250C.

Solution: In (a) the volume cannot change, so no work is done and W=0. In

(b) the gas drives back to atmosphere and therefore W=–PexΔV. We can neglect the

initial volume because the final volume (after the production of gas) is so much

larger and ΔV≈Vf=nRT/Pex, where n is the amount of H2 produced. Therefore:

W=– PexΔV = –Pex nRT/Pex = – nRT

Because the reaction is Fe(s) + 2HCl(aq) → FeCl2(aq) + H2(g) we know that 1 mol H2

is generated when 1 mol Fe is consumed, and n can be taken as the amount of Fe

atoms that react. Because the molar mass of Fe is 55.85 g mol –1, it follows that

kJKmolJKgmol

gW 2.215.298314.8

85.55

50 11

1

2. Determine the maximum work at 20oC of isothermal compression of one

mole of ideal gas to half its volume.

Solution: From equation 1.4 we have:

Wmax = – 2.303 1.987 293.15 lg 2 = – 403.78 cal/mol

LAWS OF THERMODYNAMICS

§ 3. The Zeroth Law of thermodynamics

If system A is in thermal equilibrium with sys-

tem B, and B is in thermal equilibrium with system C,

then C is also in thermal equilibrium with A (Fig.1.2).

It is an experimental fact that two systems each of

them are found to be in thermal equilibrium with the

third system, it is in thermal equilibrium with each of

them. It is a formulation of the Zeroth law of thermo-

dynamics. It is called so, because it was defined after

definition of other laws of thermodynamics but a

statement of the Zeroth law logically precedes the other three laws.

Fig.1.2. Illustration of

the zeroth law of ther-

modynamics

18

So, there are some thermodynamic parameter common to systems being in

thermal equilibrium. This parameter is temperature. By definition, two systems

being in thermal equilibrium with each other have the same temperature.

The zeroth law allows us to assert the existence of temperature as a state

function.

§ 4. The First law of thermodynamics

The first law of thermodynamics is a postulate; it cannot be proven by logi-

cal reasoning, but follows from the sum total of human experience. A number of

consequences of this law are of great importance for physical chemistry and for

solution to various technological problems. By means of this law, we can perform

calculations of the energy balance, and in particular, the heat balance, and the heat

effect of various processes.

The first law can be formulated in several ways, which are essentially equiv-

alent to each other. One of them is the law of conversion of energy, applied to

thermodynamic processes:

Energy can be neither created nor destroyed and if energy disap-

pears in one form, it reappears in another form in a strictly equiva-

lent amount.

It is impossible to construct the first order perpetual-motion ma-

chine) i.e. a machine by which we could obtain work without spend-

ing an equivalent amount of energy.

The total energy of an isolated system is constant.

These remarks might be summarized in the form that in any process the

change in the internal energy of a system is equal to the heat imparted to the sys-

tem minus the expansion of work done by the system:

ΔU = q – W 1.6

For processes involving only infinitesimally small changes, it is written in the

form:

dU = dq – dW 1.7

This equation is the mathematical expression of the first law of thermodynamics.

The signs of each quantity in the equation depend on the conditions under which a

process takes place. In endothermic process q is positive and negative in exother-

mic process. Work is positive for work done on the system by the surroundings and

vise versa. If a system loses heat to the surroundings or does work on the surround-

19

ings, we would expect its internal energy to decrease since both are energy deplet-

ed in the process. This equation may have different expressions depending on con-

ditions under which the process occurred.

In isochoric process (dV=0) the system can not do expansion work, so

dW=0 and

dU = dq 1.8

In isochoric process, the heat imparted to the system is spent only to internal ener-

gy change.

In adiabatic process dq=0, so we can write:

– dU = dW 1.9

The system is able to do work only at the expense of internal energy.

In isothermal process, when T=const, U is constant too (because U for ideal

gas depends only on the T), so dU=0 and we can write:

dq = dW 1.10

Under such conditions, the heat imparted to the system is spent only on carried out

work. This equation has an ideal sense, because it is impossible to transform heat to

work in equal amounts.

For isobaric process p=const, so the system is able to do the work of expan-

sion and the first law has the following expression:

dq= dU + dW or dq = dU + pdV = dH 1.11

As we can see, to express the value of heat released or absorbed by the system un-

der constant pressure conditions chemists use another thermodynamic function of

the system, the enthalpy, H. (Most of physical and chemical processes, including

those that take place in living systems, occur under the constant pressure conditions

of our atmosphere). Because p, V and U are all state functions, the enthalpy is also

a state function.

Though change in enthalpy is equal to the heat effect of a process occurred

in a system, their signs are opposite, because H attributes to the system and q at-

tributes to the surroundings. So, in the exothermic process dH<0, but dq>0, be-

cause in such process the energy of the system decreases, while the energy of the

surroundings increases.

For ideal gases at constant temperature pΔV = ΔnRT, so the first law we can

write as:

ΔH = ΔU + ΔnRT 1.12

where Δn is the change of moles of the gas during the reaction.

20

Enthalpy changes accompany such processes as the dissolution of a solute,

the formation of micelles, chemical reactions, adsorption on solid surfaces, vapori-

zation of a solvent, hydration of a solute, neutralization of acids and bases, the

melting or freezing of solutes, etc.

§ 5. Heat capacity

When the heat flows into the system as a rule the temperature of the system

is proportionally increased and we can write, that q=CΔT. The proportionality co-

efficient C was called heat capacity. For infinitesimally changes:

C = dq/dT 1.13

Heat capacity is defined as amount of heat necessary to the change of the tempera-

ture of the system in one degree.

There are a few kinds of heat capacity:

Specific heat capacity is amount of heat necessary for one gram of a matter

temperature arising in one degree.

Molar heat capacity is amount of heat necessary for one mol of a material

temperature arising in one degree.

Heat capacity depends on the temperature and on the conditions under which

a process takes place. According to the thermodynamics first law, dq = dH in iso-

baric process and dq = dU in isochoric process, hence the heat capacity at con-

stant pressure, Cp, is equal to:

Cp = dH/dT 1.14

and the heat capacity at constant volume, Cv, is equal to:

Cv = dU/dT 1.15

From the first law dH>dU (dH=dU+pdV), consequently Cp>Cv, and the difference

between them is equal to the work of expansion: Cp – Cv = pdV. For ideal gas Cp–

Cv = R (from equation 1.3).


1. A cylinder fitted with a frictionless piston contains 3.00 mol of He gas at

P=1.00atm and is in a large constant-temperature bath at 400 K. The pressure is

reversibly increased to 5.00 atm. Find W, q and ΔU for this process.

Solution: Consider helium as a perfect gas. Since T is constant, ΔU is zero.

Equation 1.4 gives:

W=(3mol)(8,314 J mol–1 K–1)(400 K) ln (5.00/1.00) = 1.61104 J

21

From the first law of thermodynamics at constant temperature q= –W=1.61104 J

Of course, the work done on the gas is positive for the compression and the heat is

negative because heat must flow from the gas to the surrounding constant-

temperature bath to maintain the gas at 400 K as it is compressed.

2. Calculate W, q and ΔU for the process when 1.00 mol of water goes from

25.0oC and 1 atm to 30oC and 1 atm.

Solution: In this case, when we have increase the temperature by heating at 1

atm we could apply the equation q=CΔT. Hence: q=18.0g 1.00cal/(goC)

5.0oC=90 cal.

During the heating, the water expands slightly, doing work on the surrounding. At

constant P we have W=–p(V2–V1), where equation 1.2 was used. The volume

change is ΔV = V2 – V1 = m/ρ2 – m /ρ1, where ρ2 and ρ1 are the final and initial den-

sities of the water and m = 18.0g. A handbook gives ρ2 = 0.9956 g/cm3 and ρ1=

0.9970 g/cm3. We find ΔV = 0.025 cm3 and W=–0.025cm3atm or W=–0.0006cal

(because 1cm3atm=0.0242cal).

Thus, W is completely negligible compared with q, and ΔU=q+W=90cal. Because

volume changes of liquids and solids are small, usually work of expansion is sig-

nificant only for gases.

3. The oxidation of nitric acid to nitrogen dioxide is a key step in the for-

mation of smog:

2NO(g) + O2(g) → 2NO2(g) ΔHo = – 113.1 kJ

If 6.0 moles of NO react with 3.0 moles of O2 at 1.0 atm and 25oC to form NO2,

calculate the work done against a pressure of 1.0 atm. What is the ΔU for the reac-

tion? Assume the reaction to go to complesion. (The conversion factor is 1L atm =

101.3 J)

Solution: First we calculate the volumes of the reactants and product:

L

atm 1.0

K mol atm/K L mol

P

nRTV

L atm 1.0

K mol atm/K L mol

P

nRTV

product

treac

1472980821.00.6

2202980821.00.9

tan

The pressure-volume work is then

kJ atm L 1

J L 220-L 147atm VPW 39.7

3.1010.1

The change in internal energy can be calculated using equation 1.11: ΔU = ΔH –

PΔV. The enthalpy change shown above is for the formation of 2 moles of NO2.

Since 6 moles of NO2 are formed, the enthalpy change is 3(– 113.1 kJ). Thus:

22

ΔU = 3(– 113.1 kJ) + 7.39 kJ = – 331.9 kJ

4. The internal energy change when 1.0 mol CaCO3 in the form of calcite

converts to aragonite is +0.21 kJ. Calculate the difference between the enthalpy

change and the change in internal energy when the pressure is 1.0 bar given that the

densities of the solids are 2.71 g cm–1 and 2.93 g cm–1, respectively.

Solution: The change in enthalpy when the transformation occurs is ΔH =

H(aragonite) – H(calcite) = {U(a) +pV(a)} – {U(c) + pV(c)} = ΔU + pΔV

The volume of 1.0 mol CaCO3 (100g) as aragonite is 34 cm3, and that of 1.0 mol

CaCO3 as calcite is 37 cm3. Therefore,

PΔV = (1.0105 Pa) (34 – 37) 10–6 m3 = – 0.3 J (because 1 Pa m3 = 1J).

Hence, ΔH – ΔU = – 0.3 J

§ 6. Thermochemistry. Hess’s Law

The study of the heat amount produced or required during a chemical reac-

tion is called thermochemistry. We have already mentioned that amount of heat in

isobaric process is replaced by enthalpy and changes in enthalpy take place during

the chemical reaction. Therefore, there is a difference between enthalpy value for

initial and final states of the chemical system and this difference is a heat effect of

a process.

In 1840 year, G.Hess formulated the fundamental law of thermochemistry,

which is known by his name: the heat effect of a chemical reaction is independent

on the path of the reaction, i. e., on its intermediate stages and depends only on the

initial and final states of the system. In other words, if from given initial substanc-

es we can obtain given products in different ways, then whatever the path of the

reaction, i.e., whatever its intermediate stages, the over-all heat effect of the reac-

tion are the same for all the paths.

The importance of Hess’s law is that it’s applied in various thermochemical

calculations, particularly for chemical reactions heat effect calculating, especially

the heat effects of processes for experimental data are unavailable, or unobtainable,

or of the processes which have not been realized yet. This relates both to chemical

reactions, and to the processes of solution, vaporization, crystallization, adsorption,

etc.

In applying the law, however, one must be careful to verify that the condi-

tions, which it specifies, are satisfied. In the first place, the initial (and final) states

must be the same for both paths of the process. This means that not only the sub-

23

stances must have the same chemical compositions in both cases, but also the same

environmental conditions (temperature, pressure, etc.), state of aggregation and, for

crystalline substances, the same crystalline modification (and the degree of disper-

sion if they are in dispersed state). For example, the formation of one mole of va-

por H2O and one mole of liquid H2O from H2 and O2 at 25oC is accompanied by the

evolution of 57,798 and 68,317 kcal, respectively; the difference between these two

heats is 10,519 kcal and is equal to the heat of vaporization of one mole of water

under this conditions.

It follows from Hess’s law, that the heat effect of a reaction is equal to the

difference between the heats of formation of all the substances on the right-hand

side of the equation of the reaction and the heats of formation of all the substances

on the left-hand side (each with the proper coefficient of the equation). For exam-

ple, for the reaction

aA + bB = cC + dD

the standard enthalpy change is equal to:

ΔH0 = cH0(C) + dH0(D) –aH0(A) –bH0(B)

The standard heat of formation (ΔH0f) is the enthalpy change for the pro-

cess in which one mole of the substance in its standard state is formed from the

corresponding separated elements. (Standard states refer to the conditions 1 atm

and 25oC). For separated elements, the standard heat of formation, ΔH0, is equal to

zero.

If the heats of combustion of all the substances participating in a reaction are

known, the heat effect of a reaction is equal to the difference between the sum of

the reactants combustion heats and the sum of the products combustion heats.

The standard heat of combustion (ΔH0c) is the heat effect of the reaction of

oxidation by oxygen to the higher oxides. The standard heat of combustion for wa-

ter is equal to zero.

Calorimetric study of a reaction gives either ΔU (when V=const) or ΔH

(when p=const). Using the equation H=U+PV allows conversion between them:

ΔH0 = ΔU0 + pΔV 1.16

ΔH0 = ΔU0 + nRT 1.17


1. Find 0298H for the combustion of one mole of the simplest amino acid,

glycine, according to

24

NH2CH2COOH(s) + 9/4O2(g) → 2CO2(g) + 5/2H2O(l) + 1/2N2(g)

Solution: Substitution values of standard enthalpies from table data for each

reagent we obtain:

0298H = [1/2(0) + 5/2(–285.830) + 2(–393.509) – (–528.10) – 9/4(0)] = – 973.49

kJ/mol

2. For CO(NH2)2(s), 0

298,fH = – 333.51 kJ/mol. Find 0298,fU for this spe-

cies.

Solution: The formation reaction is C(graphite) + ½O2(g) + N2(g) + 2H2(g) →

CO(NH2)2(s) and has the change of mol numbers equal to: Δn = 1 – 9/2 = – 7/2.

Equation 1.17 gives 0298,fU = – 333.51 kJ/mol – (– 7/2)(8.31410–3 kJ/mol

K)(298.15 K) = – 324.83 kJ/mol.

3. The standard enthalpy of combustion 0298,cH of C2H6(g) to CO2(g) and

H2O(l) is – 1559.8 kJ/mol. Use this and table data of enthalpy values on CO2 and

H2O find 0298,fH of C2H6(g).

Solution: The combustion reaction is C2H6(g) + 7/2O2(g) → 2CO2(g) + 3H2O(l)

The consequence from Hess’s law is given for this combustion:

0cH = 2 0

fH (CO2) + 3 0fH (H2O) – 0

fH (C2H6) – 7/2 0fH (O2)

Substitution of the values of 0fH of CO2 and H2O and 0

cH is given at 298 K:

– 1559.8 kJ/mol = 2(– 393.51 kJ/mol) + 3(– 285.83 kJ/mol) – 0fH (C2H6) – 0

0fH (C2H6) = – 84.7 kJ/mol

§ 7. The Second law of thermodynamics

As we have seen, the first law states that the internal energy of an isolated

system is constant and that transformation between different forms of energy takes

place in equivalent amounts in the form of heat or work. The first law, however,

says nothing about the character, possibility or direction of the process. The second

law specifies first of all what processes in the system can occur spontaneously (i.e.,

without work being done on the system), what amount of work can be obtained

from such processes, and what is the limit of the spontaneous process. When the

system has attained an equilibrium state, it no longer has the ability to reverse it-

25

self. Consequently, all spontaneous processes are irreversible and their limit is in

equilibrium state. For example, we know, that heat flows spontaneously only from

a body at a higher temperature to a body at a lower temperature and the process

continues only until the temperatures of the two bodies become equal.

Every spontaneous process can be used to obtain the work; for example, fall-

ing water can rotate a turbine, a heat engine works due to heat flow etc. And the

second law studies interrelation between heat and work.

The second law, like the first law, is a generalization from human beings ex-

perience and has several statements:

heat cannot pass spontaneously from a colder body to a warmer one;

a process, the only result of which is the transformation of heat into

work is impossible;

it is impossible to construct a machine (a perpetual motion machine

of the second order) the only effect of which would be to transform

heat taken from the surroundings into work.

This means, that heat cannot be completely converted into work. All the

kinds of energy can be transformed into heat, but heat itself cannot be transformed

into work completely.

From this follows that the internal energy of the system cannot be complete-

ly converted into work; one part of energy, converting into work is called free en-

ergy, and the other part, transformed into the heat

and dispersed in the surroundings is called con-

strained energy. For example, when the gas expands

and the piston arises, one part of internal energy of

the gas is transformed into the work of the piston

arises, and the other part is transformed into heat due

to friction forces between piston and walls of the

container.

Let us discuss working principle of the heat

engine, a machine that does work by absorbing heat

from heat source. The principal question is: “How

much useful work can we get out of the engine for a

given amount of heat input?” In this case (Fig.1.3)

not all the heat qh that the working body receives

from heat source is transformed into work, but only a part of it: W=qh-qc. The other

part qc, which must be dispersed in the surroundings, is transferred to a body at a

Fig.1.3. Conversion of heat to

work in heat engine

26

lower temperature (cold sink). The work of such a machine consists not only in

taking a quantity of heat from a source and producing work, but also in simultane-

ous transferring a quantity of heat to a cold sink. Thermodynamic efficiency (η) of a

heat engine (the ratio of the heat performed to the work) is equal:

η =h

ch

h

ch

T

TT

q

qq

After mathematical changes it is stated in the following form:

011 c

c

h

h

h

c

h

c

T

q

T

q or

T

T

q

q

All the reversible cycle can be divided into an infinite number of infinitesimal

strips, and for each strip, we get:

0c

c

h

h

T

dq

T

dq 1.18

In this equation, dq is an infinitesimal portion of the heat transferring along the re-

versible cycle and T is the temperature at which this heat transfer occurs. The sum

of these infinitesimal transfers is a line integral around the cycle, and we get

0 T

dqrev

Since such integral of any reversible cycle is zero, it follows, that the value of the

line integral is independent of the path of the process and depends only on the ini-

tial and final states. Hence, dq/T is the state function. Clausius discovered this state

function in 1854 and called it entropy, S, from Greek word trope, meaning “trans-

formation”, which is related to the transformation of heat to work. Therefore:

T

dqdS rev 1.19

Entropy is an extensive state function. The commonly used units of S are J/(mol K)

or cal/(mol K).

Equation 1.19 is an analytical expression of the second law and the thermo-

dynamic definition of entropy according to which entropy is a measure of the con-

strained energy and shows the quantity of energy transferred as heat. In other

words, entropy is equal to amount of heat absorbed by cold sink at certain tempera-

ture.

This equation is right only for ideal reversible spontaneous processes. For ir-

reversible process dS is not equal dqirrev/T, because in such process a part of energy

27

is lost in the surroundings as heat (exept amount of heat absorbed by cold sink).

Hence:

T

dq

T

dq irrevrev and T

dqdS irrev

For general case, it gets as:

T

dqdS 1.20

It means that the sign of dS shows the direction of the process; in isolated system

only such process can occurred spontaneously, in which entropy arises.

Entropy also has statistical or molecular interpretation according to which

entropy is a measure of the molecular disorder of a system; entropy increasing

means increasing of molecular disorder. The statistical nature of the second law

supposes that it is applicable only to the systems composed of a very great number

of particles. This nature of entropy was first demonstrated in the works of L.

Boltzmann in 1896. By him it was shown that

S = k lgω 1.21

where k is called Boltzmann’s constant, lgω is a thermodynamic probability of the

given state of the system, and ω is the total number of microstates for the system of

certain energy level by which the microstate of the system could be realized.

Disordered state generally has a higher probability than ordered state. For

example, in the mixture of two gases the disordered, mixed state is far more proba-

ble than the ordered, unmixed state. The particles in the solid state are more or-

dered than those in the gaseous state. The water molecules are more ordered in the

liquid state than in the vapor state. Heating also increases the entropy of a system.

Let us examine some other processes. If two parts of a system are at different

temperatures, heat flows spontaneously and irreversibly from the hot part to the

cold part. Hence, it is more probable for the high-energy molecules of the hot part

to lose some of their energy to the low-energy molecules of the cold part, than vice

versa. It is therefore more probable for the internal molecular energy to be spread

out among the parts of the system, than to be an excess of such energy in one part.

These examples indicate that in the direction of the spontaneous process entropy

relates to the distribution or spread of energy among the available molecular energy

levels. So, change of entropy (ΔS) indicates a direction of the spontaneous process.

Thus, the entropy of an isolated system must increase in any irreversible sponta-

28

neous process: ΔS>0: Since the spontaneous process is finished in the state of

equilibrium, so the equilibrium is a state with maximal entropy: S=max.

Taking into consideration the above mentioned there are a few more state-

ments of the second law:

thermodynamic equilibrium in an isolated system is reached, when

the system’s entropy is maximized;

isolated system spontaneously change to more probable state (equi-

librium) with maximal entropy, i.e., with maximal disordered;

equilibrium state for isolated system is most probable, so it is the

most stable state.

A concept that follows from the second law is the principle of the degradation of en-

ergy. Since entropy of the Universe is continually increasing, energy is continually being

made unavailable for conversion to work; no energy will be available for doing work; the

entropy of the universe will be maximized, and all processes, including life, will cease. This

conception was formulated as the heat death of the universe. In the past, there has been

much philosophical speculation based on this supposed heat death. It seems clear at present,

that we need to know a lot more about cosmology before we can be certain what the fate of

the universe will be.

Entropy is an extensive parameter and state function, so its change on going

from state 1 to state 2 is given as ΔS=S2-S1. ΔS means ΔS of the system and does

not include any entropy changes that may occur in the surroundings.

Let us calculate ΔS for some processes. For calculation the standard entropy

is used, ΔS0, which is entropy under standard conditions (P=1atm and T=250C).

The entropy of phase transition. Because a change in a degree of molecu-

lar order occurs when a substance freezes or boils, we should expect that the phase

transition is accompanied by a change in entropy. The entropy of a solid substance

increases when it melts to a liquid, and it increases when the liquid phase turns into

a gas. Because at constant pressure q=ΔHtrs, the change in molar entropy of the

system is equal to

ΔStrs = ΔH/T 1.22

If the phase transition is exothermic (ΔH<0) as in freezing or condensing, then the

change of entropy is negative. If the transition is endothermic (as in melting), then

the change of entropy is positive and the system becomes more disordered. (Liq-

uids are more disordered than solids, and gases are more disordered than liquids).-

29

The entropy of a gas expansion. In isotherm processes dq=dW, and if we

set equation 1.4 in the 1.19 we get

T

V

VRT

T

dWS 1

2ln

from which

ΔS = Rln1

2

V

V = Rln

2

1

P

P 1.23

The variation of entropy with temperature. From the definition of heat ca-

pacity at constant pressure and constant volume (eqn 1.14 and 1.15) we obtain

2

1

T

TT

dTC

T

CdTS from which

ΔS = 1

2lnT

TC p and ΔS =

1

2lnT

TCv 1.24


1. Find ΔS for the melting of 5.0g of ice (heat of fusion =79.7cal/g) at 0oC

and 1atm.

Solution: The equation 1.22 gives: ΔS=ΔH/T=(79.7cal/g)(5.0g)/273K=1.46

cal/K=6.1 J/K

2. Let one mol of a perfect gas undergo an adiabatic free expansion into a

vacuum. Calculate ΔS if V2=2V1.

Solution: According the equation 1.23: ΔS=Rln2=(8.314 J/mol K)

(0.693)=5.76J/mol K

3. Find ΔS when 100g of water is reversibly heated from 25oC to 50oC at 1

atm. The specific heat capacity Cp of water is nearly constant at 1.00 cal/goC in

this temperature range.

Solution: For the 100g of water, the heat capacity is Cp=100g1.00

cal/goC=100cal/K. For the heating process equation 1.24 gives:

ΔS=1

2lnT

TCv =(100cal/K)ln

K

K

298

323=8.06 cal/K=33,7 J/K

4. Use data of standard entropy value to find oS298 for the reaction 4NH3(g) +

3O2(g) = 2N2(g) + 6H2O(l).

30

Solution: Substitution of standard entropy value gives: oS298 =2(191.61) +

6(69.91) – 4(192.45) – 3(205.138) = – 582.53J/mol/K

Gases tend to have higher entropies than liquids, and the very negative oS298 for

this reaction results from the decrease of 5 moles of gases in the reaction.

5. To calculate the entropy change in the surroundings when 1 mol H2O is

formed from its elements under standard conditions at 298 K, we use ΔH0 = – 286

kJ from table data.

Solution: The heat released is supplied to the surroundings, now regarded as

being at constant pressure, so qsur = +286 kJ. Therefore,

15

959298

1086.2

JKK

JS sur

This strongly exothermic reaction results in an increase in the entropy of the sur-

roundings as heat is released into them.

§ 8. The Third Law of Thermodynamics

At absolute zero temperature all thermal motions of the atoms of a crystal

cease and the solid hasn’t any disorders, all atoms have only one position and

hence its entropy is equal to zero. This conclusion lies in the basis of the third law

of thermodynamics, which states that the entropy of a perfect crystalline material

is zero at absolute zero temperature. It follows from the equation 1.21: at T=0 on-

ly one microstate is possible, therefore ω=1 and lnω=0, from which S=0. It has

been found for real crystals that in the nature there is no perfect crystalline struc-

ture with a perfectly ordered arrangement and any crystalline material obligatory

has some defects of the crystal structure. It means that the entropy of real crystals

doesn’t vanish at absolute zero; it only approaches to zero.

This postulate has a restricted nature, but by this one can determine absolute

entropy value of various substances at different temperatures, if we know the heat

capacities of these substances within the range from absolute zero to the tempera-

ture of interest.

31

§ 9. Thermodynamic potentials

Now let us speak about free energy, at the expense of which a system per-

forms a work during the process.

Entropy is the basic concept for discussing the direction of real change (pro-

cess), but to use it we have to analyze the entropy changes both the system and its

surroundings. Consider a system in thermal equilibrium with its surroundings at

temperature T. When a change in the system occurs and there is a transfer of ener-

gy as heat between the system and its surrounding, total entropy change is the sum

of the entropy changes both the system and its surroundings and we get an equa-

tion dSsyst + dSsur≥ 0. Because dSsur = – dqsur/T, and dqsur = – dqsyst (the heat that

enters into the system comes from surroundings) it follows that for any change

0T

dqdS

syst

syst

At constant volume dq=dU and we get

0T

dUdS

We can write it in form dUTdS or 0TdSdU

The function (U – TS) is called the Helmholtz free energy and is denoted by the

symbol A:

A = U – TS

For infinitesimal changes it is in the form

dA = dU – TdS 1.25

Consequently, for constant volume–temperature processes

dA ≤ 0

The same for processes carried out at constant pressure, when dq=dH, we

obtain another thermodynamic quantity – the Gibbs energy, which is defined as

dG = dH – TdS 1.26

And for constant pressure–temperature processes

dG ≤ 0

For this comes consequence that only such spontaneous process can take place that

lead to a decrease in free energy–Helmholtz, A (when V=const, T=const) or Gibbs,

G (when P=const, T=const); the process goes to equilibrium at which these func-

tions reach a minimum value, i.e., the criterion for equilibrium is:

A = min, dA = 0 , G = min, dG = 0

32

Some remarks on the Helmholtz energy. A negative value of dA is favored

by a negative value of dU and a positive value of TdS. So, its tendency of a system

to move towards states of lower internal energy and higher entropy. However, this

interpretation is not quite right, because the tendency to lower A is solely a tenden-

cy towards states of greater overall entropy. Systems change spontaneously if in

doing so the total entropy of the system and its surroundings increases, not because

they tend to lower internal energy. Conception of Helmholtz energy includes the

entropy change of the system (dS) and the entropy change of the surroundings and

their total tends to a maximum.

The Helmholtz energy also has a meaning of the “maximum work function”,

because ΔA is the part of the change in internal energy that we are free to use to do

work:

Wmax = ΔA

Some remarks on the Gibbs energy. The Gibbs energy is more common in

chemistry, because, at least in laboratory chemistry, we are usually more interested

in changes occurred at constant pressure than at constant volume. If G decreases as

the reaction proceeds, then the reaction has a spontaneous tendency to convert re-

actants into products. If G increases, then the reverse reaction is spontaneous. In

spontaneous endothermic reactions enthalpy increases, the system rises spontane-

ously to states of higher enthalpy and dH>0. Because the reaction is spontaneous

we know that dG<0; it follows, that the entropy of the system increases so much

that TdS is strongly positive and outweighs dH. Endothermic reactions are there-

fore driven by the increase of entropy of the system, and this entropy change over-

comes the reduction of entropy brought about in the surroundings by the inflow of

heat into the system (dSsur = – dH/T at constant pressure).

In general we can say, that if outweigh the value of TdS, the direction of the

spontaneous process determines by the change of entropy of the system. If out-

weigh the value of dH, the direction of the spontaneous process determines by the

change of entropy of the surroundings.

The decrease in free energy continues until the system reaches an equilibri-

um state, when ∆G becomes zero. This process can be thought of as a gradual us-

ing up of the system’s ability to perform work as equilibrium is approached. Free

energy can therefore be looked at in another way in that it represents the maximum

amount of work, Wmax (other than the work of expansion), that can be extracted

from a system undergoing a change at constant temperature and pressure. This non-

33

expansion work can be extracted from the system as electrical work, as in the case

of a chemical reaction taking place in an electrochemical cell, or the energy can be

stored in biological molecules such as adenosine triphosphate (ATP).

Both A and G are extensive parameters and state functions and have units of

energy (J or cal). The change of Gibbs energy in the reaction one can calculate the

same as enthalpy and entropy: ΔG is equal the difference between Gibbs energies

of products and reactants in there stoichiometric numbers of moles.

Thus, among thermodynamic potentials there are internal energy, Helmholtz

and Gibbs energies and enthalpy, which values decrease in spontaneous process,

proceeding in closed system.

Basic equations of thermodynamics

All thermodynamic state-function relations can be derived from six basic

equations. The first law for a closed system is dU = dq – dW, and if only P-V work

is possible, then dW = pdV . The second law is dS=dq/T that gives dq = TdS.

Hence, under this conditions, dU = TdS – pdV. This is the first basic equation; it

combines the first and second laws and calls the fundamental equation of thermo-

dynamics. Equation shows, that entropy is the function of the internal energy and

the volume. It applies to both reversible and irreversible processes. In the case of a

reversible process, TdS may be identified with dq, and -pdV with dW. When the

change is irreversible, TdS>dq and –pdV>dW.

The next three basic equations are the definitions of H, A, and G: dH = dU +

pdV, dA = dU - TdS and dG = dH - TdS.

Collecting these expressions, for reversible process in closed system, when

only P-V work is occured we have:

dU = TdS – pdV 1.27

dH = TdS – Vdp 1.28

dA = – SdT – pdV 1.29

dG = – SdT + Vdp 1.30


1. Calculate ΔG and ΔA for the reversible (equilibrium) process of the vapor-

ization of 1.00 mol of H2O at 1.00 atm and 100oC.

34

Solution: Because the process is in equilibrium at constant T and P, so ΔG

=0. For ΔA we have the equation ΔA = ΔU – TΔS. Use of ΔU = q + W and ΔS =

q/T gives ΔA = q + W – q = W. At constant pressure the work is equal to W = PΔV.

We can ignore the volume of liquid water (because it is much lesser than the vol-

ume of the gas), and can estimate the molar volume of the gas from the ideal-gas

law: V = RT/P = 30.6103 cm3/mol. Therefore ΔV = 30.6103 cm3/mol and W =

(– 30.6103cm3atm) (0.101325 J) = – 3.10 kJ = ΔA. (1 cm3atm = 0.101325 J).

2. Calculate 0298G formation for the reaction H2(g) + 1/2O2(g) → H2O(l).

Solution: Using the table data for 0298,fH and 0

OH,298,f 2S values we ob-

tain:

0

,298, 2OHfS = 0,298

0,298

0,298 222 2

1OHOH SSS = 69.91 – 130.684 – ½(205.138) = –

163.343J/molK

0298,fH is equal to – 285.830 kJ/mol. So, from equation ΔG = ΔH – TΔS we ob-

tain 0298G = –285.830 kJ/mol – (298,15 K) (– 0.163343 kJ/mol K) = – 237.129

kJ/mol

3. Find 0298G for the reaction 4NH3(g) + 3O2(g) = 2N2(g) + 6H2O(l) using the

table data.

Solution: 0298G is equal to the difference between of final and initial reac-

tants Gibbs energies: 2(0) + 6(–237.129) – 3(0) – 4(–16.45) = – 1356.97 kJ/mol

The negative value of Gibbs energy indicates the reaction proceeding spontaneous-

ly toward the products formation.

4. When (glucose) is oxidized to carbon dioxide and water at 25oC according

the equilibrium

C6H12O6 (s) + 6O2(g) → 6CO2(g) + 6H2O(l)

Calorimetric measurements give ΔUo = – 2808 kJ mol–1 and ΔS = +182.4JK–1 mol–

1 at 25oC. How much of this energy change can be extracted as (a) heat at constant

pressure, and as (b) work.

Solution: (a) Because Δn = 0, we know that ΔHo = ΔUo = – 2808 kJ mol–1.

Therefore, at constant pressure, the energy available as heat is 2808 kJ mol–1.

(b) Because T = 298 K, the value of ΔAo is

35

ΔAo = ΔUo – TΔSo = – 2862 kJ mol–1

Therefore, the combustion of 1.0 mol C6H12O6 can be used to produce up to 2862

kJ of work.

5. How much energy is available for sustaining muscular and nervous activi-

ty at 37oC (blood temperature) from the combustion of 1.00 mol of glucose mole-

cules? The standard entropy of reaction is +182.4 J K–1mol–1. From the table data

ΔHo for this reaction is equal to – 2808 kJ mol–1.

Solution: According the equation ΔGo = ΔHo – TΔSo we obtain

ΔGo = – 2808 kJ mol–1 – (310 K)( 182.4 J K–1mol–1) = – 2865 kJ mol–1

Therefore, the reaction can be used to do up to 2865 kJ of non-expansion work.

§ 10. The chemical potential

The Gibbs energy is function of pressure and temperature, which states from

the equation 1.30. The basic equations do not apply when the composition of the

system is changing due to interchange of matter with the surroundings or to irre-

versible chemical reaction or irreversible interphase transport of matter within the

system. We now develop equations that hold during such processes.

When the one-phase system consists of two or more components, the Gibbs

energy is function not only of P and T, but also of the composition of a system:

G = f(T, p, n1, n2,........ni)

where n are the mole numbers of the components in one-phase system. The total

differential of this is

jjjj nPTinPTnTnP dn

dG dn

dn

dGdP

dP

dGdT

dT

dGdG

,,

1

,,1,,

.............

where the subscript ni on a partial derivative means the mole numbers of the

changeable component when all the other components mole numbers, nj, are held

fixed.

The relation

jnPTidn

dG

,,

is defined as chemical potential, μ:

36

µ =

jnPTidn

dG

,,

1.31

It gives the change of the Gibbs energy of the phase with respect to the moles

of component i added at constant pressure, temperature and other components

mole numbers. The chemical potential of substance i in the phase is a state function

and intensive property of a system that depends on the temperature, pressure, and

composition of the phase.

The chemical potential of a perfect gas at pressure P can be written as

µ = µ0 + RTlnP 1.32

and for ideal solution at an activity a

µ = µ0 + RTlna 1.33

where µ0 is the standard chemical potential; it is the molar Gibbs energy of the

pure gas at 1 bar, or of ideal solution with activity equal to 1. For a pure substance,

it is equal of the Gibbs energy and so is the molar Gibbs energy. For a substance in

a mixture, the chemical potential is the partial molar Gibbs energy.

In general, when we deal with mixture system, for every substance in it we

use the term partial quantities, because their properties are different from those in

pure state. Such quantities are distinguished by a bar above the symbol for the par-

ticular property. For example

2

,,2

Vdn

dV

jnPT

In practical terms, the partial molar volume represents the change in the total

volume of a large amount of solution when one additional mole of solute is added;

it is the effective volume of one mole of solute in solution. Partial molar quantities

are of importance in the consideration of open systems, that is those involving

transference of matter as well as energy. Of particular interest is the partial molar

free energy, which also referred as the chemical potential. The chemical potential

therefore represents the contribution per mole of each component to the total free

energy. It is the effective free energy per mole of each component in the mixture,

and is always less than the free energy of the pure substance.

The chemical potential µj of substance j in certain phase must increase when

the mole fraction nj is increased by the addition of substance j at constant T and P.

The chemical potential is a measure of escaping tendency. The greater the value of

37

µj, the greater the tendency of substance j to leave this phase and flow into the

phase, where it’s chemical potential is lower. This flow will continue until the

chemical potential of substance j has been equalized in all the phases in the system.

As a substance flows from one phase to another, the compositions of the phases are

changed and the chemical potential in the phases are changed.

Therefore, substance j flows spontaneously from a phase with higher

chemical potential µj to a phase with lower chemical potential µj.

In a closed system in thermodynamic equilibrium, the chemical potential

of any given substance is the same in every phase in which that substance is pre-

sent.

§ 11. Application of thermodynamic laws

Thermodynamic methods and calculations have a great application in all the

processes accompanied with change of energy; not only in heat and mechanical

engines, but also to electrochemical cells, chemical reactions, atmospheric phe-

nomena, processes occurring in vegetable and animal organisms and many other

problems were investigated not only to their energy balance, but also to the possi-

bility, direction and limit of the spontaneous trend of the process under the given

conditions. It was used to determine the conditions of equilibrium and the maxi-

mum amount of useful work that could be obtained from a certain process.

The importance of thermodynamics in the pharmaceutical sciences is appar-

ent when it is realized that such processes as the partitioning of solutes between

immiscible solvents, the solubility of drugs, micellisation and drug-receptor inter-

action can all be treated in thermodynamic terms. It allows calculating the heat

amounts, absorbed or escaped in certain process and to know the energy expense

for there realization. Enthalpy changes accompany such processes as the dissolu-

tion of a solute, the formation of micelles, chemical reaction, adsorption on the sol-

ids, vaporization, hydration of a solute, melting or freezing of solutes etc.

In contrast to the first law of thermodynamics, which is applicable for all the

kinds of systems, the second law has some restrictions, because it has a statistical

nature and therefore applicable only to systems composed of a large number of par-

ticles, i.e., to systems the behavior of which can be described by the laws of statis-

tics. In spite of it, the second law allows to indicate the external conditions neces-

38

sary for the process of interest to proceed in the desired direction. For processes in

which work must be done, we can determine the amount of work to be carried out

and how this quantity depends on the external conditions.

The second law of thermodynamics is the law of increasing entropy that

means increasing disorder. Living organisms maintain a high degree of internal

order. Hence one might ask whether life processes violate the second law. Now

consider, can we define the entropy of a living organism? Processes are continually

occurring in a living organism, and the organism is not in an equilibrium state. The

equilibrium state for an organism means death. But the state of a fully grown living

organism remains about the same from day to day and the organism is in a steady

state. Living organisms are open systems, since they take in and expel matter and

exchange heat with there surroundings. According to the second law, we must have

ΔSsyst + ΔSsurr ≥ 0 for an organism, but ΔSsyst (ΔS of organism) can be positive, neg-

ative, or zero. Any decrease in ΔSsyst must be compensated for by an increase in

ΔSsurr. The organism takes in highly ordered large molecules such as proteins,

starch, and sugars, entropy of which is low; the organism excretes waste products

that contain smaller, less ordered molecules, entropy of which is high. Thus the

entropy of the food intake is less than the entropy of the excretion products re-

turned to the surroundings. The organism discards a matter with greater entropy

content than the matter it takes in, thereby losing entropy to the environment to

compensate for the entropy produced in internal irreversible processes.

The preceding analysis shows there is no reason to believe that living organ-

isms violate the second law. But measurement of ΔSsyst + ΔSsurr for an organism

would be a difficult task.

Review questions:

1. What does thermodynamics study?

2. Define the terms “system” and “surroundings”.

3. What kinds of system do you know? Give a few examples of a close, open

and isolated system. What kind of system is living organism?

4. Give the definition of extensive and intensive parameters of a system and

note a few examples of them. Is the volume, pressure, temperature, density exten-

sive or intensive parameters of a system?

39

5. Define the following terms: internal energy, work and heat. How does a

system exchange energy with surroundings? What is the difference between work

and heat in meaning of molecules motion?

6. Which does the sign of internal energy, work and heat depend from. What

is the sign of internal energy and work in the process of (a) gas expansion and (b)

gas compression?

7. Write the expression of work for isobaric and isochoric processes. Explain

the physical meaning of the ideal gas constant (R) using one of these equations.

8. What is the state and path function of a system? Note, which among the

following terms – internal energy, work, heat – is state or path function and why?

9. Explain the meaning of the reversible process in thermodynamics. Why

does it have an ideal meaning?

10. What does the term “maximum useful work” mean and why does it oc-

cur in reversible process?

11. Formulate the zeroth law of thermodynamics. What thermodynamic pa-

rameter is common to systems in thermal equilibrium?

11. Give a few formulations of the first law of thermodynamics. Write the

common expression of the law and its expressions under different conditions of the

process: isochoric, isobaric, isothermic and adiabatic. Which of them has ideal

meaning and why?

12. What is enthalpy and what is its difference from heat of the process?

13. Write the sign of dH and dq for exothermic process and explain the dif-

ference.

14. By what quantity distinguishes enthalpy and internal energy? For what

systems there is no such difference?

15. Give the formulation of heat capacity. What types of it do you know?

16. What is the difference between CP and CV and why? What is equal to this

difference for ideal gas?

17. Determine what a simultaneous process is. What is the limit of such pro-

cess? Is it reversible or not? With which purpose one can use such process? Give a

few examples of a simultaneous process.

18. Give a few formulations of the second law of thermodynamics.

19. What is the heat engine and how does it work?

40

20. Write the expression of the second law of thermodynamics. What is the

thermodynamic meaning of this expression? Measure of what kind of energy is

entropy?

21. What is the statistical interpretation of entropy? Write the Boltzmann

equation and explain the meaning of w.

22. How is entropy connected with the degree of ordered of a system?

Which system is more ordered – liquid, gas or solid?

23. How does entropy change in a simultaneous process? What is the value

of S and ∆S in equilibrium state?

24. Write expressions of entropy for the process of gas expansion and at var-

iable temperature.

25. Explain the term of “free energy” and the difference between Gibbs and

Helmholtz energy. Write the equations of Gibbs and Helmholtz energies.

26. Proceeding from the equation of Gibbs energy note the conditions of a

simultaneous process.

27. How does entropy change in a simultaneous process? What is the value

of G and ∆G in equilibrium state?

28. How is Gibbs energy connected with the entropy of a system and of a

surrounding?

29. Give the definition of chemical potential and note its difference from

Gibbs energy.

30. What is the tendency of chemical potential change in the same species in

a thermodynamic system?

41

CHAPTER 2

CHEMICAL EQUILIBRIUM

§ 1. Equilibrium constant

Chemical reactions move towards a dynamic equilibrium in which both reac-

tants and products are present but have no tendency to change. In some cases, the

concentration of products in the equilibrium mixture is so much greater then the

concentration of unchanged reactants that for all practical purposes the reaction is

“complete”. However, in many important cases the equilibrium mixture has signif-

icant concentrations of both reactants and products, and it is very important to state

relation between them and thermodynamic probabilities of such a system.

Equilibrium is a state in which there are no observable changes as time goes

by. Equilibrium could be physical when it states between two phases of the same

substrate (such as liquid water and water vapor) and chemical equilibrium between

left and right sides of the chemical equation.

Stable chemical equilibrium characterized by the following general condi-

tions:

1. The equilibrium of the system remains unchanged when the external

conditions are kept constant.

2. If an external force causes a slight displacement from equilibrium, the

system tends to return by itself to the equilibrium state.

3. Equilibrium has a dynamic character.

4. Tendency to the equilibrium state approaches from two opposite direc-

tions – both reactants and products.

5. Rates of both direct and reverse reactions are equal.

6. The Gibbs free energy of the system has the least value.

7. Amounts of both reactants and products are constant.

The quantitative relation between amounts of components in equilibrium

state gives by equilibrium constant. The equation of the equilibrium constant is

deduced kinetically, based on the equality both of the rates direct and reverse reac-

42

tions and on the dependence of the reaction rate on the concentrations of the reac-

tants and products, according the law of mass actions. But we must have in view,

that this approach is applicable only to comparatively simple reactions. In the gen-

eral case one should make use of the thermodynamical deduction.

Consider the homogeneous reaction

aA + bB cC + dD

where a, b, c, d are the stoichiometric coefficients of the substances respectively.

For direct and reverse reactions rates we have (according to the law of mass ac-

tions):

dc

revrev

ba

dirdir DCkV and BAkV

In equilibrium state the rates of direct and reverse reactions are equal, so

rev

dir

k

k ba

dc

BA

DCcK 2.1

We see that the equilibrium constant of a reaction is expressed as a ratio of the rate

constants of the forward and reverse reactions.

Kc is the equilibrium constant deduced for reactions in dilute solutions. For

solutions of higher concentrations the activities ai should be used in place of the

concentrations (Ka):

ab

dca

aa

aaK

In a similar manner for ideal gases the expression of the equilibrium constant

includes the partial pressures of reagents (Kp), and for nonideal gases – the fugaci-

ties f (Kf).

There is connection between Kp and Kc:

Kp = Kc (RT)Δn 2.2

where Δn is a difference between the number of moles in the products and reac-

tants.

The equilibrium constant depends only on the temperature and the nature of

the components and does not depend on the amounts of the reagents. The equilibri-

um constant helps us to predict the direction in which a reaction mixture will pro-

ceed to achieve equilibrium and to calculate the concentrations of reactants and

products in equilibrium state.

43

Chemical equilibrium in heterogeneous reactions

When the reactants are in different phases, then at constant temperature equi-

librium partial pressure of each of solid substances is a constant and equal to the

saturated vapor pressure over the pure phase of those substances. Due to it their

partial pressures are not involved in the equilibrium constant equation. For example

in reaction

FeO + CO = Fe + CO2

FeO and Fe are both in solid state, soCO

CO

pP

PK 2

Or in reaction CaCO3 = CaO + CO2 pK =2COP and, hence, the pressure

2COP should be a constant for each temperature, independence of the amount of

CaCO3 and CaO in the system.

§ 2. Equation of the isotherm of a chemical reaction

It must be a relation between equilibrium constant and Gibbs energy, be-

cause they both are characterized by the state of equilibrium. Applying the concept

of the free energy decreasing in spontaneous process to chemical equilibrium, we

can derive simple relationships between free energy and the equilibrium constant.

At constant temperature, this relation is expressed by equation of the isotherm. As

we know from the equation 1.30

dG = VdP – SdT

For one mol ideal gas at constant temperature it becomes in the form dG = VdP.

Using the ideal gas law (PV=RT) for V and integrate the equation, we obtain:

1

212 ln

2

1

P

PRTdP

P

RTGG

P

P

If P1 is equal to 1atm (standard conditions) so G1 is the standard Gibbs energy and

we get

G = G0 + RTlnP 2.3

Now consider the reaction in gaseous phase

44

aA + bB cC + dD

The change of Gibbs energy for this reaction is equal to:

BADC bBaAdDcGG

For each meaning of G the eqn 2.3 is applied and we obtain

b

BaA

dD

cC

PP

PPRTGG ln0 2.4

In equilibrium state ΔG=0 and partial pressures come to the equilibrium constant

value, so b

Ba

A

dD

cC

PP

PP = Kp and we get

ΔG0 = – RTln Kp 2.5

Inserting eqn 2.5 in eqn 2.4 we obtain

(lnRTG b

Ba

A

dD

cC

PP

PP– ln Kp) 2.6

This is called the equation of the isotherm of the chemical reaction or van’t

Hoff equation. It links the Gibbs energy and equilibrium constant with partial pres-

sures of the components. We must know, that the partial pressure values of the first

member of the equation (in brackets) are changed in every moment of the reaction

and aren’t equal to the partial pressure values in the expression of ln Kp, which

refer only to equilibrium state and have a constant values at a definite temperature.

So the difference in the brackets isn’t equal to zero.

The equation allows to know the direction of the spontaneous process depend-

ing on the substance amount. We know, that if ΔG<0, the forward direction of a re-

action is spontaneous, if ΔG>0, the reverse reaction is spontaneous, and if ΔG=0, the

reaction in equilibrium. According to the isotherm equation the condition of the for-

ward reaction to be spontaneous is ln Kp>lnb

Ba

A

dD

cC

PP

PP, and the reverse reaction to be

spontaneous is lnb

Ba

A

dD

cC

PP

PP> lnKp.

The equation 2.5 is most important in chemical thermodynamics; it is a con-

nection between thermodynamic abilities of the system and the equilibrium constant.

ΔG0 value can readily be calculated from the tabulated data and hence equation 2.5 is

45

important because it provides a method of calculating equilibrium constant without

resort to experimentation.

Different chemical reactions can be compared with respect to their ability to

proceed spontaneously. This is usually carried out for conditions in which all the

substances are in standard states. Thus, the value of ΔG0 is constant at a certain tem-

perature for a certain reaction and depends only on the nature of substances. The ΔG

and, hence, the direction of spontaneous process depends on the concentrations or

partial pressures of the substances taking part in it.

One may derive a link between equilibrium constant, and enthalpy and entro-

py. On one hand we have equation 2.5: ΔG0= –RTln Kp, and on the other hand from

thermodynamics: ΔG0 = ΔH0 –TΔS. So, we can equate the right sights and obtain:

R

S

RT

HKP

00

ln

2.7

These equations applies for standard entropy and enthalpy definitions; if we

plot the graphical dependence between ln Kp and 1/T, for exothermic process it has

the view like graphic on Fig.2.1., and for endothermic process – on Fig.2.2.

Fig. 2.1. Fig. 2.2.

Dependence the equilibrium constant on temperature for exothermic and

endothermic process

The tangent of the angle slope is equal to –ΔH0/R and the segment on the abscissa

is equal to ΔS0/R.

If we know the values of equilibrium constants at two temperatures, we can

find the value of the reaction heat effect from the following:

46

R

S

RT

HK

0

1

0

1ln

and

R

S

RT

HK

0

2

0

2ln

Subtracting one from another we obtain

21

0

2

1 11ln

TTR

H

K

K 2.8

If the heat effect and the value of equilibrium constant at one temperature are

known, one can find the equilibrium constant at any temperature.

§ 3. Equations of the isobar and isochor of chemical reaction

As the equilibrium constant depends on the temperature, we can obtain the

equation of this dependence. Differentiating the isotherm equation with respect to T

gives us the following

RT

G

P

ln

bB

aA

dD

cC

PP

PP – RlnKP – RT

dT

Kd Pln

Inserting in the equation

dT

GTHG

the values of ΔG from isotherm

equation and the expression found forPT

G

, we obtain

RT lnb

Ba

A

dD

cC

PP

PP– RTlnKP = ΔH + RT ln

bB

aA

dD

cC

PP

PP – RTlnKP – RT2

dT

Kd Pln

and after cancellation

dT

Kd Pln =

2RT

H 2.9

This equation establishes a relation between the change in the equilibrium

constant with temperature and the heat effect of the reaction. It is called the isobar

equation and valid for the processes, which take place at constant pressure.

The similar equation can be obtained for a process, which takes place at con-

stant volume, and it is called the isochoric equation

2

ln

RT

U

dT

Kd C 2.10

47

The equations describe change in the equilibrium constant with change in

temperature. Indeed, if in direct reaction heat evolves, i.e., ΔH<0, then from eqn

2.9 we conclude, that for this reaction dT

Kd Pln<0. This denotes that the equilibrium

constant decreases with a rise in temperature, i.e., the relative content of products

become less and the equilibrium is displaced towards the left. In other words, with

a rise in temperature the equilibrium is displaced in the direction of the endother-

mic reaction.

Thus, these equations are quantitative expressions of Le Chatelier principle:

a system at equilibrium, when subjected to a disturbance, responds in a way that

tends to minimize the effect of the disturbance.


1. A mixture of 11.02 mmol of H2S and 5.48 mmol of CH4 was placed in an

empty container and the equilibrium for reaction 2H2S + CH4 4H2 + CS2 was

established at 700oC and 762 torr. Analysis of the equilibrium mixture found 0.711

mmol of CS2. Find KP and ΔG0 for the reaction at 700oC.

Solution: Since 0.711 mmol of CS2 was formed, 4(0.711 mmol) = 2.84 mmol

of H2 was formed. For 0.711 mmol reacted, and 5.48 mmol – 0.71 mmol = 4.77

mmol was present at equilibrium. For H2S, 2(0.711 mmol) reacted, and 11.02 –

1.42 mmol = 9.60 mmol was present at equilibrium. To find KP, we need the partial

pressure Pi, We have Pi = xiP, where P = 762 torr and the xi’s are the mole frac-

tions. Mole fraction of each component is equal to the relation of its mmols and

mmols of the sum of all components: SH2x = 9.60/17.92 = 0.536,

4CHx = 0.266,

2Hx = 0.158, 2CSx =0.0397. Hence, partial pressure for H2S is equal to 0.536(762

torr) = 408 torr, for CH4 = 203 torr, for H2 = 120 torr and for CS2 = 30.3 torr. Equi-

librium pressure for the reaction is equal to:

KP =

42

22

2

4

CHSH

CSH

PP

PP =

203408

3.301202

4

= 0.000331

The use of ΔG0 = – RT ln KP0 at 700oC (973 K) gives

ΔG0 = – (8.314 J/mol K)(973 K) ln 0.000331 = 64.8 kJ/mol

48

2. Find KP0 at 600 K for reaction N2O4(g) 2NO2(g) using the approxima-

tion that ΔH0 is independent on T and equal to 57.20 kJ/mol. It is also known that

ΔG0 = 4730 J/mol.

Solution: From ΔG0 = – RT ln KP0, we find KP

0 = 0.148. Substitution in

equation 2.8 gives:

40600,

0600,

1063.1

609.11600

1

15.298

1

/314.8

/57200

148.0ln

P

P

K

K K K molJ

molJ K

3. In the reversible reaction CH3COOH + C2H5OH CH3COOC2H5 +

H2O from 1 mol acid and 1 mol alcohol 0,7 mol of ether is formed. Determine the

value of equilibrium constant.

Solution: Since from 1 mol acid and 1 mol alcohol 0,7 mol of ether is formed

up to the state of equilibrium, so 0,3 moles of acid and alcohol remains when equi-

librium is stated. Using the equation 2.1

KC = [CH3COOC2H5] [H2O] / [CH3COOH] [C2H5OH] = 0.70.7 / 0.30.3 = 5,4

4. The equilibrium constant of the reversible reaction FeO(s) + CO(g)

Fe(s) + CO2(g) is equal to 0.5. Find the equilibrium amounts of the CO and CO2, if

their initial amounts are: [CO] = 0.05 mol/l and [CO2] = 0.01 mol/l.

Solution: According the chemical equilibrium in heterogeneous systems

CO

CO

pP

PK 2 . If we denote the amount of reacting species up to the moment of equi-

librium as x then in state of equilibrium [CO] = (0.05 – x) and [CO2] = (0.01+ x).

Hence, using the equation of Kp we obtain: 0.5 = (0.01+x)/(0.05–x), from which

x=0.01. Therefore, [CO2] = 0.02 mol/l and [CO] = 0.04 mol/l.

Review questions:

1. Determine the equilibrium state of a system.

2. Mention general conditions of chemical equilibrium.

3. What is the quantitative characteristic of chemical equilibrium?

4. What expresses the equilibrium constant? What factors does it depends

on? What does it mean if the value of equilibrium constant is (a) more than zero,

(b) less than zero.

5. In what cases the equilibrium constant is expressed by activity?

49

6. How is the equilibrium constant distinguished from heterogeneous reac-

tions?

7. Try to derive the equation of isotherm of chemical equilibrium.

8. Write the equation of isotherm of chemical equilibrium. Between what

quantities does it state the link?

9. How one can know thermodynamic abilities of the system by this equa-

tion?

10. Under which condition of the equation does a spontaneous process occur?

11. What way is the isotherm equation applied?

12. Derive the equation states the link between equilibrium constant and en-

thalpy and entropy of a system.

13. How can we apply this equation for enthalpy change determination?

14. Write the isochor and isobar equations of the chemical equilibrium. Which

relation of equilibrium constant do they express?

15. How is it related to Le Chatelie principle?

50

CHAPTER 3

PHASE EQUILIBRIUM

§ 1. Basic definitions

Most substances, depending on external conditions (temperature and pres-

sure) can exist in three basic aggregate states: gaseous, liquid and solid. Further-

more, a solid substance can exist in two or more forms (modifications) of the solid

state that differ in internal structure and properties, such as the white and black al-

lotropes of phosphorus. This phenomenon is known as polymorphism.

Between different aggregate states of substances the phase transition can oc-

cur. Тransition from the liquid to the gaseous state is called vaporization, from the

solid to the gaseous – sublimation, from the gaseous to the liquid or solid – con-

densation. Тransition from the solid state to the liquid is called melting, or fusion,

and the reverse process – solidification or crystallization (or freezing if it occurs at

a low temperature). Transition from one modification of the solid state to another is

called a polymorphic transformation.

A phase (p) is a homogeneous portion of a system that is uniform throughout

in chemical composition and physical state and is separated from other parts of the

system by a boundary surface. Phase consisting of a single chemical substance is

called simple (or pure) phase. A homogeneous system always consists of a single

phase, while a heterogeneous one contains at least two phases. The number of

phases in a system is denoted p. A gas, or a gaseous mixture, is a single phase, also

as a crystal, or two totally miscible liquids. An alloy of two metals is a two-phase

system (p=2) if the metals are immiscible, but a single phase one if they are misci-

ble.

A constituent of a system is a chemical species that can be separated from it and is

capable of exist in the isolated state for some time. Thus, the constituents of an

aqueous solution of sodium chloride are water and sodium chloride. The ions of

sodium and of chlorine are not constituents, since they cannot exist as separate sub-

stances.

51

A component (c) is a chemically independent constituent of a system. The

number of components is the minimum number of independent species that neces-

sary to define the composition of all the phases present in the system. When no re-

actions take place in the system, the number of components is equal to the number

of constituents. Thus, pure water is a one-component system (c=1), because we

need only the species H2O to specify its composition. Similarly, a mixture of etha-

nol and water is a two-component system (c=2). When a reaction occurs in the sys-

tem, we need to decide the minimum number of species that, after allowing for re-

actions in which one species is synthesized from others, can be used to specify the

composition of all the phases. Consider, for example, the system in the equilibrium

CaO(s) + CO2(g) CaCO3(s)

Phase 1 Phase 2 Phase 3

in which there are three phases. To specify the composition of the gas phase we

need the species CO2, and to specify the composition of Phase 1 we need the spe-

cies CaO. However, we don’t need an additional species to specify the composition

of Phase 3 because its identity (CaCO3) can be expressed of the other two constitu-

ents by making use of the stoichiometry of the reaction. Hence, the system has

three constituents but only two components.

Example: How many components are present in a system in which ammoni-

um chloride undergoes thermal decomposition?

Answer: Chemical reaction is NH4Cl(s) NH3(g) + HCl(g) so the system

consists of three constituents. However, NH3 and HCl are formed in fixed stoichi-

ometric proportions by the reaction. Therefore, the composition of the system can

be expressed in terms of the single species NH4Cl. It follows that there is only one

component in the system (C = 1).

The number of degrees of freedom (f) or variance is defined as the number

of parameters (temperature, pressure) that can be arbitrary varied (within certain

limit) without changing the number or kind of phases in the system. In a one-

component one-phase system (c=1, p=1) the pressure and temperature may be

changed independently without changing the number of phases, so f=2. We say

that such system is bivariant, or it has two degrees of freedom. On the other hand,

if two phases are in equilibrium (a liquid and its vapor, for instance) in a one-

component system, the temperature (or the pressure) can be changed at will, but the

change in temperature (or pressure) demands an accompanying change in pressure

52

(or temperature) to preserve the number of phases in equilibrium. That is, the vari-

ance of the system has fallen to 1.

Systems are usually classified according to the number of phases as one-

phase, two-phase, three-phase, etc., and to the number of components as simple,

binary, ternary, etc. According to the number of degrees of freedom or variance

they are classified as invariant, monovariant, bivariant, etc.

A phase change or phase transition is a process in which at least one new

phase appears in a system without the occurrence of a chemical reaction (for in-

stance, ice melting, water freezing etc.). The reason of such transition justifies on

the basis of thermodynamic tendency of a system to equality of the chemical poten-

tial. A given component may pass spontaneously only from a phase in which its

chemical potential is greater to the one in which it is smaller. As a result, the dif-

ference between the chemical potentials of the component in both phases decrease,

and when it becomes zero a state of phase equilibrium sets in. In such case, the

temperature and the pressure will be assumed constant and the same in all the parts

of the equilibrium system.

Example: Let us consider a case of a crystal of ICN is added to pure liquid

water and the system is held at 25oC and 1 atm. Eventually a saturated solution is

formed, and some solid ICN remains undissolved. At the start of the process, is the

chemical potential of ICN grater in the solid phase or in the pure water? What hap-

pens to the chemical potential of ICN in each phase as the crystal dissolve? (See if

you can answer these questions before reading further).

Answer: At the start of the process, some ICN “flows” from the pure solid

phase into the water. Since substance flows from a phase with higher chemical po-

tential to the phase with lower one. Hence, the chemical potential of ICN in the

solid must be greater than in pure water. Since the chemical potential is an inten-

sive quantity and the temperature, pressure, and mole fraction in the pure-solid

phase do not change as the solid dissolved, the chemical potential of ICN remains

constant during the process. As the crystal dissolves, the mole fraction of ICN in

the aqueous phase increases and the chemical potential of ICN increase. This in-

crease continues until the chemical potential of ICN becomes equal to each other in

solid and liquid phases. The system is then in phase equilibrium, no more ICN dis-

solves, and the solution is saturated.

53

In some cases the system is in a metastable state; the system is in visual

equilibrium, but the value of Gibbs energy does not get minimum. Examples of

metastable states are a supersaturated vapor, or liquids cooled below their freezing

point (supercooled liquid) or heated above their boiling point (superheated liquid).

Conversion to real equilibrium is accompanied with a decrease of Gibbs energy

value to its minimum.

§ 2. Gibbs phase rule

One of the most general law of physical chemistry is the law of phase equi-

librium, which was derived by J.Gibbs in 1876. It gives a general relation between

the variance, the number of components, and the number of phases in equilibrium:

f = c – p + n 3.1

where n is the external factors effects on the state of the system. Usually two fac-

tors, temperature and pressure, can effect on the state of the system and n = 2.

Hence, the phase rule becomes in the form

f = c – p + 2 3.2

Systems are possible, however, state of which is effected by other external

factors, such as electric, magnetic or gravitation fields. In such cases n will be dif-

ferent, corresponding to another number of external factors. On the other hand, n

may be equal to 1, when the pressure or the temperature is constant.

The number of degrees of freedom cannot be negative, hence the phase rule

is sometimes written as follows:

p ≤ c + 2 3.3

Example: Find f for a system consisting of solid sucrose in equilibrium with

an aqueous solution of sucrose.

The system consist of two chemical species (water and sucrose), so c = 2,

and two phases (the saturated solution and the solid sucrose), so p = 2. Hence, f = c

– p + 2 = 2. Two degrees of freedom make sense, since once T and P are specified,

the equilibrium mole fraction (or concentration) of sucrose in the saturated solution

is fixed.

54

§ 3. The Clausius – Clapeyron equation

The Clapeiron equation gives the relative dT/dP in one-component and two-

phase system, i.e., the pressure and temperature, in which two phases can coexist.

If the phase transition occur between two phases (melting, freezing, vaporization),

the change of the Gibbs energy in each phase, according eqn 1.28, is equal to

dT2 SdPVdG and dTSdPVdG 22111

In equilibrium state changes in the Gibbs energies of the two phases are equal, so

we can equate the right sides of the equations and obtain:

S

V

dP

dT or dPVVdTSS

1212

Substitute the value of the entropy change from the second law of thermodynamics

and obtain:

VT

H

dT

dP or

H

VT

dP

dT

3.4

where ΔV is the volume change occurred in phase transition, and ΔH is the heat

effect of phase transition.

For vaporization and sublimation, relative dP/dT shows the change of the

saturated vapor pressure with change of transition temperature. For melting and

crystallization, dT/dP shows how the transition temperature changes with the

change of atmospheric pressure. The sign of this relation depends on the signs of

ΔV and ΔH. For example, in melting process the enthalpy is positive and the vol-

ume change is usually positive (exception the ice melting). Consequently, the slope

dT/dP is usually positive. It means that with atmospheric pressure increase the

melting temperature also increases.

For vaporization process the equation can be written as

H

VVT

dP

dT lv

where Vv is the molar volume of the vapor, Vl – of the liquid. Equation is mostly

used for calculating heats of vaporization or saturated vapor pressure. However, it

frequently happened that no data on the molar volumes of a vapor and this may be

a stumbling to the practical application of this equation. But in many cases the

equation can be simplified. Because the molar volume of a gas is much greater than

55

the molar volume of a liquid we can write ΔVvap = Vl. Moreover, if the gas behaves

perfectly, we can applied the equation PV = RT (for one mole gas), from which

Vvap = RT/P. These two approximations turn the Clapeyron equation into

2

2

RT

HdT

P

dP or

HP

RT

dP

dT

After integration

2

1

2

1

2ln

T

T

P

PT

dT

R

HPd

it becomes in the form

211

2 11ln

TTR

H

P

P vap 3.5

This equation is applied not only for the heat calculation of the process but also it

allows to calculate vapor pressure at any temperature if we know vapor pressure at

a certain temperature.


1. Estimate the effect of pressure at the boiling point of a liquid if ΔHvap/T

for this vaporization process is equal to 85 J K–1 mol–1 and the molar volume of a

perfect gas is about 25 L mol–1.

Solution: We use the equationVT

H

dT

dP

. Because the molar volume of a

gas is so much greater than the molar volume of a liquid, we can write ΔVvap = V(g)

– V(l) ≈ V(gas). Therefore,

1-

3-

-1

K Pamolm1025

mol K1 J 85

dT

dP 3

13104.3

This value corresponds to 0.034 atm K–1, and hence to dT/dP = 30 K atm–1. There-

fore, a change of pressure of +0.1 atm can be expected to change a boiling tem-

perature by about +3 K.

2. The saturated vapor pressure of liquid sulphur dioxide is P1 = 0.9138 atm

at t1 = – 12.0oC, and P2 = 1.091 atm at t2 = – 8.0oC. Find the vaporization heat of

sulphur dioxide.

Solution: Applying equation 3.5 and using common logarithms, we obtain:

56

2.265

1

2.261

1

987.1303.29138.0

091.1lg

vapH

vapH = 6.100 cal/mol

3. The heat of fusion of ice at 0oC is 1.436 cal/mol = 59,3 l-atm/mol; the mo-

lar volume of ice at this temperature is 19.652 ml, and that of water is 18.018 ml.

Find how the melting point of ice depends on the pressure.

Solution: Inserting these values in equation 3.4, we obtain:

007530.-

59.3

18.01819.652273.15

dP

dT

deg/atm

In other words, the melting point of ice lowers by 0.00753o when the pressure is

increased by 1 atm.

§ 4. ONE–COMPONENT SYSTEMS

For a one-component system, which is described by two intensive parame-

ters (P and T) only one species is present and c=1, so the phase rule comes to form:

f = 3 – p

If p = 1, then f = 2; if p = 2, then f = 1; if p = 3, then f = 0. Consequently, in one-

component and two-phase system the maximum number of degrees of freedom is

two, and the maximum number of phases in equilibrium is three.

In physico-chemical analyses are widely applied phase diagrams to charac-

terize various systems, which are usually plotted from experimental data. Such dia-

grams, showing dependence on the state of the system (and its phase equilibrium)

on external conditions or on the composition of the system are called the diagram

of state or phase diagram. The diagram shows, how the number and the kind of the

phases change, when pressure or (and) temperature change, as well as the relation

of dP/dT. On phase diagram each phase is represented by an open area, two-phase

equilibrium state by intersection lines of these areas, and three-phase equilibrium

state by intersection point of these lines.

We shall now illustrate how these general features appear in the phase dia-

gram of pure substance as water (Fig.3.1.). The one-phase regions are the open

areas; here p = 1 and f = 2. It means that in bivariant system both the temperature

and the pressure can vary independently (within fixed limits, which are represented

57

by lines) without causing any change in the number or kind of phases of the sys-

tem. For instance, the temperature of liquid water can vary within certain limits at

some fixed pressure without any new phase appearing. If, however, water is heated

beyond a certain temperature, it will be converting to the vapor, or if it is cooled

too much it may be transformed into ice.

Along the lines (exept at point O), two

phases are in equilibrium, hence f = 1 (mono-

variant system).

Line OC (liquid-vapor line) shows the

boiling point of water as a function of pressure.

For all the points along this line f = 1; it means,

that only the temperature or only the pressure

can vary arbitrary without causing any change

in the number or kind of phases, and the varia-

tion of one of these parameters entails a corre-

sponding variation of the other. For example,

one can vary the temperature of an equilibrium

system of liquid water and its saturated vapor,

but to maintain a state equilibrium it’s necessary to allow for a corresponding

change in pressure. For example, the system in point b it represented by pressure 1

and temperature Tb. If we change the value of the pressure we have to change the

value of the temperature only in certain magnitude to stay in two-phase equilibrium

(on line OC).

The boiling point of a liquid at a given pressure is the temperature T at

which its equilibrium vapor pressure equals to P. The normal boiling point is the

temperature at which vapor pressure of a liquid equal to 1 atm (bar).

The system moves along the liquid-vapor line toward point C with both T

and P increasing (dT/dP>0). When it reaches point C the both liquid and vapor

densities become equal to each other. Hence, a two-phase system is converted into

a one-phase system, and the liquid-vapor line ends. Point C is the critical point.

The temperature and pressure at this point are the critical temperature and critical

pressure. At any temperature above this liquid and vapor phases cannot coexist in

equilibrium, and isothermal compression of the vapor will not cause condensation.

Line OB is the solid-liquid equilibrium line for water and gives the melting

point of ice as a function of pressure. Notice that the line has a negative slope (in

Fig.3.1. The phase

diagram of water

58

contrast to most other substances) up to 2000 atm, which means that the melting

temperature falls as the pressure is raised (dT/dP<0). The reason for this unusual

behavior can be explained by decrease in volume that occurs on ice melting, which

is a result of the molecular structure of the water. The density of ice is less than that

of water, and an increase in pressure always favors the appearance with a smaller

specific volume, i.e., a greater density.

The melting point of a solid at a given pressure P is the temperature at which solid

and liquid are in equilibrium for pressure P. The normal melting point of a solid is the

melting point at P=1atm. For a pure substance, the freezing point of the liquid at a given

pressure equals the melting point of a solid.

Line OA is the ice-vapor (sublimation) line and gives the values of pressure and

temperature, in which ice and vapor are in equilibrium.

Point O, where three phases in equilibrium, is called the triple point. In this case

the system has zero degree of freedom (i.e., it is invariant). There is only one combina-

tion of pressure and temperature at which all three phases can exist in equilibrium. The

slightest change in either of these parameters causes one or two of the phases to disap-

pear.

Now consider another example of one-

component system, the phase diagram for sulfur

(Fig.3.2.). The solid sulfur can exist in two modifi-

cations – orthorhombic and monoclinic forms under

any conditions. We know that in one-component

system four phases cannot be in equilibrium be-

cause variance cannot be negative. Consequently,

that on the phase diagram appear four regions of

pure phases, six lines for two-phase equilibrium

states, the existence of three triple points and there

is no one point with four phases in equilibrium.

TWO–COMPONENT SYSTEMS

When two components are present in a system c=2 and phase rule becomes

in form f=4–p. In this case in addition two independent intensive variables as pres-

sure and temperature the third factor, the composition (expressed by the mole frac-

Fig.3.2. The phase dia-

gram of sulfur

59

tion of one component), appears. Hence, we might have a three-dimensional dia-

gram, the description of which is quite difficult. Due to it we usually keep P or T

constant and plot a two-dimensional phase diagram of two forms: pressure-

composition or temperature-composition dependence.

Now we consider phase diagrams of two-component systems, the simplest

type of which is the liquid-solid system with a eutectic.

§ 5. Liquid-solid phase diagram

Let substances A and B are miscible in all proportions in the liquid phase

and completely immiscible in the solid phase. Mixing any amounts of liquids A

and B will produce a single-phase system that is a solution of A and B. Since solids

A and B are completely insoluble in each other cooling a liquid mixture of A and B

causes either pure A or pure B freezes out of the solution. The typical appearance

of the solid-liquid phase diagram of the

temperature-composition dependence is

shown in Fig.3.3. Points tA and tB are the

freezing points of pure A and B (f=0). In the

low-temperature limit (below the line cd) we

have a two-phase mixture of pure solid A

and pure solid B, since the solids are immis-

cible. In the high-temperature limit (above

the line tAetB) we have a one-phase liquid

solution of A plus B, since the liquids are

miscible. Now consider cooling a liquid so-

lution of A and B suitable point a1. When

the point a2 reaches the solvent B begins to

freeze out, giving a two-phase region with

solid B in equilibrium with a solution of A

and B. During further cooling the composition of liquid mixture changes (it be-

comes poor at the B), so the freezing temperature lowered along the line tBe (the

region between points a2 and a4). In this region f=2–2+1=1. The curve tBe gives

the depression of the freezing point of B and allows knowing the freezing tempera-

ture for each composition. The same occurs at the left side of diagram with differ-

ence that the solid A begins to freeze out from the liquid mixture. As it is, we have

Fig.3.3. The solid-liquid

phase diagram

60

a one-phase liquid solution of A plus B above the curve tAetB and that’s why it’s

called the liquidus curve and corresponds to the temperature of the onset of crystal-

lization for each suitable composition.

The two freezing-point curves intersect at point e, where both A and B are

frozen out together. This point is called the eutectic point. At point e three phases

are in equilibrium – solution, solid A and solid B, so we have f=2–3+1=0. The

composition suitable for the eutectic point is called eutectic composition. The eu-

tectic mixture has the lowest melting point among all the rest mixtures. The tem-

perature suitable of this point is called the eutectic temperature at which the tem-

perature remains constant until all the solution has frozen and the number of phases

has dropped to 2. Below the line cd we have a mixture of solid A and solid B that’s

why this line is called the solidus curve.

One way to determine a solid-liquid phase diagram experimentally is by

thermal analysis. Here, one allows a liquid solution (melt) of two components to

cool and measures the systems temperature as a function of time; this is repeated

for several liquid compositions to give a set of cooling curves. Typical cooling

curves for the simple eutectic system of Fig.3.3 are shown in Fig.3.4.

There are six cooling curves for following compositions of A and B:

Fig.3.4. Cooling curves of thermal analysis and melting diagram

for system with eutectic

61

A,% 100 80 60 40 20 0

B,% 0 20 40 60 80 100

curves Ι ΙΙ ΙΙΙ ΙV V VΙ

The curves Ι and VΙ are for pure A and B. When pure substances are cooled,

the temperature slightly falls to point 3, where the crystallization begins and the

temperature remains constant while the entire sample freezes, because the heat

evolved during freezing will compensate for the heat loss due to natural cooling.

After the sample is frozen, further cooling causes a lowering of the temperature.

So, the cooling curves of the pure substances exhibit horizontal sections at the

melting point; otherwise the pure substances crystallize at certain constant tem-

perature.

The cooling curves of solutions are of a more complicated form. Let us take,

for example, cooling curve ΙΙ. The temperature slightly falls to point 2, where the

slopes of the cooling curve changes. In this point formation of crystals is accompa-

nied by evolution of heat and causes the temperature to fall more slowly, and leads

to a smaller inclination of the curves, although not to the appearance of a horizontal

section. It appears, when the eutectic point is reached since both components is

now crystallized from the solution, and the temperature remains constant. Further

lowering of the temperature again proceeds uniformly. Melts of other compositions

(curves ΙΙ, ΙΙΙ and V) exhibit similar curves with the only difference that the tem-

peratures in the beginning of the crystallization (point 2) are different for different

compositions.

Among all the compositions set apart the curve V, which freezes at constant

temperature, as a pure substance (forms a horizontal section). This curve is for a

liquid mixture with the eutectic composition. The eutectic composition looks like

pure substances in the following properties:

1. In the eutectic point the liquid and the solid phases in equilibrium

and their compositions are equal.

2. The freezing temperature is unchanged.

3. The number of degrees of freedom is equal to zero.

Application of thermal analyses in pharmacy

The value of drug melting temperature gives the information about their sol-

ubility; the higher this temperature the less is solubility.

62

The measurement of drug melting point allows to make their identification

and to know the degree of purity. It is because, as we have mentioned above, that

each species has a certain melting (freezing) temperature and addition of the other

species reduces it and due to this the more admixture the more temperature de-

creasing.

Drugs having the same melting points can be identified by measurement of

the eutectic temperature of the mixture of this drug with various other compounds,

which, as a rule, is different.

It’s important to determine the eutectic composition of a powder or a solid

drug mixture, because it helps to solve the problem of drug physical compatibility.

The eutectic composition has the lowest melting temperature and determination of

it allows knowing the temperature, at which drugs can stay at room temperature

without melting or wetting.

At the eutectic temperature, the solution crystallizes out forming a micro-

crystalline mixture of pure substances. That is important, if we want to obtain a

tender powder for the skin (especially for the child’s gentle skin). Besides, micro-

crystalline formation increases the solubility of poorly soluble drugs that has obvi-

ous pharmaceutical possibilities. For instance, the solid eutectic solution of the

griseofulvin dissolves six to seven times faster than pure griseofulvin.

TWO–COMPONENT SYSTEMS OF TWO LIQUIDS

When a system consists of two liquids three causes are possible: completely

miscible, partially miscible and immiscible liquids. We’ll discuss all these causes.

§ 6. The systems of two completely miscible liquids

In such systems both components are volatile and, hence, have a vapor pres-

sure. Therefore, besides total vapor pressure P is in equilibrium with the solution,

we must also take into consideration the partial pressures P1 and P2 of the compo-

nents, where P = P1 + P2. The partial vapor pressure of each component, by Ra-

oult’s law is proportional to its mole fractions:

63

220

110

1 XPP and XPP 2 where 10P and 2

0P are vapor pressures

of pure components. The total vapor pressure therefore is equal

P = 110 XP + 22

0 XP = 10P (1–X2) + 22

0 XP = 10P + X2( 2

0P – 10P ).

Example: The vapor pressure of benzene is 74.7 torr at 20oC, and the vapor

pressure of toluene is 22.3 torr at 20oC. A certain solution of benzene and toluene

at 20oC has a vapor pressure of 46.0 torr. Find the benzene mole fraction in this

solution and in the vapor above this solution.

Solution: Benzene (b) and toluene (t) molecules resemble each other closely,

so it is a good approximation to assume an ideal solution and use Raoult’s law. The

vapor pressure of the solution is

46.0 torr = Pb + Pt = Xb0

bP + Xt0

tP = Xb (74.7 torr) + (1 – Xb)(22.3 torr)

Solving, we find Xb = 0.452. The benzene partial vapor pressure is Pb = Xb0

bP =

0.452 (74.7 torr) = 33.8 torr. The benzene vapor-phase mole fraction is equal to

Pb/P = 33.8/46.0 = 0.735.

The expression above shows the total and par-

tial vapor pressure (at some fixed temperature)

changes linearly with the composition and its graph-

ical dependence

forms straight lines

(Fig.3.5.) The posi-

tions of these lines

are the following: at

XB=0 and XA=1 we

obtain P=PA=P0A

and PB=0, while for XB=1 and XA=0 we have

PA=0 and P= PB= P0B . Such systems with linear

dependence between vapor pressure and composi-

tion are called ideal solutions.

The properties of most systems, however,

deviate considerably from the ideal, both in sign

and magnitude. Fig.3.6. shows typical curves of

the vapor pressure as a function of composition for

non-ideal systems. Deviations of these curves Fig.3.6. Deviations from

ideality in real solutions

Fig.3.5.Vapor pressure

dependence on composi-

tion for ideal solution

64

from a liner relation toward greater values are called positive, toward smaller val-

ues – negative. The formation from the pure components of solutions showing pos-

itive deviations of the vapor pressure is accompanied for the most part by the ab-

sorption of heat and increase in volume. The formation from the pure components

of solutions showing negative deviations of the vapor pressure is usually accompa-

nied by the evolution of heat and decrease in volume.

There are some reasons of deviations. One of them is changes in the size of

particles, leads to changes in degree association and formation of compounds,

which influence on the properties of the solution. The other factor is the attraction

between the molecules of the components. When the forces of attraction between

not similar molecules differ a little from those between similar molecules, the sys-

tem behaves as an ideal solution. But if the components are strongly dissimilar, the

forces of attraction between unlike molecules will differ from those of the pure

components, deviations from ideal behavior will appear.

§ 7. Liquid-vapor diagrams

We can describe the liquid-vapor equilibrium in systems of two immiscible

liquids by two types of diagrams: pressure-composition and temperature-

composition, which is a reflection of each other for the same system. It is because

one of the two liquids one with a greater vapor pressure at a given temperature has

the lower boiling point.

Now consider the pressure-

composition phase-diagram in Fig.3.7.

Points P*A and P*B are the vapor pres-

sures of the pure liquids A and B. In this

point two phases in equilibrium (vapor

and liquid) and degrees of freedom are

equal 0 (f=1–2+1=0). Above the higher

curve there is only a liquid phase and

below the lower curve the system con-

tains only a vapor phase (f=2–1+1=2).

Points that lie between the two lines cor-

respond to a system in which two phases

are present – a liquid and a vapor (f=2–

Fig.3.7. Vapor pressurecomposi-

tion diagram for two component

liquid solution

65

2+1=1). The horizontal line joining the two phases in equilibrium at certain tem-

perature is called tie line. Point a indicates the vapor pressure of a mixture of com-

position xA, and point b indicates the composition of the vapor yA that is in equilib-

rium with the liquid at that pressure. Hence, we can see, that the composition of the

vapor phase differs from that of the liquid phase, and this difference is the greater

the greater the difference between the vapor pressures over the pure components. It

is expressed in Konovalov’s first law: in a binary system in equilibrium the vapor

has a higher relative content of that component, the addition of which to the system

leads to an increase in the total vapor pressure, i.e., to a lowering of the boiling

point of the system at the given pressure.

Now let us discuss a temperature-composition diagram (Fig.3.8), in which

the boundaries show the compositions of the phases that are in equilibrium at vari-

ous temperatures (P=const). Note that the liquid phase now lies in the lower part of

the diagram. Consider, what occurs with solution during heating from point M to

point M1. If we heat a system of composition corresponds of point M, it will start

boiling at the temperature t1 and vapor will first appear at that point. The composi-

tion of a liquid phase corresponds to

point a, and the composition of a vapor

– to point b. You see that the vapor is

richer in the more volatile (with lower

boiling temperature) component B ac-

cording to the Konovalov’s first law.

As with following solution vaporizes,

the remainder is enriched in component

A and a solution of thus composition

boils at higher temperature t2, at which

it is in equilibrium with a vapor of

composition b′. If the process is con-

tinued, the composition of the both a

liquid and a vapor, as well as the boiling temperature are changed up to t4, where

the composition of the vapor becomes equal to the composition of the initial liquid

(point b′′′), because all the liquid transits to the vapor.

The examined phase diagrams are treated to ideal and slightly nonideal solu-

tions. If, however, the solution has enough deviation from ideality to give maxi-

mum or minimum points on the curves a new phenomenon appears (Fig.3.9).

Fig.3.8. Changes occurred in solu-

tion during heating

66

The solution at the maximum point has a higher boiling temperature, than

the less volatile component, and the solution at the minimum point has a lower

boiling temperature, than the more volatile component. The solution of such com-

position is boiled at a constant temperature and the composition of a liquid is nod

changed since vaporization is over. Such solutions are called an azeotrope and for

them Konovalov’s second law states: the minimum and maximum points on the

boiling phase diagrams correspond to solutions for which the liquid and vapor

have the same composition..

§ 8. Distillation

Distillation is a process of separating a solu-

tion into its components by boiling them off

and collecting their condensates. There are

simple and fractional distillations. Simple dis-

tillation is mainly applied for solutions with

non-volatile solute (for instance, salt solution).

In boiling process all vapor is drawn off and at

the end one has solid residual and vapor,

which is condensed in original liquid. The

fractional distillation is based on the differ-

ence between the compositions of the equilib-

rium liquid and vapor and in general proceeds

Fig.3.10. Changes in composi-

tion of the liquid and vapor dur-

ing heated of solution

Fig.3.9. Diagrams with azeotropes

67

more easily, the greater this difference is. In fractional distillation, the boiling and

condensation cycle is repeated successively. Consider what happens when a liquid

of composition a1 is heated (Fig.3.10.). It boils when the temperature reaches T2.

Then the liquid has composition a2 and the vapor has composition a′2. The vapor is

richer in more volatile component (with the lower boiling point). If the vapor in

this example is drawn off and completely condensed, the

first drop gives a liquid of composition a3, which is rich-

er in more volatile component than the original liquid.

When the condensate of composition a3 is reheated, it

boils at T3 and gives a vapor of composition a′3, which is

even richer in more volatile component. That vapor is

drawn off, and the first drop condenses to a liquid of

composition a4. The cycle can then be repeated until due

course of almost pure A is obtained. The efficiency of

separating is expressed in terms of the number of these

cycles (theoretical plates) that is shown in Fig.3.11.

The method is

involved for practical

application on an in-

dustrial scale. Large-

scale separation is achieved more effectively by

carrying out fractional distillation as a continu-

ous process in which condensation and distilla-

tion of the separate fractions take place auto-

matically. In this form the process is known as

rectification, and the main apparatus in which it

is carried out is called rectifying column

(Fig.3.12.).

Systems with azeotrop composition can-

not be separated into their pure components in

this manner. Distillation can be continued until

the maximum or minimum points, after which a system boils without changing the

composition. As we can see in Fig.3.13, a solution with a composition intermediate

between A and C can be separated by distillation only into the pure component A

and the azeotropic solution C. And any solution with a composition intermediate

Fig.3.12. Rectifying col-

umn

Fig.3.11. The process

of fractional distilla-

tion

68

between B and C can be separated by distilla-

tion only into the pure component B and the

azeotropic solution C.

§ 9. The systems of two partially miscible liquids

Let us consider a system consisting of water and aniline. When a certain

amounts of two liquids are shaken together at room temperature, one obtains a sys-

tem of two liquid phases in equilibrium. Each contains both components, but in

different amounts: one layer is the saturated solution aniline in water and the other

layer is the saturated solution water in aniline. Further adding of water or aniline

into the system does not change the composition of layers. Consider a liquid-liquid

phase diagram of such a system (Fig.3.14.).Curve AK shows solubility solute B in

solvent A with temperature rising, and

curve KB shows solubility solute A in

solvent B. At temperature t1 we have

two saturated solutions in points m and

n in equilibrium. The point’s m and n

are maximum solubility of these liquids

in each other at temperature t1. Between

points m and n two phases (two layers)

are present in equilibrium. Out of a

curve AKB a homogeneous system is

presented – solution of A and B liquids.

Suppose the overall composition

is reflecting by point O. If the overall

composition is changed (points along the line mn) the compositions of two phases

in equilibrium is not changed at certain temperature (points m and n). This is the

first rule of Konovalov for such systems. According the second rule of Konovalov

Fig.3.13.Distillation of

system with azeotrope

Fig.3.14. A liquid-liquid phase diagram

of the systems of two partially miscible

liquids

69

in unchanged overall composition (point 0′) changing of the temperature lead to

the change in the compositions of two phases (points m′ and n′).

As the temperature is raised, the solubility of liquids is raised too and in

point K they become completely miscible. The temperature, at which complete

miscibility sets, is called the critical solution temperature. In this case it has upper

position.

In some systems it can have lower position (miscibility increases with tem-

perature decrease), and for others – both upper and lower, or neither of them

(Fig.3.15.).

The lever rule. When the system consists of two phases in equilibrium it’s

possibly to determine the quantitatively the rela-

tive amounts of phases. To find the relative

amounts of two phases α and β that are in equi-

librium, we measure the distances lα and lβ

(Fig.3.16.) along the horizontal tie line, and then

use the lever rule: the point corresponding to the

composition of a heterogeneous binary system

lies on the straight line connecting the points

that represent the coexisting phases, and divides

this line into segments that are inversely propor-

tional to the amounts of the two phases:

lnln 3.6

Example: A mixture of 50 g hexane (0.59 mol C6H14) and 50 g nitrobenzene

(0.41 mol C6H5NO2) was prepared at 290 K. What are the compositions of the

phases, and in what proportions do they occur? To what temperature must the sam-

ple be heated in order to obtain a single phase?

Fig.3.16. The lever rule

Fig.3.15. The systems with different critical boiling point

70

Solution: We denote hexane by H and nitrobenzene by N; Refer to diagram,

at which the point xN = 0.41, T = 290 K occurs in the two-phase region of the phase

diagram. The horizontal tie line cuts the phase

boundary at xN = 0.35 and xN = 0.83, so those are

the compositions of the two phases. The ratio of

the amounts of each phase is equal to the ratio of

the distances lα and lβ:

735.041.0

41.083.0

l

l

n

n

That is, there is about 7 times less nitroben-

zene-rich phase than hexane-rich phase. Heating the sample to 292 K takes it into

the single-phase region.

§ 10. The systems of two immiscible liquids

In systems consisting of two immiscible liquids, each liquid is evaporated

independently of other liquid presence and the partial vapor pressure of each liquid

independently of liquid amount remains constant and equal to the vapor pressure of

pure liquids. As a result, the total vapor pressure of the mixture equal to the sum of

the partial vapor pressures is always higher than the partial vapor pressure of each

liquid. Consequently, the mixture boils at a lower temperature than either compo-

nent would do alone because boiling begins when the total vapor pressure reaches

1 atm, not when either vapor pressure reaches 1 atm. This distinction is the basis of

the method, which is called steam distillation. It applies to some heat-sensitive,

water-insoluble organic liquids that decompose at a high temperature. Mixing them

with water steam enables to be distilled at a lower temperature than their normal

boiling point. The only problem is that the composition of the condensate is in pro-

portion to the vapor pressure of the components, so oils of low volatility distil in

low amount.

There is an equation for making calculation in this process. The composition

of vapor in this system one can express as

02

01

02

01

01

1PP

PX and

PP

PX

02

2

71

where 1X and 2X are the mole fractions of the components in the vapor. For

their relations we obtain 0

2

01

2

1

P

P

X

X

Because 21

2

21

11

nn

nX and

nn

nX 2

where n1 and n2 are amounts of moles of the components and equal to: n=g/M ,

hence we obtain

2

1

X

X0

2

01

P

P

22

11

2

1

/

/

Mg

Mg

n

n

from which

2

1

g

g0

2

01

P

P

2

1

M

M 3.7

The relation of g1/g2 is called the coefficient of water steam consumption and shows

the amount of water steam necessary for unity mass of organic liquid distillation.

Example: The vapor pressure of toluol (C6H5CH3) and water at 365 K is

equal to 4.2104 N/m2 and 5.5104 N/m2 accordingly. Find the mass of water

steam necessary for 1 kg toluol distillation.

Solution: Using the equation 3.7 give us

18

92

105.5

102.4

1 4

41

g from which g1 = 3.9 kg

§ 11. The partition law

If a small amount of a third component is introduced into a two-component

system of immiscible liquids, then after equilibrium has set in, the third component

is found to be present in both phases. We shall illustrate this on the system of water

and carbon disulphide, the components of which are practically immiscible. If crys-

talline iodine is introduced into this system, iodine can be readily detected in both

layers when equilibrium sets in. Its concentration in carbon disulphide, however, is

about 600 times greater than in water and this ratio of its concentrations remains

constant (at constant temperature). So, for each given temperature, the ratio of the

concentrations of the third component added into the system of two immiscible liq-

72

uids in equilibrium is constant magnitude independences of amount of species. This

is the partition law.

Actually, in equilibrium the chemical potential of the third component is the

same in two liquid phases: µΙ3 = µΙΙ

3. On the other hand it’s known that µ = µ0 +

RTln a. So, we can write

IIIIII aRTaRT 3,0

33,0

3 lnln

from which

RTa

a III

II

I

/`ln ,03

,03

3

3

At constant temperature all quantities of the right side of the equation are constant,

i.e.:

K = II

I

a

a

3

3 3.8.

The ratio is the mathematical expression of the partition law, and the con-

stant K is called the partition or distribution coefficient. For dilute solutions in-

stead of the activity, one can use the relation of concentrations.

In some systems, the partitioned substance has unequal particle size in each

of both solvents owing to dissociation or association of its molecules. In such cases

the expression is used in general form:

n C

CK

2

1 1 3.9

where α is the degree of dissociation and n is number of molecules in the associate.

§ 12. Extraction

Extraction is the process of removal of a dissolved substance from a solution

with the aid of an other solvent (extractant) under condition that these two solvents

are immiscible. Extraction occurs according to the partition law. It’s widely used in

the laboratory and in industrial practice both to remove undesirable components

and to obtain a desirable component of a solution in a more concentrated state.

There is equation for amount of removed substance calculating. Let V0 ml of

a solution containing m0 grams of solute is treated with V ml of extractant. The

73

amount of a solute after one extraction is mn, so the amount of solute transfer into

extractant will be m0 – mn. Therefore, the concentration of each solution after ex-

traction is equal: C1= mn/V0 and C2=(m0 – mn)/V. As the partition coefficient being

K= C1/ C2, after putting the values of concentrations we obtain:

mn = m0

VKV

KV

0

0 3.10

In general case after n numbers of extractions with equal volumes of the

same solvent, the amount of substance in grams remaining in solution is calculated

by the equation:

mn = m0

n

VKV

KV

0

0 3.11

This expression also allows determining the number of extractions necessary

to remove the solute to the required degree. The efficiency of the extraction de-

pends greatly on how it is carried out: considerably fuller extraction can be

achieved with a given amount of a solvent if the process is carried out not by the

whole solvent at once, but by successively applying it in a large number of small

portions.

Example: Two liters of an aqueous iodine solution contain 0.02 g of iodine.

How much iodine remains in the solution after extraction with 50 ml of carbon di-

sulphide (a) if the process is carried out only once with all 50 ml of carbon disul-

phide, and (b) if it is carried out five times with 10 ml portions of the solvent. The

partition coefficient of iodine K =

2

2

CS

OH

C

C= 0.00167.

Solution: (a) Using formula 3.10, we find that after a single extraction with

50 ml of carbon disulphide the amount of iodine remaining in the water is

gmn 00125.0500.200167.0

0.200167.002.0

which is 6.3% of its initial content.

(b) Using formula 3.11, we find that after fivefold extraction with 10 ml portions

the amount of iodine remaining in the water will be

gmn 0000195.0100.200167.0

0.200167.002.0

5

74

This is less than 0.1% of the initial amount, i.e., 1/65 of the amount in the preced-

ing case.

§ 13. Application of phase diagrams in pharmacy

The measurement of drug boiling point as well as melting point allows to

make their identification and to know the degree of purity and the mixed drug

composition. The value of boiling temperature of drug gives the information about

their solubility: the higher this temperature the less solubility.

Plotting boiling phase diagrams is necessary for carrying out binary liquid

drug mixtures distillation, as well as to determine conditions of drugs purification,

analysis and synthesis. They allow determining the composition of azeotrop mix-

ture, which is necessary for fractional distillation.

Knowing the Raoult’s law and plotting boiling phase diagrams is important

also for the use of drugs in aerosols form or gaseous anesthetics (ether, chloro-

phorm, etc.), or when one has to determine the vaporization or condensation tem-

peratures as well as to take into consideration solubility of the gases in the blood

and in the organism tissues.

The extraction also has wide application for isolation of active substances

from vegetative raw materials as well as uses in analytical purpose, especially in

multicomponent systems analyses.

It’s very important to determine the value of the partition coefficient, on

which the drug biological activity depends. When drugs solution is administered

into organism, its distribution there depends on the partition coefficient between

essential solvent and organism liquids and tissues.

Review questions:

1. Define the terms: phase, constituent, component, number of degrees of

freedom. Explain the difference between constituent and component.

2. How many components are present in a system in which CaCO3 obtained

from CaO and CO2.

75

3. Give the examples of (a) one-component one-phase, (b) one-component

two-phase, (c) one-component three-phase, (d) two-component one-phase, (e) two-

component two-phase systems.

4. What is the phase transition, why does it occur (based on thermodynamic

ability of a system) and what is the limit of this process?

5. Write the Gibbs phase rule equation. What is n here?

6. Write the Clausius-Clapeiron equation. For what systems and phase tran-

sitions is it applied?

7. What shows the relations dT/dP and dP/dT of this equation and from what

does their sign depend on?

8. What it means that dT/dP>0? What is the sign of this relation (dT/dP) for

ice melting and why?

9. Write the simplified equation of Clausius-Clapeiron for vaporization pro-

cess. What are the ways of its application?

10. Write the phase rule expression for one-component system. How many

phases could be in equilibrium simultaneously in such a system and what is the

maximum number degree of freedom?

11. What do phase diagrams express?

12. Draw the phase diagram of water. What kinds of phases are expressed by

each plate on the diagram and which phase equilibrium does each line shows? For

each of them calculate the number of degrees of freedom.

13. What is the triple point and what does it mean that the number of degrees

of freedom in it is equal to zero?

14. Define the terms: boiling temperature of a liquid, critical temperature of

boiling and melting temperature.

15. What is the difference between phase diagrams of water and sulfur?

16. Draw the T-composition solid-liquid diagram (melting diagram) and show

the number and the states of phases for system of two-component system, consisting

of two solid components.

17. Why the curves on it are called liquiduse and soliduse curves?

18. Define all phases, phase equilibrium states on the diagram and the number

of degrees of freedom for each state.

76

19. What composition of such a system is called eutectic? What is an interest

of eutectic point.

20. How is thermal analyses made? How the curves of thermal analyses are

are obtained?

21. How are the curves of thermal analyses of the pure substance and of the

mixture of two substances distinguished?

22. What is the influence of the added admixture on the pure substance melting

point?

23. Why is the mixture of two solid species called eutectic? How do their

freezing processes occur?

24. In what the eutectic mixture and pure substance are similar?

25. How many phases are there in the equilibrium in eutectic point and which

are they?

26. What is the thermal analysis applied for in pharmacy?

27. Draw the T-composition diagram of liquid-vapor equilibrium for two-

component completely miscible liquids. Denote all the possible phase states and

determine the number of degrees of freedom for each of them.

28. What is the tie line?

29. Formulate the first law of Konovalov and explain it by a diagram.

30. Which mixture of two liquids is called azeotrop?

31. What law lies in the bases of a fractional distillation? Explain, how does it

work?

32. Is it possible or not to divide azeotrop mixture into two pure liquids by a

fractional distillation? What is the reason?

33. Draw the T-composition diagram of liquid-vapor equilibrium for two-

component partially miscible liquids. Formulate the Konovalov’s rules and explain

them by a diagram.

34. What can be determined by lever rule?

35. Why is the steam distillation applied for the systems of two immiscible

liquids?

36. Write the expression of the coefficient of water steam consumption. What

does it show?

77

37. For the systems of two immiscible liquids formulate the partition law and

write its mathematical expression.

38. Determine the process of extraction. What law lies in the bases of this pro-

cess? Write the equation of the extraction and explain it? What are the optimal con-

ductions of the extraction?

39. List areas of phase diagrams in pharmaceutical analyses.

78

CHAPTER 4

ELECTROCHEMISTRY

§ 1. Solutions of strong electrolytes

Strong electrolytes are practically completely dissociated into ions in solu-

tions where the ion-solvent interactions occur between the ions and the solvent

molecules. Due to electric fields created by the ions, the polar molecules of the sol-

vent (water) begin to assume orientations that favorable interactions between the

ionic charge and the water dipole produce; that is, the molecules oriented relative

by to the ion arranging them around the latter. The surrounding shell of water mol-

ecules formed in this way constitutes the waters

of solvation or hydration of the ion (Fig.4.1.).

The average number of water molecules associ-

ated with an ion is called the hydration number

of the ion and normally ranges between 4 and 6.

Consequently, in the water we must take into

consideration the effective or hydrodynamic ra-

dius of an ion, which is larger than its crystal

lattice radius. The effect of the hydration radius

is manifested in a number of solution physical

properties including viscosity, conductivity, compressibility, diffusion, etc.

Small ions, which may have a closer approach to the water molecules and

thus a stronger interaction, tend to have relatively larger hydration radii than larger

ions. The degree of hydration of ions depend on their size and is particularly well

expressed when electrical conductivities of different electrolytes are compared.

Since the ionic radius increase from Li+ to Cs+, it could be expressed that lithium

ion would be the best and cesium – the poorest conductor of electricity. This is ac-

tually for the molten substance. A reverse relation is observed, however, in aque-

ous solutions. Owing to its small size the Li+ ion is more hydrated and moves slow-

ly than Cs+ ion.

In the solutions of strong electrolyte, not only the state and properties of the

electrolytes, but also the state and properties of the solvent and particularly of wa-

ter are changed. Molecules of water in hydrating layer undergo polarization and

+–

+–

+–

+–

+–

+–

+–

Fig.4.1. Hydration of ions in

water solution

79

corresponding changes in the structure and properties; decreases their mobility,

increases the degree of ordering, therefore the magnitude of entropy decreases.

The region of enhanced structuring in the solvent is referred to as the “salva-

tion zone around the ion”. This salvation or structuring effect can have a number of

important consequences for the interactions between ions, molecules, colloidal par-

ticle, and interfaces. If the orientation and mobility of solvent molecules near an

ion differ from those in the bulk, one might expect that many properties of the sal-

vation zone differ from those of the bulk. In particular, there are found to be differ-

ences in density, dielectric constant, conductivity, and other parameters.

§ 2. The Debye-Huckel theory

The object of the quantitative theory of strong electrolytes developed by

P.Debye and E.Huckel (1923) is the coordination of the electrostatic ionic interac-

tion of the ions with the properties of the solutions and it explains the deviation

from ideality in ionic solutions.

According to their model, as the oppositely

charged ions attract one another, each ion in a

solution is surrounded by ions of opposite charge

more, than the ions of same charge. As a result,

anions are more likely to be found near cations in

a solution, and vice versa (Fig.4.2.) Overall, the

solution is electrically neutral but near each ion

there is an excess of counter-ions (ions of oppo-

site charge). This spherical haze with central ion

and surrounding ions cloud is called the ionic at-

mosphere. The boundary of this atmosphere is the

place where amounts of the opposite charged ions

become equal.

The ionic atmosphere has an effect on the

motion of the ions in electrical field. On one hand,

the counter-ions in the ionic atmosphere and the

central ion move in the opposite directions to the

oppositely charged electrodes (Fig.4.3.). But at-

tractive forces between oppositely charged ions of

the ionic atmosphere reduce the mobility of the

Fig.4.2. Ionic atmosphere in

solutions of strong electrolytes

Fig.4.3.The scheme of the

electrophoretic effect

80

ions, and hence, reduce their conductivity. This effect is called electrophoretic.

On the other hand, the ions forming the atmosphere do not adjust to the mov-

ing ion infinitely quickly, and the atmosphere is

incompletely formed in front of the moving ion

and incompletely destroyed behind the ion

(Fig.4.4.). The displacement of the central ion of

the atmosphere leads to the accumulation of the

ions of opposite charge behind the moving ion.

Because two charges are opposite, retardation of

the moving ion is the result. This reduction of the

ions’ mobility is called the relaxation effect.

Both these effects decrease the drift speed

and the mobility.

In consequence of described attractions be-

tween ions in the strong electrolyte solutions instead of concentration the concept

of activity is inserted. The activity expresses the effective concentration, in which

all the attractions occurring in the solution are taken into consideration: mutual at-

traction of oppositely charged ions, hydration of the ions, and incomplete dissocia-

tion of the molecules (if it takes place). The activity is related to the molality, c, by

a = γc 4.1

where γ is the activity coefficient, depends on the composition, molality, and tem-

perature of the solution. The activity coefficient is a measure of the chemical po-

tential in the solution and measures the degree of deviation of substance behavior

from ideal. For infinitely dilute solutions γ = 1; in an ideal solutions or in infinite

dilution there are no interactions between components and the activity equals to the

concentration. In real solutions at higher concentration causes a divergence be-

tween the values of activity and concentration. The ratio of the activity to the con-

centration is called the activity coefficient, that is:

γ = activity/concentration

The activity of a particular component is the ratio of its value in a given solution to

that in the reference state.

There is no experimental way to determine separately the value of the activi-

ty coefficient for cations and anions. Therefore, we introduce the average activity

coefficient as the geometric average of the individual coefficients, and for 1,1-

electrolyte it is:

Fig.4.4. The scheme of the

relaxation effect

81

2/1

4.2

In general for the electrolyte of a compound MpXq that dissolves to give a

solution of p cations and q anions the average activity coefficient is equal to:

q p ssqp

/1 4.3

For taking into consideration all the ions present in the solution the concept

of ionic strength of the solution, I, is introduced which is defined as one-half of the

sum of the products of the concentration of all the ions in the solution and the

square of their charges:

iinn czI or czzczczI222

32

2

21

2

12

1...

2

1 4.4

Magnitude of the average ion activity coefficient may be determined exper-

imentally using several methods including electromotive force measurement, solu-

bility determinations and colligative properties. It is possible, however, to calculate

it in very dilute solutions using a theoretical method based on the Debye-Huckel

theory. The relation between the ionic strength and activity coefficient is obtained

by the Debye-Huckel limiting law:

IAzz lg 4.5

where A is the limiting coefficient and for an aqueous solution A=0,509.

Equation is called limiting law, since it is valid only in the limit of infinite dilution

and gives correct limiting behavior for electrolyte solutions as I→0. Equation is

found to be accurate when I ≤ 0,01 mol/kg.

The theory is based on a number of simplifications and as yet is ap-

plicable only to solutions of very low concentrations. The simplifica-

tions primarily consist in the following:

no account of ion solvation is taken,

the difference in size between the positive and negative ions is ig-

nored,

the solvent is treated as a structureless medium with dielectric con-

stant equal to the dielectric constant of pure water,

the ionic atmosphere is treated as a continuous electric field,

the electrolyte is completely dissociated,

only attractions between ions are taken into account,

from all the properties of ions only their charges are taken into ac-

count.

82

In spite of these simplifications, the theory makes possible to know the ac-

tivity coefficient and activity, as well as the chemical potentials, from which we

can derive limiting laws for all the other ionic systems and electrolyte thermody-

namic properties.

§ 3. Ion motion in the electric field

In the absence of an external electric field, the ions of a solution are in ran-

dom motion, since all directions are equivalent. When the external electric field is

applied, the direction to the higher potential gradient (the drop in potential per cm)

prevails. To compare the ion velocities we usually refer to a potential gradient of 1

volt/cm and in this case they are known as absolute velocities, v.

Among all the ions the velocity of both hydrogen and hydroxyl ions is strik-

ingly high. This is due to a special jumping mechanism and the continuous ex-

change between hydroxonium ions H3O+ and water molecules, as well as between

the latter and hydroxyl ions OH–.

The ions in the solutions are electricity-carrying particles. The velocities of

the cation v+ and the anion v– in a given solution are as a rule not equal. It is there-

fore interesting to know contribution of each type of ions in carrying electricity.

The quantities expressing of it is called the transport number, which defined as

follows for cations and anions:

vv

vt and

vv

vt - 4.6

As shown, the transport number depends not only on a certain ion velocity, but on

the velocity of the counter-ion. After division we obtain, that the more ions veloci-

ty, the more its transport number:

v

v

t

t

It was shown by Hittorf that the difference in velocities of cations and anions

leads to a difference in the amount of electricity they transfer, but this does not vio-

late electroneutrality of the solution, because amounts of cations and anions dis-

charged on the electrodes is equal. But the electrolyte concentration change in the

anode and cathode spaces is changed with different extents:

v

v

c

c

an

cath

4.7

83

ELECTRICAL CONDUCTIVITY

Electrical conductivity is a transport phenomenon in which electrical charge

moves throw the system. There are two types of electrical conductivity. The flow

of electricity in the first type of conductors (metals in the solid and molten state) is

due to movement of the electrons. It’s very high and is not accompanied by any

chemical change. The transport of electricity in the second type of conductors (the

strong electrolyte solutions and melts) is due to movement of ions. It’s accompa-

nied by chemical processes.

In strong electrolyte solutions, two types of electrical conductivity are dis-

tinguished: specific and molar (or equivalent).

§ 4. Specific conductivity

The conduction, L, of a solution is the inverse of its resistance R, which is

equal to:

S

lR

where ρ is the resistivity; it is the resistance of a body 1 m long and 1 m2 in cross

section. Therefore

l

S

RL

11 4.8.

1 is called specific conductivity and is defined as χ. It’s equal to the conductivity

of a rod of a substance 1 m long and 1 m2 in cross section. For solutions it is the

conductance of 1 m3 of a solution. As R is expressed in

ohms, Ω, the conductivity is expressed in Ω-1m-1.

The specific conductivity depends on the con-

centration of the ions, temperature, properties of the

ions and the nature of the solvent.

The dependence on the concentration we can

see on Fig.4.5. The more electrolytes are added to the

solution the more conductivity increases. On reaching,

a certain concentration the conductivity can fall with a

further increase in concentration. In the solutions of

Fig.4.5.Dependence χ on

the concentration

84

strong electrolyte due to oppositely charged ions interactions the mobility of the

ions and hence their conductivity is reduced. In case of weak (but well soluble)

electrolytes, the degree of dissociation will begin to fall with increase in concentra-

tion (according the Ostwald’s law).

When the temperature is risen, the conductivity increases due to reduced hy-

dration of the ions and the lower viscosity of the solution. On the other hand con-

ductivity might decrease due to increase of random motion of ions.

The influence of the nature of a solution depends on the viscosity of the lat-

ter and the degree of solvation. The more these properties are expressed the smaller

conductivity.

From properties of ions, their value of charge and the size have a great im-

portance. The more value of charge and the smaller the size of ions the more con-

ductivity.

§ 5. Molar conductivity

It was impossible to obtain any simple general relations between the concen-

tration of a solution and its specific conductivity. Much better results were obtained

on using the molar (or equivalent) conductivity, λ, which is the conductivity of a

solution that contains 1 mol of dissolved electrolyte and is placed between elec-

trodes 1 cm apart. This quantity is defined by the expression

V1000 or 1000c

4.9

where c is the number of gram-moles of electrolyte per liter, V=1/c is called dilu-

tion, and is equal to the volume in liters of a solution that contains 1 mol of dis-

solved electrolyte. The unit of molar conductivity is cm2Ω-1mol-1. In spite of the

unchanged one mol concentration, the molar con-

ductivity increases with the dilution (i.e., with a de-

crease in concentration), tending to a certain limit-

ing value (Fig.4.6.). It occurs due to a decrease in

concentration, interionic attraction forces also de-

crease and in the limit of infinite dilution interionic

forces fall to zero and the ions move independently.

The limiting value of the molar conductivity is

called the molar conductivity at infinite dilution

V

¥

Fig.4.6.Dependence λ on

the dilution of solution

85

and is designated by ¥ . At certain temperature, it has a constant value for each ion

and is an important characteristic of an electrolyte.

There is connection between the molar conductivity and the degree of disso-

ciation, which is shown by Arrenius’s equation:

anc vvF 4.10

where vc and van are the absolute velocities of the cation and anion, F is the Fara-

day’s constant (equal to 96500), and a is a degree of dissociation.

At infinite dilution a=1 and the Eqn.4.9 comes to form

anc vvF ¥

where product of Fv is called ionic conductivity ( ¥ ) or ionic mobility (l). So,

the above expression we can write in the following form:

¥

¥

¥ anc ll 4.11

The molar conductivity of an electrolyte at infinite dilution equals to the sum

of the ionic conductivities (nobilities). This is called the law of independent migra-

tion of ions in infinite dilute solutions, or also called Kohlrausch’s law.

§ 6. Conductometric determinations

The determination of conduct-

ance is known as conductometric

analysis. It comes to determination of

the electrolyte solution resistance (R)

by Kohlrausch’s bridge (Fig.4.7.) un-

der altemating current (using of direct

current leads to the electrolysis and

changes

in con-

centration). R1 is standard resistance (is known), Rx is

the desired resistance of the electrolyte solution which

is contained in the conductometric cell, like the one

shown on Fig.4.8. The cell’s constant is defined as l/S;

a relation of distance between electrodes and their

cross section. It’s determined in advance. R2 and R3 are

the resistance of the wire ac and could be replaced by

Fig.4.7. Kohlrausch’s bridge

Fig.4.8. Kohlrausch’s

bridge

86

the length l1 and l2 (d is the mobile contact point). By moving mobile contact along

the wire one finds the point in which two parts of the bridge (ab, ad and bc, cd) are

in compensation state (amperemetre A does not display current flow along bd

branch). In such equilibrium state the following relation takes place: R1/Rx = R2/R3

or R1/Rx = l1/l2, from which

Rx = R1

1

2

l

l 4.12

If R is known, we are able to calculate specific conductivity from equation 4.8:

S

l

Rx

1 4.13

This method finds a wide application and in particular:

1. Determination of molar conductivity by equation 4.9.

2. Determination of conductivity for industrial application of electrolysis.

3. Degree of ionization by relation a = λ/λ0 (from Eqn.4.10).

4. Dissociation constant of weak acids using the Ostwald’s dilution law:

C

K , from which

1

2CK

5. Solubility of difficultly soluble electrolytes basing on the fact that a satu-

rated solution of difficultly soluble electrolytes can be regarded as infinite dilutes

(due to little amount of ions into it) and eqn.4.9. is written as 1000c

¥

in which the value of ¥ is known for each ion, χ is measured by the mentioned

method, so the concentration of the solution and hence solubility is determined.

6. Determination of the electrolyte concentration by conductometric titration

(measuring the conductance during the titration process).

Conductometric titration is based on the fact that ions of the titrant (solution with

known concentration) enter into reaction with ions of the solution (with unknown

concentration) being titrated to form molecules of a weakly dissociated compound

or a difficultly soluble substance. In the solution, equivalent amounts of other ions

will replace of the ions removed. When the new ions have different mobilities,

such a substitution will cause a change in the conductance of the solution. For ex-

ample, if an aqueous HCl solution is titrated with NaOH solution of known con-

centration, hence on gradual addition of NaOH conductivity of the original solu-

tion decreases before the endpoint because H+ ions communicate with OH– ions

and are replaced by Na+ ions according the reaction:

87

NaOH + HCl = NaCl + H2O

Then conductivity increases after the endpoint as a result of the increase in OH-

ions concentration, because there is no any more H+ ions in the original solution.

If the change in conductance versus the

amount of added sodium hydroxide is plotted

(the abscissas showing the number of millili-

ters of added base, and the ordinates - the con-

ductance), the lowest point on it is point of

equivalency (Fig.4.9), i.e. the moment of

complete neutralization of sodium hydroxide

by hydrochloric acid. One knows the volume

of the added base by diagram (Vequiv.) and then

finds the concentration of the unknown solu-

tion by equation: CbaseVbase = CacidVacid.

On titrating a weak acid (CH3COOH)

with a strong base (NaOH) the conductance in-

creases very slowly up to the point of equivalen-

cy, because a weak acid is replaced by equiva-

lent amounts of its salt according the reaction

CH3COOH + NaOH = CH3COONa + H2O

After this point conductivity increase as a result

of the increase in OH- ions due to following ad-

dition of the base (Fig.4.10).


1. The conductivity χ of a 1.00 mol/dm3 aqueous KCl solution at 25oC and 1

atm is 0.112 Ω–1cm–1. Find the KCl molar conductivity in this solution.

Solution: Substitution in 4.9 gives

121311

11210

00.1

112.0

molcm dm 1

cm

dm mol

cm

C 3

3

3-KCl

2. The molar conductivity of 0.0100 M CH3COOH (aq) at 298 K was meas-

ured as λ = 1.65 Ω–1cm2mol–1. Determine the degree of ionization and pKa of the

acid.

1/R

V

a

Vequiv.

Fig.4.9. Conductometric titra-

tion curve of strong acid by

strong base

1/R

V

a

Vequiv.

Fig.4.10.Conductometric titra-

tion curve of weak acid by

weak base

88

Solution: From the table data we find λo = 39.05 Ω–1cm2mol–1. Therefore,

equation a = λ/λ0 gives a = 0.0423. It follows from equation

1

2CK that Ka =

1.910–5, implying that pKa = 4.72.

3. The conductivity χ of a 0.05 mol/dm3 aqueous CH3COOH solution at

25oC is 0.00032 Ω–1cm–1. Determine the degree of ionization of the acid if the ionic

mobility of H+ and CH3COO– are 315 Ω–1cm2 and 35 Ω–1cm2.

Solution: We can find the degree of ionization from the relation a = λ/λ0.

According the Kohlrausch’s law λ0 = lH+ + lCH3COO– = 315 + 35 = 350. The other

hand 121311

4.610

05.0

00032.03

molcm dm 1

cm

dm mol

cm

C 3

3

3-COOHCH

Hence a =

0.018.

Review questions:

1. What occurs to strong electrolytes in water solution?

2. What is an effective radius of ion?

3. Explain how ions are arranged in solution according to Debye-Huckel

theory.

4. Explain the reason of electrophoretic and relaxation effect during ion

movement in electric field.

5. Why in strong electrolyte solutions concentration is replaced by activity?

6. What is equal to the activity coefficient? How does it change (more or less

unite) with dilution of concentration? Explain this phenomenon.

7. Write the equation of ionic strength and Debye-Hucke limiting law.

8. What types of conductors are known?

9. Determine specific conductivity. What factors does it depends on?

10. Draw the diagram of specific conductivity dependence on solution concen-

tration and explain the curve position. Why specific conductivity decreases in high

concentration of solution?

11. How does specific conductivity depend on ion charge and size?

12. Determine molar conductivity. Does it depend on concentration? Why?

13. Draw the diagram of molar conductivity dependence on dilution of solu-

tion and explain it. What is molar conductivity at infinite dilution and why does it

have a constant value?

89

14. Write Arrenius equation which links molar conductivity and electrolyte

ionization degree.

15. Write the expression of Kohlrausch’s law. Why is it called the law of inde-

pendent migration of ions? How can it be obtained from Arrenius equation? What is

ionic mobility?

16. What is measured by conductometry method? Explain the principal of

working the Kohlrausch’s bridge. What is measured by it directly?

17. List the ranges of conductometry application. How is electrolyte ionization

degree measured by this method? From which equation and how is corresponding

expression obtained?

18. Explain the method of conductometric titration. Why does conductivity of

solution decrease in case of a strong acid titration by a strong base? Why does it

increase in case of a weak acid titration by a strong base?

19. Why does conductivity of solution increase after equivalent point in all

cases of such titration?

§ 7. ELECTRODE PROCESSES

Electrode processes are oxidation and reduction processes, which take place

on the electrodes. An electrode (or half-cell) is the plate of metal with electron

conductivity that is dipped into a solution with ionic conductivity containing the

same metal ions Due to charges moving between these two phases the electrical

potential difference is formed at the interfacial boundary (between metal and solu-

tion) which is called potential difference or electrode potential.

Let us imagine that a metal plate, for instance zinc, is immersed in water.

The zinc ions act by the polar water molecules due to which they are pulled off

from the metal and begin to pass into the layer of water adjoining the surface of the

plate. Because the positive charges leave the metal, the latter is charged negatively

due to excess of electrons. The electrostatic attraction now arises between the ions

passing into solution, the oppositely charged metal plate inhibits the further ions

pulling off from the metal as some of them back on the plate, and eventually equi-

librium is set in. In equilibrium state a double electrical layer sets: the negatively

charged layer of electrons on the plate and the positively charged layer of zinc ions

in the water. A potential difference will thus arise between the metal and the sur-

rounding aqueous medium (Fig.4.11).

90

Similar effect

is obtained when the

metal is immersed

not only in pure wa-

ter, but in a solution

of its salt. An in-

crease in concentra-

tion of the metal ions

obviously promotes

transition of the ions

from the solution to the metal, and equilibrium is set at another potential of the

metal charge. Metals, the ions of which have strong tendency to pass into solution

are also charged negatively in such a solution, but to a lesser degree than in pure

water.

On the other hand, metals, the ions of which have little tendency to pass over

into solution may even become positively charged owing to the fact that the initial

rate of deposition of positive ions on the metal is greater than that of the passage

of ions from the metal into the solution (Fig.4.12). For example, if we are im-

mersed copper in the CuSO4 solution the Cu2+ ions passages on the plate from the

solution forming the

positively charged

layer. Due to electro-

static attractions be-

tween these ions and

the oppositely

charged 24SO ions,

the latter come near

the plate and form a

negatively charged layer.

Thus, in each case of a metal immersed into the solution of its salt the adja-

cent solution has an opposite charge to that of the metal. A double electric layer

thus arises, i.e. a definite potential difference (potential jump) at the metal-solution

interface. The value of this electrode potential is different for different metals and

depends on the nature of the metal and of the concentration of a solution.

The charge of a metal (positive or negative) depends on the different abili-

ties of the various metals to send ions into the medium, the different tendencies of

Fig.4.11. Mechanism of the electrode potential for-

mation on the zinc electrode

Fig.4.12. Mechanism of the electrode potential for-

mation on the cooper electrode

91

the ions to hydrate. In other words, it depends on the chemical potentials ratio be-

tween a metal and a solution.

The electrical work (transfer of charges between phases) done in this process

will be

–ΔG = zFε 4.14

where ε is the value of an electrode potential.

On the other hand, the chemical work (oxidation and reduction) could be ex-

pressed by equation of isotherm

zMe

Me

a

aKRTG lnln 4.15

where zMea is the activity of the ions in the solution and Mea - in the plate.

In electrochemical equilibrium the right sides of these both equations are equal,

and to take notice that Mea = 1, we obtain

zMea

zF

RTK

zF

RTlg

303,2lg

303,2 4.16

At a certain temperature, all the quantities of the first term of the equation

are constant; hence, we could define it as one constant ε0, which is called normal,

or standard electrode potential. So, we can write

zMea

zF

RTlg

303,20 4.17

This equation is called Nernst equation; it shows the dependence of the

electrode potential on the solution activities and the nature (by ε0) of the metal (or

other conductor).

When zMea =1 then ε = ε0, therefore the standard potential of an electrode is

the potential appearing when the activity of the ions in the solution (determining

the electrode reaction) is equal to 1.

It is not possible to measure the value of the electrode potential of the single

electrode, so one can define the potential of one of the electrodes as having a zero

potential and then assign to others by there difference. The specially selected elec-

trode is the standard hydrogen electrode for which the activity of the ion in the

solution is equal to 1(aH+=1) and the hydrogen gas pressure is equal to one atmos-

phere.

92

TYPES OF ELECTRODES

There are the following types of electrodes: first and second kinds, redox

(reduction-oxidation) and ion-selective.

§ 8. The electrodes of the first kind

The electrodes of the first kind are such systems wherein the electrode is in

contact with varying concentrations of its ions with which it is in reversible reac-

tions, i.e. in potential determination reactions only one type of ions takes part (ei-

ther cations or anions). They include metallic, nonmetallic, gaseous electrodes.

Metal-ion electrodes. Here, a metal M is in electrochemical equilibrium with

a solution containing Mz+ ions. They are marked as Mz+/ M. Half-reaction in the

system is Mz+ + ze–↔ M. The potential is determined by

zMea

zF

RTlg

303,20

For example, in case of zinc electrode it has the following form:

222 ln// Zn

o

ZnZnZnZna

zF

RT

Examples include Cu2+/Cu, Hg22+

/Hg, Ag+ /Ag, Pb2+ /Pb and Zn2+/Zn. Met-

als reacting with a solvent cannot be used.

Example: Determine the potential of zinc electrode if the zinc plate is dipped

into the ZnCl2 solution with the concentration 0.005 mol/dm3. The molar conduc-

tivity of the solution is 870 Ω–1cm2 and λo is 113.70 Ω–1cm2. It is known from table

date that oZn/Zn2 = – 0.76 V.

Solution: According the equation above 222 ln// Zn

o

ZnZnZnZna

zF

RT .

The activity of the solution (active concentration) is equal a = Ca, where C is the

concentration of the solution, and a is the degree of ionization of the salt, which we

find from the relation a = λ/λ0 = 870 Ω–1cm2 / 113.70 Ω–1cm2 = 7.652 Ω–1cm2.

Hence, 7.6520.005 ZnZn

lg0591.076.0–/2 = –0,84 V

93

Nonmetal electrodes. The most important examples are the bromine and io-

dine electrodes: Pt | Br2| Br– and Pt | I2 |I–. In these electrodes, the solution is satu-

rated with dissolved Br2 or I2.

They also include gas electrodes. Here, a

gas is in equilibrium with ions in solution. For

example, the hydrogen electrode is Pt|H2|H+ and

its half-reaction is H2↔2H+ + 2e–, which is quite

similar to reactions occurring on the surface of

metallic electrodes reversible respectively to

cations. This electrode usually consists of the

platinum, coating the surface with a layer of

platinum black (for surface increasing) im-

mersed in a solution containing hydrogen ions

and around which a current of hydrogen gas

flows (Fig.4.13). Platinum here plays only the

role of an inert carrier and may be replaced by

palladium, gold and other inert metals. The po-

tential of this electrode depends on the hydrogen ion concentration of the solution,

on the hydrogen gas pressure and temperature. As the standard potential of hydro-

gen electrode is equal to zero so Nernst equation has the following form

HHH

azF

RTlg

3,2

2/2 4.18

Because numerical value of zF

RT3,2 is equal to 0,059, so we can write

2H/H2 = 0,059 Halg or

2/2 HH = –0,059 pH 4.19

The hydrogen electrode is very sensitive to the operating conditions. To ob-

tain correct and stable results, it is necessary, in particular, to have hydrogen and

platinum surface of high purity.

§ 9. The electrodes of the second kind

In the electrodes of the second kind in potential determination reactions both

cations and anions take part and the electrode is reversible for both of them. The

metal of the electrode is coated with a poorly soluble salt of this metal and is in a

solution containing a well soluble electrolyte with the same anions.

Fig.4.13.Hydrogen elec-

trode

94

Such an electrode, in particular, is the calomel electrode (Fig.4.14). It con-

sists of a paste of mercury and calomel (Hg2Cl2) in a

solution of KCl (usually saturated). It is marked as Hg |

Hg2Cl2, KCl. The following reaction takes place on the

electrode:

2Hg + 2Cl– = Hg2Cl2 + 2e–

Its potential is determined by the relation:

εcal = ε0cal – Cl

azF

RTln 4.20

This equation is similar to Eqn.4.17, but it differs

in the sign before the Claln term, in conformity with

the fact that the formation of negative ions from neutral

atoms occurs by reduction, whereas positive ions are formed by oxidation. The

value of this electrode potential, as we can see from the equation, depends on the

amount of Cl– ions. It is due to the presence of two salts in the system. One of

them, calomel, is weak electrolyte and its dissociation degree is quite small:

Hg2Cl2=2Hg++2Cl–. The other dissociates completely: KCl=K++Cl–. Hence, the

amount of Hg+ ions (due to which electrode potential formed) depends on the

amount of Cl– ions, which comes from the KCl dissociation. The more Cl– ions the

more Hg+ ions links by them. Therefore, the value of this electrode potential de-

pends on the KCl concentration in the solution and for saturated KCl solution the

εcal is equal to 0,2490 (≈ 0,25) at 20oC.

Calomel electrode is stable in operation and yield accurate results, due to this

is applied as a standard comparative electrode for the other electrode potential de-

termination. But, owing to the toxicity of mercury, in recent years wider use is be-

ing made of the silver chloride electrode, which working mechanism is similar to

the calomel electrode: Ag|AgCl|Cl–.

§ 10. Redox electrodes

Although oxidation or reduction occurs on the electrodes during all in elec-

trode process, the name oxidation-reduction electrodes is used in a more restricted

meaning. Redox electrodes are those for which no change in the substance of the

electrode takes place during operation, but which serve only as a source or receiv-

er of electrons taken up or given off by substances oxidized or reduced on the elec-

trode surface. Such electrodes differ from those discussed previously in that the

Fig.4.14. The scheme of

calomel electrode

95

oxidation and reduction products remain in solution instead of being evolved at the

electrode. They are marked as (Pt)|Ox, Red.

A simple illustration of such an electrode is platinum (or other inert metal)

plate immersed in a solution of ferric and ferrous

chlorides: FeCl3 and FeCl2 (Fig.4.15). By combining

such an electrode with another one, a galvanic cell is

formed in which oxidation of Fe2+ to Fe3+ or reduc-

tion of Fe3+ to Fe2+ takes place, depending on the

nature of the other electrode. If the other electrode is

more positive with respect to the considered redox

electrode, the reaction Fe2+ ↔ Fe3+ + e– will go to

the right (oxidation), if negative, the reaction will go

to the left (reduction).

The potential of such an electrode, as of the

others, is determined by the equation

d

OxdOx

odOx

a

a

zF

RT

Re

Re/Re/ ln 4.21

The value of standard redox potential is determined by redox electrode abili-

ties to oxidation or reduction. On this basis, electrochemical series of metals is ob-

tained, where the metallic elements are arranged in the order of their reduction

power respectively to the hydrogen (measured by their standard redox potential in

aqueous solution. A metal lower in the series (with a lower standard potential) can

reduce the ions of metals with higher standard potentials. For example, to deter-

mine whether zinc can displace magnesium from aqueous solutions at 298 K, we

note that zinc lies above magnesium in this electrochemical series, so zinc cannot

reduce magnesium ions in aqueous solution. Zinc can reduce hydrogen ions, be-

cause hydrogen lies higher in the series.

Oxidation-reduction systems may also contain organic components. Such is

the quinhydrone electrode, widely used for pH measurements. Quinhydrone is a

crystalline compound of hydroquinon and quinon: C6H4O2∙C6H4(OH)2. It is spar-

ingly soluble in water and on dissolving partially is decomposed into quinon and

hydroquinon. If sufficient quinhydrone is added to a solution to form a saturated

solution, constant and equivalent concentrations of quinon and hydroquinon are

established. The latter is dibasic acid which is somewhat dissociated according to

the equation

C6H4(OH)2 ↔ C6H4O22– + 2H+ + 2e–

Fig.4.15. Illustration of

work of ferric and ferrous

chlorides electrode

96

forming an anion that has the same composition as quinon and is transformed into

the latter losing its charge, i.e. oxidation process occurres:

C6H4O22– ↔ C6H4O2+ 2e–

The overall reaction may be presented by the following equation

C6H4O2 + 2H+ + 2e–↔ C6H4(OH)2

which shows that equilibrium in this oxidation-reduction system depends on the

hydrogen ion activity and the value of this electrode potential can be determined by

the equation

2

246

246lg2

059,0

OHHC

HOHCquin

oquin

Because the concentrations of quinon and hydroquinon are equal in quinhydrone,

we obtain

pH or Hlg quinquino

quin 059,07,0059.0 4.22

(because quino =0,7 and pH=-lg H )

§ 11. Ion-selective membrane electrodes

Ion-selective membrane electrode consists of glass, crystalline, or liquid

membrane (with ions of alkali metals into it) which is in contact with an electrolyte

solution and the potential difference between the membrane

and solution appears due to ion exchange between them. The

difference of such electrodes from others is that there is no

oxidation or reduction processes during an electrode poten-

tial formation. An electrode potential is formed due to ex-

change between ions of the membrane and of the solution.

The most widely used membrane electrode is the glass elec-

trode (Fig.4.16), which is sensitive to hydrogen ion activity

in the solution, and has a potential proportional to pH. An

Ag-AgCl electrode and an internal filling solution of aque-

ous HCl are sealed in as a part of the glass electrode. It con-

tains a thin glass membrane of special composition. Glass

contains a three-dimensional network of covalently bound Si

and O atoms with a net of negative charge, plus positive

Fig.4.16. The

glass electrode

97

ions, for example, Na+, Li+, Ca2+, in the spaces in the Si-O network. The positive

ions of the alkali metals can move through the glass, giving it a very weak electri-

cal conductivity. The thinness (0,005cm) of the membrane reduces its resistance.

The responsiveness of a glass electrode to the hydronium ion activity is a result of

complex processes at the interface between the glass membrane and the solutions

on either side of it. A clue to the mechanism comes from a detailed inspection of

the glass membrane, for each face is coated with a thin layer of hydrated silica

(Fig.4.17). The hydrogen ions in the test solution modify

this layer to an extent that depends on there activity in

the solution, and the charge modification of the outside

layer is transmitted to the inner layer by the Na+ and Li+

ions in the glass.

When the glass electrode is immersed in solution

an equilibrium is set up between ions (such as Na+, Li+,

Ca2+, NH4+,Mg2+,F–,NO3–) in solution and ions in the

glass surface. This charge transfer between glass and

solution produces a potential difference on the glass–

solution boundary surface.

Ion-selective membrane electrodes allow measur-

ing of the activities of certain ions that are difficult to

determine by traditional analytical methods. Glass mi-

croelectrodes are used to measure activities of H+, Na+, K+ in biological tissues.

§ 12. The diffusion and membrane potentials

The diffusion potential (or the liquid junction potential) is the potential dif-

ference appearing at the junction between two solutions differing in either the na-

ture or concentration of the solute. These potential jumps are not very high. Their

origin is due to the difference in the motilities and hence in the diffusion rates ions

of different species. We shall consider only the simplest case when the contacting

solutions contain the same electrolyte and differ only in its concentration. Consider

two HCl solutions of differ concentrations (Fig.4.18).The passage of HCl from so-

lution of higher to solution of lower concentration is due to diffusion of the ions.

But the H+ ion possesses a higher mobility than the Cl– ion and at first H+ ions dif-

fuse though the junction in large numbers than Cl– ions. As a result a potential dif-

Fig.4.17. The membrane

structure of a glass elec-

trode

98

ference arises, the more dilute solution becoming posi-

tively charged and the more concentrated – negatively

charged.

If, now, we set a semipermeable membrane (per-

meable for one type of ions, for instant H+, and imper-

meable for the others) between such solutions a diffu-

sion potential value becomes higher and it’s called the

membrane potential, the value of which could be meas-

ured by the equation

2

1lna

a

llF

llRT

ac

ac

4.23

where lc and la are ion mobility’s, and a1 and a2 are their

activities ( a1 > a2).

§ 13. Bioelectrochemistry

The membranes of biological cells are semipermeable membranes, and as

there are different ions on both sides of the membrane, hence the potential differ-

ence (transmembrane potential) is measured between the interior and exterior of

the biological cell. Since interphase potential differences exist in living organisms,

living organisms are electrochemical systems.

When an impulse propagates along a nerve cell or when a muscle cell con-

tracts, the transmembrane potential changes becomes positive for a moment. Nerve

impulses are transmitted by changes in nerve-cell membrane potentials. Muscles

are caused to contract by changes in muscle-cell membrane potentials. Our percep-

tion of the external world through the sense of sight, hearing, touch, etc., our

thought processes, and our voluntary and involuntary muscular contractions are all

intimately connected to interphase potential differences. To understand life re-

quirements is to understand how these potential differences are maintained and

how they are changed.

The existence of transmembrane potential differences means that there is an

electrical double layer at the membrane of each cell. Consider the heart muscles.

As these muscles contract and relax, the potential differences across their cell

membrane continually change, and hence the total dipole moment of the heart

changes, and so do the electric field and electric potential produced by the heart.

Fig.4.18.The diffu-

sion potential for-

mation

99

An electrocardiogram (ECG) measures the difference in electric potential between

points on the surface of the body as a function of time. Changes in these potential

differences arise from the changes in the heart dipole moment. An electroencepha-

logram (EEG) records the time-varying potential difference between two points on

the scalp and reflects the electrical activity of nerve cells in the brain.

A biological cell membrane is permeable to all three ions of K+, Na+, and Cl–

, but it is more permeable to K+ than to Na+ or Cl– ions. These three ions are the

main inorganic ions. The concentration of K+ inside the cell is about 20 to 30 times

more than on the outside, and is maintained at that level by a specific pumping op-

eration fuelled by ATP and governed by enzymes. An active-transport process that

uses some of the cell metabolic energy to “pump” continually Na+ out of the cell

and K+ into it maintains the observed steady-state concentrations of Na+ and K+.

We can use the Nernst equation to see which ions are in electrochemical

equilibrium across the membrane. The observed transmembrane potential shows

that the potential difference between the two sides is predicted to be about 77 mV.

The transmembrane potential difference plays a particularly interesting role

in the transmission of nerve impulses. A nerve impulse is a brief change in the

transmembrane potential. Potassium and sodium ion pumps occur throughout the

nervous system, and when the nerve is inactive there is a high K+ concentration

inside the cells and a high Na+ concentration outside. The potential difference

across the cell wall is about 70 mV. When the cell wall is subjected to a pulse of

about 20 mV, the structure of the membrane adjusts and it becomes permeable to

Na+. This adjustment causes a decrease in membrane potential as the Na+ flood into

the interior of the cell. The change in transmembrane potential difference triggers

the adjacent part of the cell wall, and the pulse of collapsing potential passes along

the nerve fiber at 103 to 104 cm/s, depending on the species and the kind of nerve.

Behind the pulse, the sodium and potassium pumps restore the concentration dif-

ference ready for the next pulse.

100

§ 14. Galvanic cell

A galvanic (or electrochemical) cell

is any appliance that creates electric cur-

rent by means of a chemical reaction. In

other words it produces electricity as a re-

sult of the spontaneous oxidation-reduction

process occurring inside it. An electrochem-

ical cell consists of two electrodes separated

by a porous partition. Each electrode and the

medium in which it is immersed is called a

half-cell. Let us consider a system of two

metals such as zinc and copper plates each

of them dipped in the solution of their salt

and separated by a porous membrane

(Fig.4.19). Each of metals is in equilibrium

with the ions of its solution and has a certain

value of an electrode potential corresponding to its equilibrium state. As we con-

sider above, in equilibrium state there is an excess of electrons on the plate of zinc

and of copper ions on the plate of the copper.

If now we connect the plates by means of

a wire (the electric circuit connecting the two

electrodes outside cell is often called external

circuit), the difference in potential between the

plates will cause a corresponding amount of

electrons to flow from the zinc to copper plate,

where reduction of copper ions takes place

(Fig.4.20). This disturbs the equilibrium of the

double layer at both plates, another portion of

Zn2+ ions will pass from the zinc electrode into

the solution, and another portion of Cu2+ ions

will pass from the solution to the copper elec-

trode (in amounts suitable for their equilibrium

state). Thus, there again appears a difference in

the charge of the plates causing a transition of

the electrons from the zinc plate to the copper

Fig.4.19. A construction of galvanic

cell consisting of zinc-cooper elec-

trodes

Fig.4.20. Direction of electrons

flow in zinc-cooper galvanic

cell

101

one, and further passage of the ions as described above. As a result a spontaneous

process takes place, in which the zinc plate dissolves and metallic copper is depos-

ited on the copper plate owing to discharge of the copper ions. The passage of elec-

trons along the wire from the zinc to the copper is responsible for the electric cur-

rent. The current can be quantitatively measured by connecting electrical measur-

ing instrument in its path.

A galvanic cell based on the above principles can serve as a current source.

Such a cell was proposed by J.Daniell and B.Jacobi in the middle of the last centu-

ry.

All galvanic cells involve oxidation-reduction reactions occurring on the

electrodes. Oxidation is a loss of electrons. Reduction is a gain of electrons. The

electrode at which oxidation occurs is called the anode; the electrode at which re-

duction occurs is called the cathode. The oxidation of zinc Zn – 2e– → Zn2+ takes

place on the negative electrode; the reduction of copper Cu2+ + 2e– = Cu – on the

positive electrode. Hence, the overall cell oxidation-reduction reaction will be:

Zn + Cu2+ = Cu + Zn2+

In a cell with two electrolyte solutions in contact, as in the Daniel cell, there

is an additional source of potential difference, the liquid-junction potential (consid-

ered above) across the interface of the two electrolytes. Junction potential is small,

but cannot be neglected in accurate work. By connecting the two electrolyte solu-

tions with a salt bridge (Fig.4.21), the junction potential can be minimized (but not

completely eliminated). A salt bridge consists of a gel made by adding agar to a

concentrated aqueous KCl solution. The

gel permits diffusion of ions but elimi-

nates convection currents. It allows con-

necting the excess of Zn2+ in the solution

of the zinc electrode, and the excess of

SO42– in the solution of the copper elec-

trode (occurs due to functioning of the

cell).

A galvanic cell is represented by a

cell diagram, where each symbol repre-

sents the phase in which the substance is

found. All phases comprising the cell are

written consecutively (beginning at the negative electrode) in a single line, and the

interfacial boundaries between them are set off by a vertical line. A double vertical

Fig.4.21. Connection of two electrodes

solutions by salt bridge

102

line assumed that the junction potential has been eliminated (by salt bridge). If

there are two or more species present in the same phase, they are separated by a

comma. For Daniel cell it is formulated as follows:

Zn | ZnSO4 | | CuSO4 | Cu

A cell is often written to show just the reacting species rather than the com-

plete formula of the solute in each half-cell:

Zn | Zn+2 | | Cu+2 | Cu

The work producing in the galvanic cell is quantitatively expressed by the

electromotive force and designated e.m.f. The electromotive force is the maximum

potential difference of a cell. The maximum potential difference between the elec-

trodes of the cell (or minimum current flow) is obtained when the cell is closely

operating under to reversible conditions. The general thermodynamic conditions of

reversibility as applied to galvanic cell may of e.m.f. exceeds only by an infinites-

imal amount an applied external e.m.f. of opposite sign (oppositely directed exter-

nal e.m.f.). Under such condition, potential difference of a cell is maximal, and a

current flow is minimal. Therefore, for Daniel’s cell e.m.f. is equal to:

22 // ZnZnCuCuE

After putting the meaning of the electrode potentials in the above equality, we ob-

tain:

2

22

2 lnln //Zn

ZnZno

CuCuCu

o azF

RTa

zF

RTE

or

2

2

22 lg2

3,20

/

0

/

Zn

Cu

ZnZnCuCu a

a

F

RTE

The sign of normal functioning and current giving cells (to the direction in

which the reaction proceeds spontaneously and can do work) e.m.f. is positive;

hence to determine e.m.f. we subtract a negative electrode potential from positive.

To take into consideration conditions of reversibility for the reverse direction of the

reaction (when the external e.m.f. exceeds infinitesimally of e.m.f. of the cell), the

e.m.f. of the galvanic cell is then given a negative sign.

Besides galvanic cells consisting of two different electrodes, there are cells

consisting of the same two electrodes but with different concentrations of solutions.

Such cells are called concentration cells. Current is generated in them due to the

difference in and its functioning up to concentrations between the different regions

of the solution become equal.

103

Let us consider an example of such type of cell consisting of two zinc elec-

trodes:

Zn | ZnSO4 (a1) | | ZnSO4 (a2) | Zn

where a1> a2. Both of them are negative, but the charge value of the electrode in

lesser concentration is more negative (more Zn2+ ions are able to pass from a plate

to a solution). Hence, the e.m.f. such of cell is equal:

21

//

lg2

059,0

lnln 22

22

1

2

21

aa

azF

RTa

zF

RTE

aa ZnZnZn

o

ZnZnZn

oaa

2

1lg059,0

a

a

zE 4.24

Equation shows, that the e.m.f. of the concentration cell doesn’t depend on

the nature of the metal (on εo), and depends only on the relation of solution concen-

trations and the charge value of the ion.

§ 15. Potentiometric determination

The determination of any properties by measuring the e.m.f. of a galvanic

cell by using a potentiometer (Fig.4.22) is known as a potentiometric determina-

tion. This can be done by the compensation method, where the Ex (definable) of

the cell is balanced by an opposing potential difference E1 (known), so as to make

the current passing throw the cell equal to zero. The method allows potentials to be

measured under reversible operating con-

ditions of the cell, when minimum current

flows throw the cell.

Source of electricity 1 of the known

e.m.f. such as a storage battery is connect-

ed to the two ends of wire AB. The wire

must be of the same resistance throughout

its entire length, i.e., it must be sufficiently

homogeneous and of the same cross sec-

tion. As a result the drop in potential along

the length from A to B is proportional to

this length. The wire is stretched along a scale. A branch circuit 2 is formed by cell

Fig.4.22. The scheme of a po-

tentiometer

104

x to be measured (current flow of which is opposite to that of the storage battery)

and connected to a sensitive galvanometer 4. By sliding contact C, a position is

found at which the e.m.f. of the cell is exactly compensated and no current flows in

the branch circuit. This means that the potential drop along the segment AC of the

wire is exactly equal to the e.m.f. of the unknown cell Ex. Knowing the length of

this segment, the length of the rest of the wire and the e.m.f. E1, the e.m.f. of the

cell being measured is obtained from the relation

AB

ACEEx 1 4.25

§ 16. Applications of e.m.f. measurements

1. Determination of thermodynamic functions and the equilibrium con-

stant. After E0 of the cell determination by potentiometry ΔGo and Ko of the cell’s

chemical reaction can be found from

ΔGo = –zFE0 4.26

followed by ΔGo = – RT ln Ko, from which

ln Ko = zFE0/RT 4.27

Using the equation from thermodynamics PT

GS

and putting in it

the meaning of the ΔGo from eqn. 4.26, we obtain the equation for standard entropy

determination:

T

EzFS

oo

4.28

The ΔHo can be found from ΔGo = ΔHo–TΔSo

2. Determination of activity coefficient. Since the e.m.f. of a cell depends on

the ions activity in solution, it is easy to use measured e.m.f. values to calculate

activity coefficient. For example, for the cell

(Pt) | H2 | HCl | AgCl | Ag (Pt)

with HCl as its electrolyte, the cell reaction is

½ H2 + AgCl ↔ Ag + H+ + Cl–

and the e.m.f. is given by equation:

E = Eo – 0,059 lg(γ±C)2

So the activity coefficient can be calculated from the measured E (Eo is known) at a

certain molality of the solution.

105

3. Determination of pH. For such determination one has to assemble the

galvanic cell, one electrode of which is comparative (with known and unchanged

value of the electrode potential), and the other with potential, depending on the

concentration of H+ ions in solution (with sought pH). By measuring the e.m.f. of a

constructed cell, which is equal to the difference between electrode potentials, one

can find out the magnitude of the pH depending potential and from this the magni-

tude of the sought pH. It could be the following cells:

Concentration hydrogen cell, consists of two hydrogen electrodes, the con-

centration of H+ ions of one of them is known

(–) (Pt) H2 | H+x| | H+

stand| H2 | (Pt) (+)

The e.m.f. of this cell is determined by

stx

xHdsx

pHpH 0,059E

or aaEst

059,0

lg059,0lg059,0tan

from which

stpHE

pH 059,0

4.29

Calomel-hydrogen cell consists of a calomel electrode and a hydrogen elec-

trode, dipped in the solution with unknown molality of H+ ions

(–) (Pt) H2| H+| | KCl,Hg2Cl2 | Hg (Pt) (+)

0,059

0,243-EpH

and pHlga EHhydrcal

059,025,0059,025,0

4.30

Quinhydrone-calomel cell:

(–) (Pt) Hg| KCl,Hg2Cl2 || H+,quinhydrone | H2(Pt) (+)

pHaEHcalquin 059,0457,025,0lg059,07,0

from which

059,0

45,0 EpH

4.31

Calomel-glass cell is one of which today is used almost universally in pH-

meters. The cell diagram is:

(–)Ag | AgCl,HCl | glass| solution with unknown pH | | KCl,Hg2Cl2 | Hg (+)

0,059

0,25-EpH

aE

o

H

oglcal

lg059,025,0

4.32

106

where εo depends on the kind of the glass and has a different value for different

glasses. Because it is impossible to predetermine the magnitude of the potential of

each glass electrode, so each pH meter before use must be standardized by a buffer

solution of known pH.

The wide popularity of potentiometric pH determination is due to their high

accuracy and applicability in systems for which colorimetric and other methods do

not give satisfactory results.

4. Potentiometric titration

This method is also widely used to determine the total concentration of solu-

tions, particularly an acid or base. Essentially the method is based on measuring the

potential of an electrode (immersed in the solution undergoing titration and linked

with another electrode with known value of electrode potential) by measuring the

e.m.f. of obtained galvanic cell during titration. The e.m.f. gradually a change as

the reagent is added until the point of equivalency is reached, at which a jump of

e.m.f. occurs. The point of equivalency is defined most sharply when a strong acid

is titrated with a strong base. The slope of a plot of pH (or e.m.f.) versus volume of

added reagent is a maximum at the equivalent point (Fig.4.23). It is more conven-

ient to construct a plot of ΔE/ΔV (or ΔpH/ΔV) versus volume (Fig.4.24), on which

it is easier to determine a point of equivalency.

Fig.4.23. A potentiometric titration curve

for a strong acid being titrated by a

strong base

Fig.4.24. A differential potentiometric titra-

tion curve for the same case

107


1. Use standard electrode potential values from table data to find 0298G and

0298K for the reaction Cu2+(aq) + Zn(s) → Cu(s) + Zn2+(aq), occurring in the cell:

Solution: We know that the standard e.m.f. of the cell is equal to the differ-

ence of standard electrode potentials:

VVVEEEZnZnCuCu

101.1)762.0(339.00

/

0

/

0298 22

0298G = – nFE0= – 2(96485 C/mol)(1.101 V) = – 212.5 kJ/mol

Use of ΔGo = – RT ln Ko gives

73,85)15.298)(/314.8(

/212500ln

0298

KmolKJ

molJ

RT

GK

370 102K

2. The e.m.f. of the galvanic cell consists of hydrogen and calomel (with sat-

urated KCl solution) electrodes and dipped in the solution with unknown pH is

equal to 0.76 V. Determine pH of the solution.

Solution: The pH of the such galvanic cell is found from the equation 4.30:

0,059

0,243-EpH . Hence, pH = (0.76 – 0.243) / 0.0591 = 8.74

3. The galvanic cell consists of two hydrogen electrodes each of them im-

mersed in the solutions with pH values 0.5 mol/dm3 and 0.2 mol/dm3 accordingly.

Find the e.m.f. of this cell.

Solution: It is concentration cell and the e.m.f. of it is described by the

equation 4.24: 2

1lg059,0

a

a

zE . Therefore,

2.0

5.0lg

1

059,0E = 0.023

Review questions:

1. Define the following terms: electrode processes, electrode, and electrode

potential.

2. Explain the mechanism of electrode potential formation on (a) zinc elec-

trode and (b) copper electrode. What kind of forces takes place in the process?

3. Due to what action negative charge is formed on the zinc electrode and

positive charge on the copper electrode?

4. What does the sign formed on the electrode depend on?

5. Write the Nernst equation for electrode potential determination. What

does the value of electrode potential depend on?

108

6. How is the value of zinc electrode potential changed if the solution of the

electrode is diluted?

7. Define the term of standard potential and standard hydrogen potential.

Does their value depend on the solution concentration?

8. List types of electrodes you know.

9. Define the first kind of electrodes and write the Nernst equation for metal-

ion and hydrogen electrodes.

10. Define the second kind of electrodes. Explain the working principle of of

calomel electrode. Why does the value of its electrode potential depend on the

amount of Cl– ions?

11. In which way are redox electrodes distinguished from the rest elec-

trodes? Explain the working principle of such electrode by pairs Fe3+/Fe2+ and

Cr2+/Cr3+.

12. On what principle is the work of ion-selective electrodes based? What is

determination glass electrode applied for?

13. Why is the diffusion and membrane potential formed?

14. What system is called galvanic cell? Explain the working principle of

cell by the example of zinc-copper cell. What is the limit of its work?

15. Does the work of cell (electric current obtaining) occur spontaneously or

not? What charged particles are moved along the external chain?

15. Which electrode is the anode and which is the catode in galvanic cell?

What charge do they have?

16. Why is the salt bridge applied during the cell working and what is it?

17. Write the cell diagram for zinc-copper cell. In what order it is usually

written?

18. What quantity is the quantitative characteristic of the work of a galvanic

cell? Define this term.

19. When a galvanic cell works under reversible conditions? What is value

of potential difference in such case?

20. What electrodes a concentration cell consists of? How does such cell

work? What quantities does such cell e.m.f. depend on? What is the working limit

of such cell? Write the equation of e.m.f for such cell.

21. What is measured in potentiometric method? What is it applied for?

22. Write pH determination by calomel-hydrogen and quinhydrone-calomel

cells.

109

§ 17. Electrolysis

Electrolysis is a process of an oxidation–reduction chemical reactions pro-

ceeding under the action of an electric current. In such the second kind (in contrast

to the first kind one as galvanic cell) of electrochemical cell (electrolytic cell) elec-

trical energy from an external source is used to produce a chemical reaction. Chem-

ical transformations in electrolysis may vary widely, depending mainly on the na-

ture of the electrolyte and solvent and also on the material of the electrode and the

presence of foreign substances in the solution. The following is a general outline of

the process.

Ions of the electrolyte on reaching the corresponding electrode (cations – the

cathode and anions – the anode) interact with the latter, losing some or in most case

all of their charge to become neutral atoms or atomic groups. In contrast to the gal-

vanic cell in electrolytic cell the anode is the positive electrode and always an oxi-

dizing agent, and the cathode is negative electrode and always a reducing agent.

Let an aqueous solution of HCl be subjected to electrolysis on inert elec-

trodes. The Cl– ions migrates to the anode and the H+ ions to the cathode. On reach-

ing the anode, the Cl– ions transfer their excess electrons to it and are converted

into neutral atoms according to the reaction Cl– → Cl + e–. At the cathode the H+

ions take up the electrons they lack, becoming neutral atoms according to the reac-

tion H+ + e– → H. The neutral Cl and H atoms are unstable and combine in pairs

into diatomic molecules Cl2 and H2. As a result chlorine gas is liberated at the an-

ode and hydrogen gas at the cathode.

The rate of an electrochemical reaction occurred in the system is determined

by the amount of the substance produced in a unit of time, what depends on the

current flow throughout the cell. Hence, measure of the rate of the process is a cur-

rent density (i), which is equal to an electricity amount flowing throughout of the

electrode unit surface.

The current intensity depends on the voltage (potential difference) applied to

the electrodes. On increasing the voltage, the current intensity is increased, and, on

the contrary, by lowering voltage the current is reduced. The lowest voltage re-

quired for a given electrolytic reaction is called the decomposition voltage or de-

composition potential.

110

§ 18. Electrode polarization

During electrolysis process a certain potential difference appears between

the electrodes that are oppositely directed to the potential difference applied exter-

nally. This effect is called polarization. A distinction is made between chemical

and concentration polarization.

Chemical polarization is due to the formation of a galvanic cell by the elec-

trolysis products. For instance, in the electrolysis of an aqueous solution of CuSO4

with platinum electrodes, copper is liberated at the cathode and oxygen at the an-

ode. This converts the original electrolytic bath into a galvanic circuit as Cu |

CuSO4 |O2 (Pt) the e.m.f. of which opposes electrolysis, i.e., polarizes the elec-

trodes. In the same way during the electrolysis of an H2SO4 solution the hydrogen

evolved at the catode and the oxygen at the anode form the circuit as Pt (H2) |

H2SO4 | O2 (Pt) that causes polarization.

Concentration polarization is due to the difference in the concentration of

the electrolyte at the anode and cathode regions arising during the electrolysis. For

example, in the electrolysis of an AgNO3 solution with silver electrodes, the elec-

trolyte concentration in the cathode region diminishes (due to Ag+ ions reduction)

and in anodic region increases. This causes a concentration cell as Ag | AgNO3 (a1)

| AgNO3 (a2) | Ag e.m.f. of which is opposed to the applied potentials.

Both types of polarization prevent and reduce a process of electrolysis and

enlarge expense of energy necessary for electrolysis occurring.

As a result of polarization the overvoltage phenomenon occurs on the elec-

trodes during the electrolysis. At very low current density the overvoltage (ήo) on

an electrode equals to the difference between the deposition potential (i.e., the po-

tential of the polarized electrode) and the equilibrium potential of the given elec-

trode:

ήo = εdep – εequ 4.33

Overvoltage, in particular hydrogen overvoltage, is not only of theoretical,

but also of practical importance. This can readily be seen from the fact that electro-

lytic deposition of such metals as Fe, Pb, Zn, which stand higher than hydrogen in

the electromotive series, is possible only because these metals have a lower over-

voltage than the hydrogen overvoltage on them, especially at high current densities.

The potential of hydrogen evolution therefore becomes higher than the deposition

potential of these metals. When higher current densities at higher overvoltage are

used, substances in a more active state may be obtained.

111

In some cases, for instance, in the electrolytic production of hydrogen, over-

voltage, on the contrary, is an undesirable phenomenon that leads to an increased

consumption of electric energy.

§ 19. Polarography

Curves expressing the dependence of current density on the applied voltage

are known as polarization curves, since they make possible evaluation of the

changes in polarization of the electrode with the voltage. The construction of polar-

ization curves lies at the basis of a special method of electrometric analysis called

polarography. In this method the anode is practically non-polarized and makes use

of polarization process in a cathode the surface of which is continuously renewed.

Therefore the applied voltage goes only to polarize the cathode and sent current

through the solution. So, by measuring the current strength at different voltages,

one can determine the polarization on the cathode. For to construct polarization

curves different voltages gives on the cathode (in growing order) and the current

strength flows through the solution is measured. Then plot the diagram of there

dependence. For example, let us consider the polarization curve of the electrolytic

process of the aqueous solution of CuSO4 (Fig.4.25). There are two possibilities of

reduction processes realization on the cathode: for Cu2+ ions and for H+ ions. But,

because of standard potential of Cu2+ has more positive value, than H+, the Cu2+ is

reduced on the cathode at low level of

the cathode voltage. The ab part of the

curve corresponds to the process of

growing the voltage on the cathode

without current flow through the solu-

tion, as the voltage doesn’t reach the

value of decomposition potential for

Cu2+ in which the cathode process

begins. It begins at point b, where the

Cu2+ ions start reducing and the cur-

rent flowing through the solution.

During the reduction process current

strength becomes more and more (bc

section) until the point c, where the

amount of Cu2+ ions in solution come to the end, so there is now more growing of

Fig.4.25. The polarization curve of the

electrolytic process of the aqueous solution

of CuSO4

112

current strength. This value of current strength (cd) is called boundary or diffusion

current (Ib). It’s characterized by a maximal possible rate of the process (reduction)

in certain conditions. Now, as there are other

ions (H+) can reduce at the higher value of

cathode potential, corresponds to the point d,

and the next section of the curve shows the

reduction process of H+ ions.

Anyway, the polarization curves have

wavy character each wave of which suits a

certain ion (Fig.4.26).

Polarographic method is widely used

in analytic analyses in ion nature and con-

centration determinations and in particularly

in pharmacy for mixed drugs, or drugs in

very small amounts, or insoluble (in water) drug analyses.

For ion nature determination, one has to find out the height of a certain

wave (h) and then the potential value suitable for the half of the height (Fig.4.25).

This value has a constant meaning for every ion under certain conditions.

For ions concentration determination we use dependence between Ib and the

concentration:

Ib = Kc 4.34

The value of K is found by polarographic measurement with solutions of

known concentrations.

Review questions:

1. What process is called electrolysis? How is it different from galvanic cell?

2. What is the charge of catode and anode in electrolytic system?

3. What is polarization and why does it occur in electrolysis?

4. Define the method of polarography.

5. Draw an example of polarographic curve and explain the construction

process.

6. How are qualitative and quantitative analyses carried out by polarographic

curves?

Fig.4.26. Wavy character of the

polarization curves

113

CHAPTER 5

CHEMICAL KINETICS

The study of a process rate is called kinetics or dynamics. The branch of ki-

netics that studies the rates and mechanisms of chemical reactions is chemical

kinetics. A reacting system is not in equilibrium, so reaction kinetics is not a part

of thermodynamics but is a branch of kinetics.

In the industrial synthesis of compounds, reaction rates are as important as

equilibrium constant. For example, the thermodynamic equilibrium constant tells

us the maximum possible yield of NH3 obtainable at any given T and P from N2

and H2, but if the reaction rate between N2 and H2 is too low, the reaction will not

be economical to carry out. Thus, to understand and predict the behavior of a

chemical system, one must consider both thermodynamic and kinetic possibilities

of a system.

§ 1. The rate of reaction

The overall stoichiometry of the reaction does not demonstrate the process,

or mechanism, by which the reaction actually occurs, because most of reactions

proceeds by many steps often with a reaction of intermediates formation (which

does not appear in the overall reaction). Each step in the mechanism of a reaction is

called an elementary reaction. A simple reaction consists of a single elementary

step. A complex (or composite) reaction consists of two or more elementary steps.

Different reactions take place at different rates. Some of them, as in the det-

onation of explosives, terminate in fractions of the second; others continue for

minutes, hours or days, and still others, such as those occurring in the earth’s crust,

may continue for tens, hundreds or thousands of years. Not only can there be a

great difference in the rates of different reactions, but the rate of a given reaction

may also vary greatly, depending on the conditions under which it occurs. These

problems are highly important from both theoretical and practical standpoint.

The rate of a chemical reaction is quantitatively characterized by the

change in moles number of the reactants proceeds in a unit of time in a unit of the

volume:

114

dt

dn

avV

11 5.1

where t is the time, v is the volume, and a is a stoichiometric number of a sub-

stance. In most systems the volume is constant and one can replace the number of

moles in a certain volume by concentration and define the reaction rate as the

change of the concentration of reactant or product proceeds with time.

One may use the finite changes in concentration C2 – C1 occurring in the

time t2 – t1 and in this way determine the average rate of the reaction for that time

period:

12

12

tt

CCV

Usually the rate of a chemical reaction does not remain constant, but changes

with time, so it’s more convenient the change in concentration to refer to infinites-

imal time and thereby the instantaneous rate of reaction at the given moment can

be determined as a derivative of the infinitesimal change of a concentration with

respect to infinitesimal time change:

dt

dCV 5.2

In the following we shall consider only instantaneous reaction rate.

We can have several possibly different rates to describe the same reaction

(expressing each substance taking part in the reaction). The sign of dC is negative

for reactants and is positive for products because the concentration of the reactants

decrease and the concentration of the products increase during a reaction. So, if the

rate is expressed by the concentration of reactants we put the sign “minus” in front

of the relation, and the sign “plus” for products, because the rate of reaction is al-

ways considered to be positive. For instance, in reaction A+2B→3C+D the rate

may be expressed as:

dt

Bd

dt

Ad

dt

Cd

dt

Dd

2

1][

3

1

The reaction rates in all cases depend on the three basic factors – concentra-

tion, temperature, catalysts. Besides for certain reactions it depends on the other

factors: on surface condition for heterogeneous reactions, on flask form for chain

reactions, on electrical yield voltage for electrochemical reactions etc.

115

§ 2. The reaction rate dependence on the reactants concentration

According the rate law, the rate of a given reaction at constant temperature

is proportional to the product of the reactant concentrations, each concentrations

being raised to a power, according the coefficient in front of the formula of the

substance in the equation of the reaction. The rate law of a reaction is determined

experimentally, and in general cannot be inferred from the chemical equation of a

reaction.

This rule expressed the effect of the concentration on the reaction rate. For

instance, in the reaction as A → B it has the following form:

Ak

dt

Ad 5.3

and for reaction as mA + nB → pC is:

nm

BAkdt

Ad 5.4

The proportionality factor k is a constant for a certain temperature and a cer-

tain reaction and is called the rate constant of the reaction. It is numerically equal

to the rate of the reaction when the concentrations of the initial components equal

unity. The rate constant does not depend on the concentration of the reagents but

depends on the temperature.

Practical application of the rate law is the following: once we know the val-

ue of the rate constant, we can predict the rate of the reaction basing on the compo-

sition of the mixture. Moreover, as we can see later, knowing the rate law, we can

go on to predict the composition of a reaction mixture in each stage of a reaction.

Moreover, a rate law is a guide to the mechanism of the reaction, for any proposed

mechanism must be consistent with the observed rate law.

Taking into account their kinetics, chemical reactions are classified either

according to their molecularity or to their order.

The number of molecules that simultaneously react in the elementary reac-

tion determines the molecularity of a reaction. Molecularity is defined only for el-

ementary step and should not be used to describe overall reactions that consist of

more than one elementary step. The elementary reaction A → products is uni-

molecular (CaCO3 → CaO + CO2); A + B → products, and 2A → products are

bimolecular (H2 + I2 ↔ 2HI); A + B + C → products, 2A + B → products, and 3A

→ products are trimolecular (2NO + H2 → N2O + H2O). No elementary reactions

involving more than three molecules are known, because of very low probability of

116

simultaneous collision of more than three molecules. When the equation of the re-

action indicates that a large number of molecules participate, this usually means

that the process must proceed in a more complicated manner, namely, two or more

consecutive stages each of which is due to the collision between two, or, rarely,

three molecules. Most elementary reactions are unimolecular or bimolecular, tri-

molecular reactions being uncommon because of the low probability of three-body

collisions.

The order of a reaction expresses the dependence of the rate on the reactant

concentrations. For a general reaction of the type

aA + bB = cC + dD

the rate law takes the form baBAkV . The sum of the powers to which all re-

actant concentrations appearing in the rate law are raised is called the overall re-

action order. Reactions are classified as the first-order, the second-order, and the

third order. There are also zero-order and fractional order reactions. The above

reaction is the first-order in A as well as in B (if a=1 and b=1). The overall order is

the sum of the individual orders and this reaction therefore is the second-order.

It is most important to distinguish molecularity according to the order. Reac-

tion order is an empirical quantity, obtained from the experimental rate law, and

related to all the chemical processes (with all steps). The molecularity refers to an

elementary reaction proposed as an individual step in a mechanism. They coincide

in single-step reactions. Bimolecular simple reaction is the second-order one be-

cause its rate is proportional to the rate at which the reactant species meet, which in

turn is proportional to their concentrations. Therefore, if the reaction is an elemen-

tary bimolecular process, then it has the second-order kinetics but, if the kinetics is

of the second-order, then the reaction might be complex.

The order of a reaction must be determined by experiment, it cannot be de-

duced from the overall balanced equation.

A useful indication of the rate of chemical reactions is also half-life of a re-

action, τ1/2, which is equal to the time required for the concentration of a reactant to

decrease to half of its initial concentration.

117

KINETIC EQUATIONS

The rate of a reaction is described by kinetic equations according their order,

which shows the rate dependence of reactant concentrations in each certain reac-

tion.

§ 3. Zero-order reactions

Some (very rarely) reactions obey a zero-order rate law, and therefore have a

rate independent on the concentration of the reactant:

okdt

dC 5.5

Only a few heterogeneous reactions can have rate law, which is zero-order

overall. Thus, the catalytic decomposition of phosphine (PH3) in hot tungsten at

high pressures has the rate law of zero-order. The PH3 decomposes at a constant

rate until it has almost entirely disappeared.

§ 4. First-order reactions

For the first-order reaction as A → B the rate dependence on the concentra-

tion has the following form:

Ckdt

dC1

We can write it in the form as under:

dtkC

dC1

After integration we obtain

consttkC 1ln

Under conditions t=0 and C=Co we find that const=ln Co

Insert the meaning of the constant of integration in the above equation it may get

the form –lnC = k1t – lnCo, from which

C

C

tk oln

11 5.6

It is the kinetic equation of the first-order reactions, where the unit of the rate con-

stant is t–1. From the equation we can see that the ratio of concentrations is propor-

tional to –k1, and in the first-order reactions the reactant concentration decreases

118

exponentially gradually with a rate determined by k1. One can find the value of k1

by plotting a diagram of lnC versus of time (Fig.5.1), where tga= – k1/2,303.

To characterize the rate of the first-order re-

actions, in addition to the rate constant, the half-

life is often use. Assuming that C=Co/2, we obtain

from Eqn.5.6 2/

ln1

2/1

1

o

o

C

Ck

from which

2/1

2ln

k

Inserting the value of ln2=0,6932, we obtain

1

2/1

6932,0

k 5.7

This relation shows that the half-life of the first-order reactions is inversely propor-

tional to the rate constant, and independent of the initial concentration of reactants.

This means that in equal times the same fraction of the initial substance will react.

§ 5. Second-order reactions

The rate of such reaction depends on reactant concentration raised to the

second power or on the concentrations of two different reactants, each raised to the

first power:

212 CCkdt

dC

where C1 and C2 are two different reactants. In case C1= C2 we can write

22Ck

dt

dC or dtk

C

dC22

and after integration

consttkC

2

1

When t=0 and C=Co we obtain const=1/C and after inserting it in the upper equa-

tiom we abtain

oCtk

C

112 from which

CC

CC

tk

o

o 1

2 5.8

Fig.5.1. Determination of

the first-order rate constant

119

The unit of the second-order rate constant is C-1t-1, hence the rate of such re-

actions depends on the number of the collisions of

the reactant molecules, and the plot of 1/C versus t

gives a strait line (Fig.5.2), from slope of which one

can find a value of k2.

In the other case, when the initial concentra-

tions of reactants are not equal we denote them as a

and b, and the concentration of time t after the reac-

tion starts as x. Hence, the rate law has the following

form:

dtk xbxa

dx or xbxak

dt

dx22

The left side of equation we can write in the form

dtkxa

dx

xb

dx

ba2

1

After integration we obtain

consttkxbxaba

2lnln1

When blnalnb-a

1const 0x and 0t

After inserting of the constant meaning:

b

a

batk

xb

xa

baln

1ln

12

from which

xba

xab

batk

ln

112 5.9

Or in the other form

210

120

2010

2 ln11

CC

CC

CCtk

5.10

The half-life of the second-order reactions is equal to:

o2oo

oo

2

2/1Ck

1

C2/1C

C2/1C

k

1

5.11

Obviously that in the second-order reactions, in contrast to the first-order re-

actions, the half-life depends on the initial concentration of a reactant; τ1/2 doubles

when the reactant concentration is cut in half. Thus, it takes twice as long as the


the second-order rate con-

stant

120

reaction proceeds from 50 to 75 percent completion as from 0 to 50 percent com-

pletion.

§ 6. Third-order reactions

The rate law of the third-order reactions in case when the concentrations of

reactants are equal is:

33Ck

dt

dC

After writing in the form –dC/C3=k3dt and integrating we obtain

consttkC

322

1

When t=0 and C=C0, const=1/2 20C . Inserting the meaning of the constant we ob-

tain

22

22

32

1

CC

CC

tk

o

o 5.12

The value of the rate constant of the third-order reaction is t–1C–2.

The half-life is equal:

203

2/12

31

Ck 5.13

§ 7. Determination of the reaction’s order

For reaction rate calculation and certain kinetic equation application we need

to know the order of the reaction. There are some methods of it:

1. The method of substitution. For a certain reaction the concentrations of the

reactants is determined during the reaction (in certain intervals) and then their

meanings in the kinetic equations (separately in the first-, second-, and third order)

are put. Where we obtain the same meanings of the rate constant the reaction is that

order.

2. The graphic method. Having carried out the same determination as in the

first method one can constructs a diagram of lnC, 1/C, and 1/C2 versus t. Where

dependence is shown by straight line, the order is the same.

3. The half-life determination. For different initial concentration of the same

reactant their half-life is found out and dependence character is determined: for the

121

first-order reactions there is not any dependence, for the second-order reactions

dependence is inversely proportional, and for the third-order reactions 20C is in-

versely proportional to the half-life.

Example: It’s found for reaction 2A → A that in case of the following ini-

tial concentrations of compound A: [A]/(mmol/dm3) = 68, 60, 50, 40, 30 the fol-

lowing values of half-life time was correspondingly observed: 114, 132, 163, 198,

266. If we plot the [A] versus half-life time it shows inverse proportion, hence, it is

a second order reaction.


1. Investigation of the a-radioactive polonium isotope of mass 210 showed

that in 14 days its activity diminished by 6.85%. Determine the rate constant of its

decay and its half-life, and calculate the time it will take to decompose by 90.0%.

Solution: In treating radioactive processes the amounts of the substances in-

stead of their concentrations are used in equation 5.6. Taking the initial quantity to

be 100 %, we find in conformity with the conditions of the problem that in 14 days

93.15 % will have remained undecomposed, whence the rate constant is obtained

with the aid of equation 5.6.

00507.015.93

0.100lg

0.14

303.2k so that

1

2/1

6932,0

k = 137 days. The period of time during which the substance will un-

dergo decay by 90.0%, i.e., 10 % will be left, follows from the equation 5.6:

0.10

0.100lg

00507.0

303.2t 454 days

COMPLEX REACTIONS

Complex reactions are those consisting of one or more simple reactions re-

lated to one another in one way or another depending on the nature of their interre-

lation and on the ratio of their reaction rates. Species formed in an earlier elemen-

tary step and consumed in a later elementary step are called intermediates. They

appear in the mechanism of the reaction (that is, the elementary steps) but not in

the overall balanced equation.

122

The theory of these reactions is based on the following principle: when

several reactions occur simultaneously, each takes place independently from others

and each obeys the kinetic equations of the simple reactions. Typical forms of such

interrelations are simultaneous, consecutive, conjugate, reversible, chain and pho-

tochemical reactions

.

§ 8. Simultaneous (parallel) reactions

Simultaneous reactions are those in which a species can react in different

ways to give a variety of products. For instance, in the nitration of phenol with ni-

tric acid the nitro group may simultaneously occupy either the ortho or the para

position. We consider the simplest case that of two competing irreversible the first

order reactions of the type

k1 B

A

k2 C

Accentuate the initial concentration of species A as a, and taking part in the

reaction in time t as x, so for the products it is be x1 and x2. Hence, the rate law for

every reaction is equal:

xakdt

dxV and xak

dt

dxV 2 2

21

11

As x = x1 + x2, so dx = dx1 + dx2 and dx/dt = dx1/dt + dx2/dt

Therefore, the rate of simultaneous reactions is equal the sum of rates of

competing reactions.

So, we can write:

xakkxakxakVVV 212121 or

xakkdt

dx

21

After integration we obtain:

xa

a

tkk

ln

121 5.14

It is the first order reaction but with the sum of the two reactions rate con-

stants instead one, so we couldn’t determine the rate constant for each reaction sep-

arately. But it is possible to determine their relation from knowing the sum of the

rate constants and the relation between product amounts: dx1/dx2=k1/k2.

123

§ 9. Consecutive reactions

Consecutive reactions are those of the type

PIAkk 21

Frequently the product of one reaction becomes the reactant in the subsequent reac-

tion. This is true in multistep reaction mechanisms. In the scheme above, I is

formed from A and is called an intermediate product, which decays to P. It is the

simplest type of consecutive reactions, which occurs in two stages, each of which

is a simple unimolecular reaction. In general case the number of stages may be

more than two, and the stages may be not unimolecular, but those of more involved

type. Calculation of the rates of consecutive

reactions in general form is very complicated.

We shall point out only that if one of the stag-

es proceeds at a considerably lower rate than

all the others, the overall rate of the reaction

will be determined by the rate of this stage.

This step is called rate-determining. The

curves on Fig. 5.3 show the dependence of the

concentration of each of the substances on the

time: the decrease in concentration of the ini-

tial substance, the change with time of the

concentration of the intermediate substance,

and the growth in concentration of the prod-

uct. According to the rate law, each stage is

described by following equations:

Bk

dt

Cd

BkAkdt

Bd

Akdt

dA

B

BA

A

Consecutive reactions are very widespread. Among them the hydrolysis of

dicarboxylic acid esters, glycol esters, or saponification of diethyl oxalate by sodi-

um hydroxide, etc.

Fig.5.3. Dependence the rate of

each of stage on concentration in

consecutive reactions

124

§ 10. Coupled reactions

In this case two chemical reactions occurred in a system when a chemical

species takes part in both reactions. It could be defined as

A + B → M

A + C → N

Those reactions take place only together, since one reaction influences the

equilibrium position of the second reaction, i.e., is induced by the other. This phe-

nomenon is called chemical induction. In such case substance C is an inductor of

the first reaction, A, common to both reactions, is an actor, and substance B is an

acceptor.

The reason of coupled reactions is that the second reaction occurs with ab-

sorption of energy, which arrived from the first reaction. A different kind of reac-

tion coupling is important in biology. For example, in living organisms thermody-

namically unfavourable processes such as the synthesis of large biochemical mat-

ters (for example, amino acids, proteins, RNA, and DNA) from small molecules

occur, which involves in the processes of the transport of chemical species from the

regions of low to the regions of high chemical potential, muscle contraction per-

forms mechanical work etc. These reactions coupled with the hydrolysis of adeno-

sine triphosphate (ATP) to adenosine diphosphate (ADP) and inorganic phosphate,

has ΔG<0, which is thermodynamically favourable. Hence, the important biochem-

ical matters synthesis is possible due to energy absorption from coupled ener-

gylosting reaction. The thermodynamically unfavourable resynthesis of ATP from

ADP is proceeds by coupling with the oxidation of glucose, for which ΔG<0.

The coupled reactions are characterized by induction coefficient, which is

equal to the relation of the number of moles acceptor and inductor:

J = nacc/nind 5.15

§ 11. Reversible reactions

They are those taking place both in forward and reverse directions:

A ↔ B

The rate of a reversible reaction equals to the difference between the rates

of the forward and reverse reactions.

125

The rate of the forward reaction will diminish with a decrease in the concen-

trations of initial substances. Hence the dependence of the rate on time is shown by

above curve in Fig.5.4. If there were no reverse reaction, the curve would reach the

axis of abscissas when the initial substance was

completely consumed. The other curve depicting

the rate of the reverse reaction begins at the

origin of the coordinates and grows as the for-

ward reaction takes place. Obviously on further

reaction the two curves for the rates of the for-

ward and reverse reactions should intersect at

some point, where there rates becomes equal

(the equilibrium point).

Let this reversible reaction be of the first

order in both forward and back directions. If we

accentuate the initial concentration of species A

as a, and of species B as b, and taking part in the reaction in time t as x, then the

overall rate at any moment is equal to:

2121221121 kkxbkakxkbkxkakxbkxakdt

dx

If the right part of the equation is divided and multiplied to the 21 kk , we’ll ob-

tain

21

21

21 kkxkk

bkak

dt

dx

5.16

In equilibrium state dx/dt = 0, so:

¥

x

kk

bkak

21

21 5.17

where ¥x is the equilibrium concentration of the reactant (these reactions come to

the end in the equilibrium state with equilibrium concentrations of reactants in con-

trast to the others ending when initial reagents completely converts to the final

ones). Inserting Eqn.5.17 in the 5.16 we obtain:

kkxxdt

dx21 ¥ and after integration

xx

x

tkk

¥

¥ln1

21 5.18

Fig.5.4. Dependence of the rate

on time for direct and reverse re-

actions

126

It is the first order equation, where instead of the one rate constant there is a

sum of the two reverse processes constants, and instead of the initial concentration

of the reagent there is an equilibrium concentration.

§ 12. Chain reactions

This type of reaction contains a series of steps with participation of active

particles such as free atoms and radicals with unsaturated valences, ions, excited

molecules, etc. In chain reactions, the initial molecules react with active particles

formed a series of intermediate steps, in each of them the final product and the ac-

tive particles of the same type is formed. Regeneration of the intermediate allows

this cycle to be repeated over and over again. Thus, a small amount of active parti-

cles produces a large amount of product. Most combustions, explosions, and

polymerizations are chain reactions and usually involve free radicals as intermedi-

ates.

One of the best-understood chain reactions is that between H2 and Br2. The

overall stoichiometry is known:

H2 + Br2 → 2HBr

In spite of seeming simplicity, the observed rate law for this gas-phase reaction is

22

2/1

221

/1 BrHBrk

BrHk

dt

HBrd

which shows the difference between overall stoichiometry and real mechanism of

the reaction.

Chain reactions proceeds in three steps:

Step 1 is the initiation step, in which the formation of active particles occurs

in various ways. This may be the result of the thermal dissociation of an easily

splitted molecule (for example Br2 ↔ Br + Br), or of the impact of two energy-

reach molecules; it may also be the result of impact of a molecule with the walls of

a vessel, or, in particular, its chemical reaction with atoms or ions. The rate of the

initiation step is proportional of the concentration of species (Br2) and the intensity

of the initiator (by which the formation of active particles occurs.

In the propagation step (Step 2) the chain carriers (radicals) produced in the

initiation step attack other reactant molecules, and each attack gives rise to a new

carrier:

Cl + H2 → HCl + H

127

H + Cl2 → HCl + Cl

Cl + H2 → HCl + H

H + Cl2 → HCl + Cl etc.

This is an example of non-branching chain, when in every step one radical

attack to the other reactant molecules and as a result one radical is obtained. The

chain length (n) determined by the numbers of the elementary steps and equal

n=V/Vo, where V is the rate of the product formation, and Vo is the rate of the chain

initiation.

There are chain reactions when one active molecule may lead to the for-

mation of two or more new active molecules. So the attack results in the production

of more than one chain carrier, and as a result the branching chain is formed:

• • • O + H2O → HO + OH • • • O + H2 → HO + H

The rate of such reactions may quickly grow, the process usually ending in

an explosion.

The rate of chain developing determined by the number of collisions be-

tween reactants and radicals molecules, and the form and volume of the vessel,

where the reaction occurred.

In the termination step (Step 3) the combination of the radicals occurred and

the chain comes to the end. It proceeds by trimolecular (not bimoleculare) reaction

with participation of any neutral molecule on which the radicals excess of energy

passes on:

Cl + Cl + M → Cl2 + M

where M can be any atom or molecule. The energy released on formation of the

chemical bond becomes vibration energy of the diatomic molecule, and unless the

third body is present to carry away this energy, the molecule dissociate to the back

to atoms during its first vibration.

The chain termination may occur also in every case, when active particle

collisions with the walls of the container or with molecules of inert substances con-

tained in it. In such collisions, active particles lose their excess energy and become

inactive.

Chain reactions are widespread. For instance, a chain mechanism may lay at

the basis of many reactions for the oxidation of hydrocarbons, in particular the

128

highly important industrial processes for the synthesis of aldehydes, alcohols, ac-

ids, etc.

§ 13. Photochemical reactions

These reactions are produced by light. Absorption of a photon of light may

raise a molecule to an excited electronic state, where it will be more likely to react

than in the ground electronic state. The number of photons absorbed equals the

number of molecules making a transition to an excited electronic state.

Photochemical reactions are of tremendous biological importance. Most

plant and animal life on earth depends on photosynthesis. The process of vision

depends on photochemical reactions. Other important case of photochemical reac-

tions are the formation of ozone from O2 in the earth’s stratosphere, the formation

of photochemical smog from automobile exhausts, the reactions in photography,

and the formation of vitamin D and skin cancer by sunlight.

The initial step of a photochemical reaction is

A + hν → A*

in which the absorption of a photon occurs by a molecule to raise it to an excited

electronic state. Then this excited molecule could take part in some primary reac-

tions:

A* → B intramolecular transition

A* → 2A dissociation with two radicals formation

A* + B → AB chemical interaction with other reactant

A* + M → A + M deactivation by collision with inert molecule

A* → A + hν deactivation by fluorescence

The quantitative characteristic of these primary reactions are the primary

quantum yield which is equal to the number of excited moles, taking part in the

primary reactions, divided by the number of absorbed photons. It couldn’t be more

then 1 (varies between 0 and 1), because the number of excited moles couldn’t be

more then the number of absorbed photons.

The products generated in the primary reactions, could take part in the sec-

ondary (following) reactions. For example, the radicals formed from dissociation

of excited molecule could take part in the chain reaction. When the primary and

secondary reactions occur, the completely photochemical process is characterized

by overall quantum yield, which is equal to the number of product formed in the

secondary reaction divided by the number of absorbed photons. It varies from 0 to

106. The overall quantum yield less than 1 are due to deactivation of excited mole-

129

cules (discussed above), and it will be quite large (more than 1), when the chain

reaction occurres after formation of radicals. For instance, absorption of light by

Cl2 puts it into an excited electronic state that “immediately” dissociates into Cl

atoms (primary reaction). The Cl atoms then start a chain reaction, yielding many

HCl molecules for each Cl atom formed.

§ 14. Kinetics of heterogeneous reactions

A heterogeneous process always takes place at an interfacial boundary or in

its immediate vicinity. The rate of such a process, all the other conditions being

constant, will depend on the area and state of the surface. Heterogeneous processes

in the systems consisting of one component come to a transition of the component

from one phase to another without any change in chemical composition of the

phase. All these processes are reversible and the relation between the rates of the

forward and reverse directions is determined by the distance of the given system

from the state of equilibrium. The nearer both phases are to equilibrium, the lower

is the overall rate of the process.

Reaction at the interfacial boundaries of the system consisting of two or

more components causes composition differences in the surface and bulk layers of

a given phase that, in turn, gives rise to the process of leveling of these composi-

tions, i.e., ultimately to a leveling of the composition of the entire phase. The pro-

cess of a leveling of the composition could proceed by stirring or by diffusion.

A heterogeneous process occurred by consecutive processes of the diffusion

of the reactants towards to the reaction surface, and then the chemical reaction

takes place on the surface of interfacial boundaries. And to estimate the effectively

of possible ways of affecting the rate of a heterogeneous reaction, it is very im-

portant to know which of its stages is the slowest under the given conditions, and

hence determines the rate of the reaction as a whole. In some cases this stage is

diffusion of one of the reaction components from the bulk to the interfacial bound-

ary and accordingly is described by the law of diffusion rate. In others it is the

chemical interaction at the interface and is described by kinetic equations.

§ 15. The reaction rate dependence on temperature

The rate of most chemical reaction increases with an increase in temperature.

There is a rough rule of van’t Hoff according to which the rate of a reaction in-

130

creases about two to fourfold with each 10oC rise in temperature. If we denote the

rate constant of the reaction at certain t temperature as Kt and the rate constant of

the same reaction at the 10oC higher temperature as Kt+10, so this rule will have the

following expression:

Kt

K t 10

where γ is called the van’t Hoff temperature coefficient and equal from 2 to 4. If a

temperature arises n times 10oC, so the expression is

n

t

nt

K

K 010 5.19

where 10

12 ttn

This rule has shortcomings: it is not accurate, not applicable at higher tem-

peratures and doesn’t show all possible rate dependences on the temperature. In

spite of it the rule has a large practical application in many cases, particularly in

determination of drug stability, hence their shelf life.

In 1889, Arrhenius found experimentally that for many reactions a plot of

lnK against 1/T gives a straight line. This behavior is expressed mathematically by

the Arrhenius equation

CT

Bk ln 5.20

where k is the rate constant, B and C are characteristic constants of the given reac-

tion. Later based on many experimental data Arrhenius created the theory of active

collisions. According of this theory the rate of a reaction is proportional to the rate

of the collisions between reactant molecules. However, the relationship between

rate and molecular collision is more complicated and not all collisions lead to reac-

tions. Collisions alone do not guarantee that a reaction will take place. A collision

will be successful only if the kinetic energy of reactant molecules exceeds a mini-

mum value of the energy called the activation energy (Ea) of a reaction. The spe-

cies temporarily formed by the reactant molecules as a result of the active collision

before they form the product is called activation complex.

We must remember that this theory applicable only for simple stage of a re-

action, and not for whole complex reaction.

Now consider a simple reaction as type A → B to understand the theory. Ac-

cording to the rate law, the rate of this reaction is equal

131

Ak

dt

Bd 5.21

According to the Arrhenius theory this reaction must occurred by the activation of

the initial molecules, so must be an activating stage for product formation:

BAEA actact

Initial molecule A obtains activation energy with active A molecule (Aact) for-

mation, from which the product B is obtained. Only active molecules, i.e., mole-

cules having the necessary excess energy at the time of collision can enter into the

corresponding chemical reactions. The first stage of this process is reversible;

hence we can express it by equilibrium constant, which is equal:

A

AK act

C from which AKA Cact 5.22

We can apply the equation of isochor for equilibrium constant:

2

ln

RT

E

dT

Kd actC 5.23

The rate of the second stage is equal:

actAconstdt

Bd 5.24

or inserting Eqn.5.22

AconstK

dt

BdC 5.25

Now, comparing Eqns.5.21 and 5.25 we can equate their right sides:

CC constKk or AconstKAk 5.26

After taking their logarithm and differentiate by temperature we obtain:

dT

Kd

dT

kd Clnln 5.27

Comparing this with Eqn.5.23 we obtain:

dT

kd ln2RT

Eact 5.28

This is the differential form of Arrhenius equation. After integration under

condition that activation energy is constant the Arrhenius equation is obtained,

which gives relation between rate constant and temperature:

CRT

Ek act ln 5.29

132

If we compare this equation with Eqn.5.20 we see that constant B includes

the meaning of the activation energy.

This equation can be written in the form

RTEAek / 5.30

where the exponential factor RT/Ee can be interpreted as the fraction of collisions

that have enough kinetic energy to lead the reaction. The pre-exponential factor A

is a measure of the rate at which collisions occur irrespective of their energy (all

collisions are successful) and called the frequency factor. So, it is clear, that a rate

of a reaction depends on the number of active molecules in the system which one

can calculate from the Boltzmann law:

RTEoact eNN / 5.31

where No is the number of all molecules, e the base of the natural logarithm scale.

Thus, the central conception of the Arrhenius’s theory is the conception of

the activation energy. The activation energy is the excess energy that a molecule

must have at the time of collision to be able to enter into the given chemical reac-

tion. The role of activation energy for reaction rate is very important. We can think

of activation energy as a barrier that prevents less energetic molecules from react-

ing. If elements of a physicochemical system have sufficient energy to overcome

the barrier, reaction will occur. Normally, only a small fraction of the colliding

molecules have enough kinetic energy to exceed the activation energy. These mol-

ecules can therefore take part in the reaction. The increase in the rate with tempera-

ture can now be explained: since more high-energy molecules are present at the

higher temperature, the rate of product formation is also greater at the higher tem-

perature.

The fact that Ea is given by the slope of the plot of lnk against 1/T (from Ar-

rhenius equation) means that the higher the activation energy, the stronger the tem-

perature dependence of the rate constant. If a reaction has zero activation energy,

its rate is independent on temperature. In some cases activation energy is negative,

which indicates that the rate decreases as the temperature is raised. We shall see

that such behavior is a signal that the reaction has a complex mechanism.

Note from equation 5.30 that low activation energy means a fast reaction

and high activation energy means a slow reaction.

To understand physical meaning of the activation energy in detail let us

consider course of a reaction A → B on the Fig.5.5. In the figure the energy of the

molecular system is plotted on the axis of ordinates and the course of the reaction

133

on the axis of abscissas. If the forward reaction (transition from reactants to prod-

ucts) is exothermal, the overall energy of the products is less than that of the reac-

tants, i.e., as a result of the reaction the system passes to a lower energy level. But

this transition must pass through a sort of

energy barrier, or transition state, which is

the activation energy of this reaction. To

understand this process, activated complex

theory is applied. On Fig.5.5 it is shown in

general how the potential energy of the reac-

tants changes in the course of a bimolecula-

re elementary reaction. Initially, only reac-

tants A and B are present. As the reaction

event proceeds, A and B come into contact,

distort, and begin to exchange or discard

atoms. The potential energy rises to a max-

imum, and the cluster of atoms that corre-

sponds to the region close to the maximum

is called the activated complex. After the maximum, the potential energy falls as

the atoms rearrange in the cluster, and it reaches a value characteristic of the prod-

uct. The climax of the reaction is at the peak of the potential energy. Here two reac-

tant molecules have come to such a degree of closeness and distortion that a small

further distortion will send them in the direction of the products. This crucial con-

figuration is called the transition state of the reaction.

§ 16. Calculation of activation energy

Consider two ways of such calculation:

1. We can plot the slope of lnk against 1/T (from Eqn.5.29) and determine

the value of activation energy by angle of curve

(Fig.5.6):

RtgaE

R

Etga

a

a

3,2

3,2

2. If we write Eqn.5.29 for two temperatures

Fig.5.6. Determination of the

activation energy

Fig.5.5. The potential energy of

the reactants changes in the course

of a bimoleculare reaction

134

CRT

Ek

CRT

Ek

a

a

2

2

1

1

ln

ln

and then subtract the first equation from the second, we obtain

212

1 11ln

TTR

E

k

k a 5.31

This equation give abilities to calculate the activation energy if we know two

rate constants at two temperatures, or to calculate the unknown rate constant at eve-

ry temperature if we know the rate constant at a certain temperature and the activa-

tion energy.


1. Calculate Ea for a reaction the rate constant of which at room temperature

is doubled by a 10oC increase in T.

Solution: Equation 5.31 gives: Ea = RT1T2 (ΔT)–1 ln (k/k1) = (1.987 cal mol–1

K–1)(298 K)(308 K)(10 K)–1 ln2 = 13 kcal/mol = 53 kJ/mol

2. In the first-order reaction, the half-life at 323 K is equal to 100 minutes

and at 353 K is equal to 15 minutes. Calculate the van’t Hoff temperature coeffi-

cient.

Solution: According the equation 5.7 K323 = 0.69/100 = 0.0069 and K353 =

0.69/15 = 0.046. Using the van’t Hoff equation n

t

nt

K

K 010 where

10

12 ttn

=

3 give us γ3 = 0.046/0.0069 = 6.6. Hence γ = 1.88

3. The rate constant of the first-order reaction is equal to 2.4 min–1 at 30oC

and 4.8 min–1 at 40oC. At what temperature the value of the rate constant will be

0.6 min–1?

Solution: Using the van’t Hoff equation allows to know the temperature co-

efficient: γ = K40/K30 = 4.8/2.4 = 2. Applying the van’t Hoff equation again we get:

2n = K40/Kx = 4.8/0.6 = 8. Therefore, n = 3 and from 3 = (40–x)/10 we obtain that x

= 10oC.

CATALYSIS

Basic principles

135

Catalysis is the term given to the phenomenon in which certain substances,

called catalysts, increase the rate of a reaction while they themselves, although

participating in the reaction, in the end remain chemically unchanged. This effect

may be very strong, catalysts be able to change the rate of a reaction in million

times or even more. Catalysts display high specificity. This means that they are of-

ten selectively accelerate only one of a number of possible reactions of the reac-

tants.

There are also inhibitors (or negative catalysts), which decreases the rate of

a reaction. The retarding effect of negative catalysts is frequently due to their poi-

soning, i.e., lowering of the activity of the positive catalysts.

The mechanism of catalytic effect is the following: the catalyst forms an in-

termediate substance the catalyst with the reactants, facilitating the formation of the

products by lowering the activation energy.

Catalysts do not affect the equilibrium of a given reaction, but only facilitate

its establishment. Although a catalyst cannot change the equilibrium constant, a

homogeneous catalyst can change the equilibrium composition of a system.

Depending on a reagent and catalyst aggregate states catalysis are divided in-

to homogeneous, heterogeneous and enzyme (or microheterogeneous).

§ 17. Homogeneous catalysis

In homogeneous catalysis a catalyst is in the same phase as the reaction mix-

ture. A well-known example of homogeneous catalysis is the oxidation of sulfur

dioxide with the aid of nitrogen oxides as catalyst. The mechanism of homogene-

ous catalysis is explained by activated complex theory. Let us consider reaction A

+ B → C + D. Without a catalyst, this reaction, according of transition state, occurs

by the following scheme:

A + B → AB# → C + D

where AB# is the activated complex.

In presence of a catalyst process proceeds as following:

A+K k1

k2

AK

AK + B 3kABK#

ABK# 4k C + D + K

136

In the first stage the reactant A interacts with the catalyst (this stage is reversible),

then the intermediate product AK with another reactant B is formed an activated

complex, which is destroyed by product formation and catalyst regeneration. Acti-

vated complex ABK# has a lower activation energy then AB# (due to this the rate of

the reaction increases) what we can see on the

Fig.5.7. In how many times changes the rate

of the reaction with participation of the cata-

lyst in contrast to catalytic reaction is changed

we can calculate by the equation

RTE

uncat

cat ek

k / 5.32

As we can see it depends on the difference

between activation energies of the two reac-

tions – with and without catalyst.

Now let us deduce a kinetic equation for

homogeneous catalysis. According to the reaction scheme above the rate of the

product C accumulation is equal to:

ABKkdt

Cd #4 5.33

Activated complex ABK # is the intermediate species, the concentration of

which is very small and unknown, so we need in another approximation to express

it by initial reactants. In general multistep reaction mechanism usually involves one

or more intermediate species that do not appear in the overall equation. These in-

termediates are very reactive and therefore do not accumulate to any significant

extent during the reaction. Therefore it is frequently a good approximation to take

their rate equal to zero for each reactive intermediate. This is the steady-state (or

stationary state) approximation. The steady-state approximation assumes that (af-

ter the induction period) the rate of the formation of an intermediate essentially

equals to its rate of destruction, so as to keep it at a near constant steady-state con-

centration. According this approximation the rate of the third stage (where com-

plex ABK #appears) is equal to the rate of the forth stage (where com-

plex ABK #destroys):

#43 ABKkBAKk

from which

Fig.5.7. The difference between

value of activation energy in reac-

tion with and without catalyst

E

E

reaction coordinate

A+B

C+D

ABK¹

AB¹

137

ABK # =

4

3

k

BAKk 5.34

AK also is the intermediate and for it we apply the steady-state approximation;

the rate of the first stage is equal the sum of the rates of the second and the third

stages:

BAKkAKkKAk 321

from which

Bkk

KAkAK

32

1

5.35

Putting Eqn.5.35 in the 5.34, and then in the 5.33, we obtain

Bkk

KBAkk

dt

Cd

32

31

5.36

which is the kinetic equation of the homogeneous catalysis. It is related to the rate

of the process with initial reactants and catalyst concentrations.

We can do simplification of this equation for two extreme cases;

If k2>>k3 the intermediate AK almost entirely transformed into the initial

species in this case it is called the Arhenius’s intermediate and the equation comes

to the following form:

BAK

k

kk

dt

Cd

2

31 5.37

Because all the members of the expression Kk

kk

2

31 are constant so one can re-

place it by one constant, k, and write:

BAk

dt

Cd 5.38

which is the second order relatively to the initial species.

If k2<< k3 the intermediate AK is completely transformed to the products and

called van’t Hoff’s intermediate and the equation has the following form:

AKk

dt

Cd1 5.39

which is the first order relatively to the initial species A, and zero order with re-

spect to the initial species B.

§ 18. Enzyme catalysis

138

Most of the reactions that occur in living organisms are catalyzed by mole-

cules called enzymes (or ferments). They are biological catalysts consisting of pro-

tein molecules with molar mass from thousands to millions of grams. An enzyme is

highly specific in its action; many enzymes catalyze only the conversation of a par-

ticular reactant to a particular product; other enzymes catalyze only a certain class

of reactions (for example, ester hydrolysis). Enzymes speed up reaction rates very

substantially, and in their absence most biochemical reactions occur at negligible

rates.

The molecule an enzyme acts on is called the substrate. The substrate binds

to a specific active site on the enzyme to form an enzyme-substrate complex.

While binding to the enzyme, the substrate is con-

verted to the product, which is then released from

the enzyme. Some physiological poisons act by

binding to the active site of an enzyme, thereby

blocking (or inhibiting) the action of the enzyme.

The structure of an inhibitor may resemble the struc-

ture of an enzyme’s substrate.

Now we consider the simplest mechanism of

an enzyme catalysis, which is

E + S ES → E + P

where E is the free enzyme, S is the substrate, ES is

the enzyme-substrate complex and P is the product.

k1, k–1 and k2 are velocity constants for the corre-

sponding reactions. The enzyme is consumed in step

1 and regenerated in step 2. This process is illustrated in Fig.5.8.

The initial rate of the product formation is

ESk

dt

PdV 20 5.40

Hence ES is the intermediate and its concentration is much less than of S, the

steady-state approximation can be used for ES (i.e., 0/ dtESd ):

ESkESkSEk 211

from which

21

1

kk

SEkES

Fig.5.8. The mechanism of

enzyme catalysis

139

If 0E is the initial enzyme concentration than ESEE 0 . Replacing it, we

obtain

211

01

kkSk

SEkES

Setting it in the 5.40 gives

211

0120

kkSk

SEkkV

Dividing numerator and denominator on the k1 gives

1

21

020

k

kkS

SEkV

Replacing one constant Km (Michaelis constant) instead of 1

21

k

kk we obtain

mKS

SEkV

02

0 5.41

which is called the Michaelis-Menten equation for enzyme catalysis.

When so little S is present that mKS , the rate of the process becomes

first order relatively to the substrate. When mKS , the rate is zero order.

Figure 5.9 plots V0 against S for fixed 0E . In the limit of high concentra-

tion of the substrate, virtually all the entire enzyme is bound to substrate in the

form of the ES complex, and the rate becomes maximum that is independent on the

substrate concentration. In this case ESE 0 and the equation 5.40 has the fol-

lowing form

02 EkVm 5.42

The quantity k2 is called the turnover num-

ber of the enzyme; it is the maximum num-

ber of moles of the product produced in a

unit time by 1 mole of the enzyme.

If we put the quantity 20 / kVE m

from 5.42 in the 5.41 we obtain

SK

SVV

m

m

0 5.43

One can find the value of the Vm and

Km by Fig.5.9. Vm is suitable of the limit of high concentration of the substrate. To


value of Km and Vm

140

find out Km we take the reaction velocity is one-half maximal velocity V0=Vm/2 and

put it in the 5.43:

SK

SVV

m

mm

2 from which SKm

Practically it is difficult to rich the max-

imal velocity and the value of the Vm and Km

one can determine and by plotting a diagram

1/V0 versus 1/S from the inverse of equation

5.43 (Figure 5.10).

mSm

m

VV

K

V

11

0

§ 19. Heterogeneous catalysis

Heterogeneous catalytic reactions are the reactions in which the catalyst is

in a separate phase, and the reaction takes place on its surface. This means that

the nature of the surface and its size, the chemical composition of the surface layer,

its state and structure should have a direct bearing on the activity of the catalyst.

The majority of industrial chemical reactions are run in the presence of solid

catalysts. Most products of the chemical and related industries are produced by

heterogeneous (usually gaseous) catalytic reactions. Examples are the Fe-catalyzed

synthesis of NH3 from N2 and H2; the hydrogenation of oils; the oxidation of CO

on a Pt or Pd ets.

The surface of a solid is not smooth and uniform and only some areas on the

surface possess catalytic activity. They called active centers of a catalyst. The ac-

tivity of a catalyst may increase and its lifetime extended by the addition of sub-

stances called promotors. Small amounts of certain substances that bound strongly

to the catalyst can inactivate (or poison) it.

The catalytic process catalyzed by solid catalysts in fluid-phase reactions

consists of the following steps:

1. Diffusion of the reactant molecules to the solid surface.

2. Sorption of at least one reactant species on the surface.


value of Km and Vm

141

3. Chemical reaction between molecules adsorbed on adjacent sites of

the surface.

4. Desorption of the products from the surface.

5. Diffusion of products into the bulk fluid.

In many cases, one of these steps is much slower than all the others, and only

the rate of the slowest step needs to be considered (limiting-stage rate). By this rea-

son overall rate of heterogeneous reaction could be calculated by equation of diffu-

sion rate (if the slowest is the first step) or by the rate of adsorption (if the slowest

is the second step) or by kinetic equation (if the slowest is the third step). Found by

this way (by experimental methods) the rate of the reaction is called seeming rate.

The rate suitable to the kinetic equation of the certain reaction (step 3) is called

true rate of a reaction. In case when the slowest is the third step the seeming and

the true rates coincides to each other. In other cases they are different.

We shall consider mainly solid-catalyzed reactions of gases in which step 3

is much slower then all the other steps. Since we are assuming the adsorption and

desorption rates to be much greater than the chemical reaction rate for each species,

adsorption-desorption equilibrium is maintained for each species during the reac-

tion (see Chapter 6). We can therefore use the Langmuir isotherm, which is derived

by equating the adsorption and desorption rates for a given species.

Step 3 may consist of more than one elementary chemical reaction. Since the

detailed mechanisms of the surface reaction are usually unknown, we adopt the

simplifying assumption of taking step 3 to consist of a single gaseous unimolecular

reaction. Then the reaction rate per unit surface area will be proportional to the

number of adsorbed molecules of a gas per unit surface area, and this in turn will

be proportional to θ – the fraction of adsorption sites occupied by reactant mole-

cules:

V = k θ

where k is a rate constant. According the Langmuir isotherm

KP

KP

!

where K is the adsorption constant, and P is the pressure of the gas (reactant).

Hence

KP

KPkV

1 5.44

At low pressure the reaction is the first order:

kKPV

142

At high pressure it is the zero order:

V = k

At high pressure the surface is fully covered with reactant, so an increase in pres-

sure has no effect on the rate.

The theories of heterogeneous catalysis throughout all the stages of their de-

velopment issue from the concept that the reaction occurs in some way through the

formation of intermediate surface compounds. There are some theories explaining

its mechanism. We consider only two – more assumed.

The multiplet theory of heterogeneous catalysis is based on the principle of

geometrical correspondence between the arrangement of the atoms on the catalyst

surface and the atoms of the reactant molecules, and of correspondence in bond

energies. The theory therefore dials not with the reaction of the molecule as a

whole with the catalyst surface, but consider the reaction of separate atoms or

groups of the reactant molecules with atoms or groups of the surface. Multiplets are

the names given to associations on the catalyst surface of several atoms or ions

regularly arranged in correspondence with the structure of the crystal lattice of the

catalyst (active centers). Catalytic activity appears when the arrangement of the

atoms or ions is in geometric conformity with the arrangement of the atoms in the

molecules of the reactants. Besides, there must be the energetic correspondence of

the bonds between the atoms in the reactant molecules and the bonds formed by

these atoms with the catalyst.

Among other theories of heterogeneous catalysis is the electron-chemical

theory. In recent studies the mechanism underlying the action of certain semicon-

ductor and metal catalysts is approached from the standpoint of possible electron

transitions between various energy levels on the catalyst surface corresponding to

different states of the catalyst.

Applications of reaction kinetics abound. Thus, the productivity of the ap-

paratus and hence the amount of final product in a chemical process is intimately

connected with the reaction rate. It is therefore essential to know the rate at which a

given reaction takes place under the given conditions and what changes must be

brought about for the reaction to proceed at the desired speed. Frequently, in organ-

ic preparative reactions, several possible competing reactions can occur, and the

relative rates of these reactions usually influence the yield of each product. What

happens to pollutants released to the atmosphere can be understood only by kinet-

ics of atmospheric reactions. Reaction rates are fundamental to the functioning of

143

living organisms. Biological catalysts (enzymes) control the functioning of an or-

ganism by selectively speeding up certain reactions. From a theoretical point of

view, the importance of kinetics in chemistry is its aid in the elucidation of many

important aspects of reactions and in obtaining a deeper insight into their mecha-

nism.

A practical application of a rate law is the following: once we know the law

and the value of the rate constant, we can predict the rate of the reaction from the

composition of the mixture. Moreover, knowing the rate law, we can go on to pre-

dict the composition of the reaction mixture at a later stage of the reaction. Moreo-

ver, a rate law is a guide to the mechanism of the reaction, for any proposed mech-

anism must be consistent with the observed rate law.

The application of chemical kinetics in pharmacy is grate; there is a branch of ki-

netics – pharmacokinetics, which dials to the rates of the processes, occurred in the

organism under influence of drugs, rates of drug distribution between different or-

gans, rates of drug activity loss, time and conditions of drug storing (their shelf-

life), etc.

Review questions:

1. What does the subject kinetics study?

2. Give the formulation of the rate of a chemical reaction and write its ex-

pression. In which cases do we put the sign “+” or ”–“ in front of this expression?

3. What factors does a reaction rate depend on?

4. How is the rate dependence on concentration expressed? Write an exam-

ple for the reaction 2NO2 + O2 = 2NO3

5. What is the physical meaning of the rate constant and what factors does

it depends on?

6. Define the following terms” molecularity, order, half-life of the reaction.

Write an example of bimolecular second-order simple reaction.

7. What could be the numbers of the molecularity and reaction order? Why

couldn’t the molecularity be more than three?

8. What is the difference between molecularity and order of a reaction and

in what cases could they coincide?

9. Can the order from the stoichiometric equation of the chemical reaction

be found? In what cases could it be done?

10. What dependence expresses the kinetic equation of the chemical reac-

tion?

144

11. What does it mean: kinetic equation of the first order or of the second

order?

12. What are the units of the rate constants of the first order and of the sec-

ond order reactions? What does it mean?

13. How does the half-life of the first order and of the second order reactions

depends on the initial concentration of the reactant?

14. What quantities do we have to obtain experimentally for the reaction or-

der determination by method of (a) substitution, (b) graphic and (c) half-life?

15. What kinds of reactions are called complex?

16. How one can determine the rate of (a) simultaneous, (b) consecutive, (c)

reversible reactions?

17. Why are coupled reactions also called the reactions of chemical induc-

tion? What is the reason of such reactions? What does the induction coefficient

show?

18. Define the chain reactions. What particles are called active radicals?

Which steps do the chain reactions are included?

19. Write the examples of branching and non-branching reactions.

20. By what reasons the chain reaction could come to the end?

21. What reactions are called photochemical? What steps do they include?

22. What steps characterize primary and secondary quantum yield?

23. Why primary quantum yield couldn’t be more than a unit and secondary

quantum yield could be both more and less than a unit?

24. How is the kinetics of heterogeneous reactions distinguished from that of

the homogeneous and why? How to determine the rate of such reactions?

25. How the rate of a reaction depends on the temperature? Formulate the

rule of van’t Hoff.

26. Explain the theory of active collisions of a reaction and write the Arrhe-

nius equation.

27. What is the activation complex and what does the rate of the reaction by

Arrenius depend on?

28. Formulate the term the activation energy. Is the rate of a reaction high or

low if the activation energy is high? Why?

29. Give the methods of activation energy calculation.

30. Define the terms “catalysis” and “catalyst”. What is the mechanism of

catalytic effect?

31. How does catalyst influence on the equilibrium state?

145

32. Write the stages of homogeneous catalysis and its kinetic equation.

33. What is the meaning of the steady-state approximation and what sub-

stances is it applied for?

34. What is the specificity of enzyme catalysis? What substances do we call

enzyme (ferment) and substrate?

35. Write the equation of Michaelis-Menten. What is the maximal rate of the

enzyme catalysis reaction? How can one find the value of the maximal rate and

Michaelis constant?

36. How does heterogeneous catalysis reaction occur and through what stag-

es. What is the limiting-stage rate?

37. Why are there two kinds of the reaction rate here – seeming and true?

Explain their meaning. When could they coincide?

146

PART TWO

SURFACES AND COLLOIDS

INTRODUCTION

In 1915 Wolfgang Ostwald described the subject matter of colloid and sur-

face science as a “world of neglected dimensions”. The reason for such a descrip-

tion stemmed from the unique nature of interfaces and related colloidal phenomena

– they could not be readily interpreted based on “classical” atomoc or solution the-

ories. There is the world referred to by Ostwald – the region between two phases

(interphases) involves colloids – represents a bridge not only between chemical and

physical phases, but also plays a vital but often unrecognized role in other areas of

chemistry, physics, biology, medicine, engineering, and other disciplines.

It would be practically impossible to list all of the human activities that in-

volve surface and colloidal phenomena, but a few examples have been listed: soaps

and detergents, herbicides and pesticides, foods (ice cream, butter, mayonnaise,

etc.), pharmaceuticals, cosmetics and topical ointments, inks, paints. They take part

in many industrial and physiological processes such as blood transport, respiration,

arteriosclerosis, cell membranes permeability, enzymes activity, adhesion, wetting,

etc. One can see that we and our world simply would not function or even exist in

absence of interfacial and colloidal phenomena.

147

CHAPTER 6

SURFACE PHENOMENA

In a system composed of two or more phases, molecules at or very near to

the region of phase contact have different molecular environment than molecules in

the bulk of either of phases. Molecules in the inner layers of a substance experience

the same average force of attraction in all the directions by surrounding molecules,

while molecules in the surface layer experience different forces of attraction from

inside the substance and from the side bordering with the surface layer of the me-

dium (Fig.6.1). For this reason the properties of the surface layer differ somewhat

from those of the substance in the bulk. In turn, the properties of the surface layer

may affect other properties of the substance. Thus, at a liquid-air interface the mol-

ecules of liquid in the surface layer

are attracted more strongly by the

underlying molecules than by the

molecules of the gas, hence total

force acting on the surface molecule

is directed down and called inner

pressure (P). For this reason, sys-

tems tend to assume a configuration

of minimum surface area. Thus an isolated drop of liquid is spherical, since a

sphere is the shape with a minimum ratio of surface area to volume.

This effect is slight if the substance has a relatively small surface area. But

as the degree of dispersion and due to it the summery surface area increases, the

surface properties become more expressed and are quite considerable for a highly

developed surface (for example, colloidal systems).

Surface effects are of tremendous industrial and biological importance.

Many reactions occur most readily on the surfaces of catalysts, and heterogeneous

catalysis is important in the synthesis of industrial chemicals. Such subjects as lu-

brication, corrosion, adhesion, detergency, and reactions in electrochemical cells

involve surface effects. Many industrial products, drugs, biological systems are

colloids. The problem of how biological cell membranes function belongs to sur-

face science.

Fig.6.1. Different forces of attraction on a

surface and inside the substance

A

B

P

liquid

air

148

§ 1. Surface tension

Due to the unique environment of the molecules in the surface layer and

presence of the inner pressure positive work, it is required to increase the area of an

interface between two phases. Therefore, the more interphase area the more work

needed for its formation, and as it should be the more surface Gibbs free energy:

dG = σ dS 6.1

where σ is the constant of proportionality and called the surface tension. From

equation it’s equal to:

dS

dG 6.2

hence the surface tension is the Gibbs free energy of a unit of surface, or it is the

maximum useful work done for the formation of a unit of surface area. It could be

defined also as force acting parallel to the surface and perpendicular to a line of

unit length anywhere in the surface. The units of surface tension are mN m–1.

The surface tension depends on the molar volume, the polarity of the mole-

cules and the nature of the substances. The surface tension of water is much higher

than of most common liquids. Molten salts and metals have still higher values. As

for polarity, there is the rule of Rebinder according to which: the more polarity dif-

ference of two phases the more surface tension of their boundary surface. There is

linear dependence between surface tension and temperature for many substances,

particularly for liquids, because when a substance is heated it usually expands, and

the forces of attraction between the molecules in both the inner and surface layers

weaken. Hence the surface tension diminishes with a rise in temperature up to the

critical or absolute boiling point, the temperature at which the surface tension be-

comes zero. Above it a substance can no longer remain in the liquid state.

§ 2. Sorption

We have observed above that systems with a large surface area have a large

surface free energy. From thermodynamics we know that all systems aspire to the

state of the minimum free energy by spontaneous processes taking place in its.

Hence, according to the equation 6.1 this aspiring for surface free energy could be

realized by decreasing of the surface area or by decreasing of the surface tension.

Decreasing of the surface area becomes apparent in tend of liquids to assume a

149

configuration of minimum surface area, or in tend of colloid particles join in more

big aggregates, etc.

Tend of the surface tension to decrease becomes apparent in that addition of

substances having much lower surface tension in the pure state leads to there ac-

cumulation on the surface area and to a sharp depression of the surface tension.

This is due to the circumstance that a substance, which lowers the surface tension,

has a greater concentration in the surface layer than in the bulk of the substance.

The difference in concentration of a component in the surface layer and in the bulk

of the substance is called adsorption. In other words adsorption is the enrichment

of a component with a lower surface tension in the interphase region compared

with a bulk region, or it is attachment of such particles to a surface. The process

of adsorption is one of the principal ways in which high-energy interfaces can be

altered to lower the overall energy of a system.

The substance that adsorbs is called the adsorbate and the underlaying mate-

rial on whose surface adsorption occurs is called the adsorbent. “Adsorption”

should be clearly differentiated from “absorption”, in which physical penetration of

one phase (adsorbate) is involved (for instance, gas absorption by liquid). With ad-

sorption process simultaneously the reverse of it – desorption occurs and in the end

the equilibrium is stated.

The quantitative characteristic of adsorption is specific adsorption, Г (capital

gamma), which shows moles of adsorbate (x) adsorbed per gram (or unit surface),

of adsorbent:

Г = x/m or Г = x/S 6.3

Adsorption is classified into physical adsorption and chemical adsorption;

the dividing line between these two tipe is not always sharp. In physical adsorption

the molecules of the adsorbate are held to the surface by relatively weak intermo-

lecular van der Waals forces. In chemosorption, a chemical reaction occurs at the

surface, and the adsorbate is held to the surface by relatively strong chemical

bonds. Physical adsorption is nonspecific. The distance between the surface and the

closest adsorbate atom is shorter for chemosorption than for physysorption. The

enthalpy changes for chemosorption are usually substantially greater in magnitude

than those for physical adsorption. The principal test for distinguishing chemosorp-

tion from physysorption used to be the enthalpy of adsorption. Values less negative

than –25 kJ mol–1 were taken to signify physysorption, and values more negative

than –40 kJ mol–1 were taken to signify chemosorption.

150

There are several types of interfaces that are of great practical importance

and that will be discussed in turn.

§ 3. Adsorption on the liquid surface

The surface properties of solutions differ from those of pure liquids first and

foremost in that the composition of the surface layer of a solution differs to a cer-

tain extent from that of the inner layers. The surface tension of a solution can great-

ly depend on the composition of the surface layer, and in the surface layer sponta-

neously increases those of the component which lowers the surface tension and,

hence, reduces the total Gibbs energy of the system.

The curves showing how the surface tension of the liquid-gas or liquid-liquid

interface depends on concentration and nature of the solute can be classified into

three types (Fig.6.2). Type 1 solutes produce a small rate of increase in σ with in-

creasing concentration; examples include most inorganic salt and other good solu-

ble substances, surface tension of which is higher than solution surface tension.

Hence there are no energetical abilities for ad-

sorption of these substances. Type 2 solutes

have the same surface tension with solution and

therefore do not change the surface tension of

the solution (for example, sugar). For type 3 so-

lutes surface tension shows a very rapid de-

crease as the concentration is increased, because

the outside tension of these substances is much

less than the surface tension of the solution. So-

lutes of this type will preferentially adsorb at

interfaces to lower the energy of the system. A solute that significantly lowers the

surface tension is said to be a surface-active agent or surfuctant. The presence of

the surfactant decreases the work required to increase the interfacial area resulting

in a decrease in interfacial tension. Usually they are medium-chain length organic

acids or salts consisting of the hydrophilic polar part

of the molecule (the “head’) and the hydrophobic

(straight line or ‘tail”) hydrocarbon part (Fig.6.3). In

water, the hydrophobic group may be, for example, a

hydrocarbon, fluorocarbon chain of sufficient length to

produce the desired solubility characteristics. The hydrophilic group will be ionic

Fig.6.2. Dependence of the sur-

face tension on a concentration

of a solution

Fig.6.3.Constraction of a

surfactant molecule

151

or highly polar, so that it can act as a solubilizing functionality. Molecules with

both hydrophilic and hydrophobic groups are called amphipathic or amphiphilic.

Due to such construction is explained an adsorption ability of these molecules; the

hydrophilic polar part of the mole-

cule dropped into the polar water

and hydrophobic hydrocarbon part

pushes away from the surface and

molecules gathered on the liquid

surface (Fig.6.4.).

Such asymmetric molecules are oriented on the surface in certain order; the

polar part tends to a polar solvent, and the nonpolar part tends to a nonpolar sol-

vent. For example, if the molecules contain highly polar hydroxyl or carboxyl

groups bound to a large nonpolar hydrocarbon radical (high molecular fatty alco-

hols or acids), then they, depending on the

boundary surface nature, are oriented

there as shows at Fig.6.5.

As the concentration of the mole-

cules in the surface layer increases and the

distance between them becomes suffi-

ciently small, they may form a monomo-

lecular, regularly oriented layer on the

surface.

Adsorption in the surface layer of a solution is described quantitatively by

Gibb’s equation, obtained by thermodynamic treatment. If we denote the concen-

tration of the solute in the solution by C and the degree of adsorption in the surface

layer by Г then Gibbs’s equation can be written in the form:

RT

CГ

TdC

d

6.4

The quantity TdC

d

(change in surface tension of the solution with change

of solute concentration) is usually taken as the measure of surface activity. This

equation shows that when σ decreases with an increase in concentration, i.e.,

TdC

d

<0 (for surfactants), then Г>0, and the concentration of the solute is greater

in the surface layer than in the main body of the solution (positive adsorption). In

Fig.6.5.Different orientation of the

surfactants on the boundary surfaces

of different nature

coal

water

silicagel

benzol

air

water

Fig.6.4. Orientation of the surfactant mole-

cules on the air-water boundary surface

152

the opposite case, i.e., when TdC

d

>0, then Г <0, and the concentration is lower

in the surface layer than in the bulk of the solution (negative adsorption).

Example: Calculate the value of specific adsorption for the surfactant solu-

tion of 0.028 mol/dm3 concentration if at 293 K the 0.056 mol/dm3 change in con-

centration leads to the – 4.3310–2 Nm–1 change in surface tension.

Solution: Use the equation 6.4: Г 22

1009.0056.0

1033.4

293082.0

028.0

§ 4. Application of surfactants in medicine and pharmacy

The applications of surfactants in science and industry are legion, ranging

from primary processes such as the recovery and purification of row materials in

the mining and petroleum industries, to enhancing the quality of finished products

such as paints, cosmetics, pharmaceuticals, and foods. On a more personal level,

the functioning of our most important life processes – cell structure, respiration,

blood flow, muscle function, and many disease processes are based on interfacial

phenomena. Small alterations in the functioning of those phenomena can literally

be the difference between life and death. Here there are only a few of application

areas:

– They are used to obtain emulsions because of decreasing water/oil in-

terphase tension.

– Surfactants adsorption on hydrophobic surfaces of insoluble substances

reverses their solubility and adjuvant to obtain suspensions.

– Surfactants are used for stabilization of different disperse systems.

– Because of surfactants abilities to form surface membranes they are

used as drug covering.

– Some of drugs are surfactants by themselves.

§ 5. Adsorption of gases on solids

Solids have an ability to adsorb molecules, atoms or ions from a surrounding

medium on their surface to a certain degree. This adsorption is a spontaneous pro-

153

cess when it leads to a decrease in Gibbs free energy of the surface. In other words,

those substances are adsorbed on the surface that reduces the surface tension with

respect to the surrounding medium.

A solid surface will almost certainly be heterogeneous in terms of the distri-

bution of its excess surface energy, meaning that adsorption will not be a uniform

process, while liquids it is assumed to be so. The surface of a solid, especially of a

good adsorbent, is not smooth, but has numerous submicroscopic elevations and

depressions. The saturation degree of the valence forces is not the same on various

parts of the surface and, hence, their ability to interact with the atoms and mole-

cules of the surrounding depends on their location. For this reason adsorption oc-

curs not on the whole surface, but on its active parts, which is called active centers

of the surface.

For the given adsorbent and gas, all other conditions being equal, the amount

of gas adsorbed is in direct proportion to the adsorbing surface area. Hence, to ob-

tain a large adsorption effect, one must have the largest possible surface of adsor-

bent. The best adsorbents are, therefore, materials with a large surface area, such as

highly porous or finely divided (highly dispersed) materials. The first place among

the adsorbents used in practice belongs to the different kinds of specially prepared

carbons. For instance, in one gram of a highly adsorbing carbon (activated carbon)

the inner surface of the pores has a surface area of 400-900 m2. Besides carbon

other substances are used as adsorbents such as the gel of silicic acid (silica gel),

alumina, kaolin, and certain alumosilicates.

The amount of gas adsorbed by a given

amount of adsorbent depends on the nature and on

the pressure of the gas. All the other conditions

being equal, the effect of pressure on the amount

of gas adsorbed can be depicted by the curve,

which is called adsorption isotherm. The nature of

the adsorption isotherm mainly depends on the

character of the adsorbent and gas and exist a wide

variety of isotherm types. One of them we consid-

er on Fig.6.6. As a rule, the amount of gas adsorbed increases with pressure in-

crease. The influence of pressure, however, varies on different parts of the adsorp-

tion isotherm. It is especially strong in the low-pressure region (section І of the iso-

therm), where the amount of adsorbed gas is directly proportional to the pressure.

With a further increase in pressure, the amount of adsorbed gas also increases, but

Fig.6.6. Adsorption iso-

therm for case of gas-solid

adsorption

154

more slowly (section ІІ), while starting. With section ІІІ the curve approaches a

straight line parallel to the axis of abscisses. This corresponds to gradual saturation

of the adsorbent surface. When saturation is reached, a further increase in pressure

has practically no effect on the amount of adsorbed gas, and adsorption in this case

is called limit adsorption (Г∞). Its value is constant in every certain condition.

One of the analytical expressions for the adsorption isotherm is the empiri-

cal Freundlich equation

nKPm

x1

6.5

where P is equilibrium pressure of the gas, K and n are empirical parameters, con-

stant for given adsorbent and gas at constant temperature. The Freundlich equation

is generally used in logarhythmic coordinates

Pn

Km

xlg

1lglg

Equation gives a straight line on a graph of lg x/m versus lgP (Fig.6.7.), from

which one can find values of constants.

This equation does not give good results

in the low and high pressure regions, but it is in

good agreement with experimental data over a

wide range of intermediate pressures. The

Freundlich equation also has other shortcom-

ings; it is not supported by any theory and its

constants have not a theoretical meaning.

Due to it in 1918 Longmuir used a simple

model of a solid surface to derive an equation

for an isotherm based on the monomolecular adsorption theory, which has a few

assumptions:

– a solid has a uniform surface

– adsorbed molecules don’t interact with one another and can’t move on a

surface

– adsorbed molecules are localized at active centers of adsorbent

– adsorption is restricted to monolayer coverage

– a gas behaves as ideal one

To derive the equation let all the numbers of adsorption sites (active centers)

on the solid surface be equal to 1, and the numbers of adsorption sites occupied by

adsorbat at equilibrium be θ. θ is called the degree of busyness and is equal to the

Fig.6.7.Determination of

constants of Freundlich equa-

tion

155

relation Г/ Г∞. In this case the number of unoccupied adsorption sites is equal to

1–θ. The rate of adsorption is assumed to be proportional to the partial pressure of

the adsorbate (gas) and the number of unoccupied adsorption sites:

1PKV adsads 6.6

where adsK is the adsorption rate constant.

The rate of desorption is proportional to the number of occupied adsorption sites:

desdes KV 6.7

At equilibrium the rates of adsorption and desorption will be equal, so we can

write:

1PKads = desK

from which

desads

ads

KPK

PK

If we divide numerator and denominator on desK , replace K instead of the relation

adsK / desK , and put the meaning of θ, we obtain the equation of the Langmuir iso-

therm:

1 ¥

KP

KP 6.8

This equation gives a good description of the isotherm in the low and high-

pressure regions. Indeed, at very small values of P the term KP becomes much

smaller than unity and can be neglected in the denominator. The equation then be-

comes to form Г= Г∞KP, which expresses a direct proportionality between the

amount of adsorbed gas and its pressure (as was mentioned above). In the high-

pressure region, on the contrary, the term KP becomes much greater than unity, and

the latter can be neglected in the denominator. As a result, the equation takes the

form Г= Г∞, which shows that the quantity of adsorbed gas does not change with

pressure. This actually does correspond to the adsorption isotherm at sufficiently

high pressure.

There are two constants in the equation; K and Г∞. Г∞ is the maximum possi-

ble adsorption for certain case; its value depends on the numbers of active centers

on the solid surface and on the size of adsorptive molecules. K is the equilibrium

constant of the adsorption process in equilibrium. To know the physical meaning of

the K we accept that Г = Г∞ /2 and after putting it in the equation obtain K=1/P.

Hence, K is proportional to the gas pressure under conditions that Г = Г∞ /2.

156

Because of its simplicity and wide utility, the Langmuir isotherm has found

wide applicability in a number of useful situations. Like many such “classic” ap-

proaches, it has its fundamental weaknesses, but its utility generally outweighs its

shortcomings.

§ 6. Adsorption on solids from solution

Interactions between solid surfaces and solutions are of fundamental im-

portance in many biological systems (joint lubrication and movement, implant re-

jection, etc.), as well as in mechanics, in agriculture, in communications (ink and

pigment dispersions), in electronics, in energy production, in foods (starch-water

interactions in bakery dough), and so on.

Adsorption on solids from solution is a much more complex phenomenon

that adsorption from the gas phase, as only because the solvent, as well as the so-

lute, may be adsorbed. For knowing which of them will be adsorbed in certain con-

ditions one have to take into consideration three principal factors: concentration of

the solution, the nature of the solid surface and wettability of solid surface.

The experimental evaluation of adsorption of this case involves the meas-

urement of changes in the concentration of the solute in the solution after adsorp-

tion has occurred. Adsorption dependence on

the concentration (adsorption isotherm) is

shown on Fig.6.8. Co is the initial concentra-

tion of the solution, and C is the concentration

after adsorption, hence their difference shows

an amount of the adsorbed solute. As m is the

weight of the adsorbent hence (Co– C)/m is

equal to Г. As shown in the diagram, at first

adsorption of the solute increases with increas-

ing of concentration, but after certain concentration it becomes decreased. There-

fore, we can do a conclusion, that adsorption of the solute is occurs well from di-

lute solution, and the adsorption of the solvent occurs well from concentric solu-

tion.

The nature of the solid surface involved in the adsorption process is a major

factor in determining the mode and extent of solute adsorption. It involves three

mainly groups of solids: nonpolar, polar and strongly charged surfaces. As for po-

larity of the substances, adsorption of the solute is occurs well from such solution

Fig.6.8. Adsorption isotherm

for case of solution-solid ad-

sorption

157

where there is a great difference in polarity between the solute and the solvent. In

other words the worse solubility of a solute the better its adsorption from solution

is.

The wetting of a surface by a liquid and the ultimate extent of spreading of

that liquid are very important aspects of practical surface chemistry. Wettability of

certain solid surface by liquid depends on the interaction forces between the same

molecules in each phase and different molecules of the two phases. When a drop of

liquid is placed on a solid sur-

face, the liquid will either spread

across the surface to form a thin

film, or spread to a limited ex-

tent, or remain as a discrete drop

on the surface (Fig.6.9.). The

quantitative measure of the wetting process is taken to be the contact angle (θ),

which the drop makes with the solid. In the case of a liquid that forms a uniform

film (i.e. θ = 0o), is said the solid to be completely wetted by the liquid, or that the

liquid wets the solid. If a finite contact angle is formed (i.e. θ > 0o), some investiga-

tors describe the system as being partially wetted. The more value of the angle the

worse wetted this surface is. Between 30o and 89o the system would be “partially

wetted”, and 90o and above nonwetting. If we return to discuss a case of the ad-

sorption from solution, the worse wet the solid surface the better adsorption of the

solute from solution is.

The wetting of solid drugs has great importance in pharmacy. Wettability

could be changed by surfactant adsorption. Surfactant adsorption results in the ori-

entation of the molecules with their hydrophobic groups toward the aqueous phase;

therefore, the surface becomes hydrophobic and less easily wetted by that phase.

In many practical situations it’s important to know how fast wetting and

spreading occurs. Typical examples would be detergency, in which a liquid or solid

soil is displaced by the wash liquid; petroleum recovery, in which the liquid petro-

leum is displaced by an aqueous fluid; textile processing, in which air must be dis-

placed by a treatment solution. Contact angles can be extremely useful as a spot

test of the cleanliness of sensitive surfaces such as glass or silicon wafers for mi-

croelectronics fabrications.

Fig.6.9. Wettability dependence on the solid

surface nature

158

Example: The initial concentration of the solution is 0.35 mol/dm3. After ad-

sorption from 50 ml of the solution by 2 g adsorbent its concentration reduces to

0.25 mol/dm3. Find the value of specific adsorption.(R = 0.082)

Solution: According the determination of specific adsorption Г = x/m where

x = Co – C. For 50 ml of the solution Co = 500.35/1000 = 0.0175 and C =

500.25/1000 = 0.0125. Therefore, Г = (0.0175–0.0125)/2 = 2.510–3.

§ 7. Electrolytes adsorption on solids from solutions

In adsorption from such solutions ions can be adsorbed as well as neutral

molecules. This leads to some peculiar phenomena, when not only adsorption forc-

es but electrostatic ones as well take place in the process. Adsorption of ions can

occur only on the polar surfaces.

There are two mechanisms of electrolytes adsorption – equivalent and ex-

change.

In equivalent adsorption both cations and anions are adsorbed in equivalent

amounts. First are adsorbed one type of ions either cations or anions. It depends on

the nature of the solid surface; first selective adsorbs the ions which is the same of

the composition of the solid. For example, if the solid is AgCl and in the solution

we have Ag+ and NO3– ions, first Ag+ ions will be adsorbed because the same ions

are in the solid phase. Due to it the solid surface obtains the corresponding charge

and the ions having the opposite charge also come near to the first layer of ions due

to electrostatic forces of attraction. As a result double electrical layer is formed.

In ion-exchange adsorption an exchange of ions occurs between the adsor-

bent and the solution due to ion chemical potential difference between these two

phases. The adsorbent is adsorbed by certain ions from the solution simultaneously

sending other ions but of the same charge from its surface to the solution. In the

result the charge of the surface is not changed. Adsorbents available for ion ex-

change are called ion-exchangers. Some of them, such as sulphurated chaircoals

and ion exchange resins, are called cation exchangers, can exchange cations in the

solution with hydrogen ions. Others, called anion exchangers, exchange various

anions for hydroxyl ions. Applying both kinds of exchangers successively it is pos-

sible to demineralize water practically completely without distillation. The ex-

changers themselves are easily recovered: the cation exchangers – by washing with

an acid solution, the anion exchangers – with base solutions.

159

One of the important properties of the exchangers is the exchange capacity.

It is equal to the numbers of ionic groups (which could exchange ions) of the unit

weight of the exchanger (mmol/g).

Example: How many grams of Ca2+ ions remaine in the solution of calcium

salt after it’s passed through a 15g cation exchanger as adsorbent if the original

solution was 600 ml of 0.08 mol/dm3 concentration? The exchange capacity of the

cation exchanger is equal to 1.2 mmol/g.

Solution: The moles amount in the original solution will be equal to

6000.08/1000 = 0.048 mol/dm3. 15g cation exchanger can adsorb 1.215 = 18

mmol = 0.018 mol of Ca2+ ions. The difference between them is the amount of Ca2+

ions which remain in the solution after adsorption, hence 0.048–0.018 = 0.03mol or

it is equal to 0.03mol40 = 1.2g of Ca2+ ions.

Review questions:

1. In which systems do surface phenomena exist?

2. What is the surface tension and by what reason does it appear on the liq-

uid surface? Write the expression of a surface tension.

3. Formulate the process of adsorption and explain by what reason it hap-

pens.

4. Define the following terms: adsorbent, adsorbat, desorption, adsorption,

absorption.

5. What quantity is the quantitative characteristic of adsorption?

6. In which cases is specific adsorption expressed as Г=x/m?

7. What kind of substances is adsorbed on the liquid surface? Why?

8. Draw a diagram of surface tension isotherm and explain it. When surface

tension of liquid does becomes independent on the concentration of a solution?

9. Which substances are called surfactants? What structure do their mole-

cules have?

10. On what depends the position of surfactants on the boundary of two

phases? Draw their position on the water-coal boundary.

11. Write the Gibbs equation for liquid surface adsorption. What relasion is

called surface activity and what does it show?

12. Is surface activity of surfactants positive or negative?

13. What is the peculiar property of solid surface adsorption?

160

14. Draw the adsorption isotherm for case of gases adsorption on the solid

surface. Explain the curve position. When does specific adsorption becomes inde-

pendent on the gas pressure? How is specific adsorption called in this case?

15. Write the Freundlich isotherm equation and denote its shortcomings.

16. Why is the Longmuir theory of adsorption called monolayer adsorption?

17. Write the Longmuir isotherm equation and apply it for large and small

values of gas pressure.

18. What is the meaning of constants (Г∞ and K) in this equation? Is the val-

ue of Г∞ less or more if the numbers of active centers on the solid surface are big

and size of the adsorbate molecules is small?

19. What is the complexity of the adsorption from solution on solid surface?

What factors does this case of adsorption depend on?

20. Draw the adsorption isotherm for this case of adsorption and explain it.

21. What is dependence of such adsorption on the substances polarity? Is ad-

sorption of solute from solution less or more if the solubility of solute is worse?

22. What phenomenon is called wettability? In which way does adsorption

from solution on solid surface depend on it?

23. What forces take place in case of electrolyte adsorption on solid surface?

24. What types of electrolyte adsorption exist? Explain their mechanism.

25. What are ion-exchangers and what types of them are distinguished?

26. Formulate the exchange capacity. What is its unit?

§ 8. Chromatography

Chromatography is the method of separation, division and analysis of sub-

stances from the complex mixture, based on the adsorption phenomena.

Russian botanist M. Tsvet discovered (1903) that many solid materials of

widely different chemical nature have the ability of selective and consecutive ad-

sorption of various substances from a solution. He had passed the extract of chlo-

rophyll obtained from the plant throughout the adsorbent (CaCO3) which was

packed into the glass cylinder. During passing the chlorophyll was divided into

components (Fig.6.10.).

This method was called chromatographic adsorption analysis, because

when a solution of a colored substance passes through a column containing an ad-

sorbent, the origin mixture is resolved into a series of colored bands or zones.

However, the method is also used under the same name to separate colorless mate-

161

rials. At present new methods and techniques of chromatographic analysis are de-

veloped.

In all the variants of chromatographic methods,

the mixture to be treated percolates an adsorbent and is

divided on the components according their adsorption

activity. Thus, the chromatographic method can now be

defined as the separation of materials by distribution of

the components between two phases – a motionless layer

(stationary phase) of solid adsorbent with a large sur-

face area, and a stream of liquid solution or gas mixture

(mobile phase) that filters through it.

Chromatography is a widely applied almost in all

sciences for a multitude of reasons! First of all it can be

applied for almost all kinds of substances and for all

amounts. Even very similar components, such as pro-

teins that may only vary by a single amino acid, can be

separated by chromatography. For these reasons chroma-

tography is very well suitable to be used in the field of

biotechnology for mixtures of proteins separating. It has

great precision and easily can be robotized. Chromato-

graphic methods are now being used not only for analyt-

ical purposes (qualitative and quantitative analysis), but

also for preparative purposes – to extract highly valuable components of a mixture

and to obtain materials of high purity by removing small amounts of undesirable

impurities.

§ 9. Classification of chromatographic methods

The different chromatographic procedures can be classified into some types.

1. Depending on the technology of the process chromatographic procedures

are classified into three types:

Column chromatography, in which the mixture (liquid or gas) passes

through the special chromatographic column (usually glassy) packed with a solid

adsorbent in granulated form. Selective and consecutive adsorption of components

occurs during the mixture passing along the column.

Fig.6.10.Chromato-

graphic column

162

In paper chromatography the division of the components from mixture oc-

curs on chromatographic paper. A drop of the

mixture is placed on the start line which is drawn

by pencil parallel of the one end of the paper to 2

cm above. Then this end of the paper is pulled

down to the solvent (usually consist of two or

three liquids mixture), which wets the paper and

rises along the paper with the drop of the mixture.

During this movement, the components of the

drop are adsorbed on different levels of the paper

according to their distribution coefficient between

the stationary and mobile liquids, forming the patches on the paper. (Fig.6.11.).The

process continues up to the solvent reaching the 2/3 the paper height (finish line)

and the later the paper is taken out of the solvent. After drying the paper one can

analyzes the obtained patches by different methods. One of the qualitative charac-

teristics of the chromatography is the retinue factor (Rf), which is equal to the rela-

tion between distance of patch from the start and distance of solvent from the start.

In these cases for the first substance it is equal to AC/AB and for second substance

AD/AB.

In some cases the qualitative analysis could be done by “witnesses” method.

On the start line places a few drops of known solute solutions is placed (which

supposes to be into the investigating mixture). If we obtain the patches of them on

the same level with the patches of the investigating mixture, so these are the same

species).

Thin-layer chromatography is the same as paper chromatography but with

such difference that instead of paper as an adsorbent the powder material on the

glass plate is used.

2. Chromatographic procedures can be classified into the following four

types considering the aggregate state of the adsorbent and adsorbat:

Liquid adsorption chromatography utilizes a mobile liquid phase that is ad-

sorbed onto the surface of a stationary solid phase.

Gas adsorption chromatography is the same of liquid chromatography ex-

cept that a mobile phase is gaseous. This method is used to separate mixtures of

gases. Separating gases are contained in a carrier gas, such as helium, to carry the

mixture through the column.

Fig.6.11. A paper chro-

matography

3 2 1A

C'

A

C

D' D

B

finish

start

163

Liquid partition chromatography involves hydrophobic, low polarity sta-

tionary phase which is chemically bonded to an inert solid such as silica. The sepa-

ration is essentially an extraction operation and is useful for separating non-volatile

components.

Gas-liquid chromatography utilizes an inert porous solid coated with a vis-

cous liquid which acts as the stationary phase. Gaseous mixture in the feed stream

is dissolved into the liquid phase and eventually vaporizes. The separation is thus

based on relative volatilities.

In gel permeation chromatography as an adsorbent is used a porous poly-

meric material about 0,1 mm in diameter that captures molecules of adsorbate (in

particularly polymers) selectively, according to their size. In this method attractive

interaction between the stationary phase and solute is lacked. It does not involve

any adsorption and is extremely fast. This method is now the most widely used one

for molar mass determination of polymers. A solution of the polymer sample is

filtered through a column and small molecules, which can permeate into porous

structure of the gel, require a long elution time, to pass through a particular length

of column, whereas larger ones, which are not captured, pass through rapidly. The

average molar mass of macromolecule may therefore be determined by observing

its elution time in a column calibrated against standard samples.

3. The processes of chromatographic analysis differ essentially in the sepa-

ration procedure used. According to this liquid adsorption chromatography is di-

vided into three methods: elution, displacement and frontal. To understand we shall

consider the simplest case of a mixture consisting of two components A and B,

where B is adsorbed more strongly than A.

In the elution method a feed mixture of liquid solution of A and B compo-

nents to be analyzed is injected into the chromatographic column inlet. Then com-

ponents migrate through the column at different speeds due to their different ad-

sorption ability and are held up in different zones of the column. Then a solvent

(the eluent), which is adsorbed more weakly than both components A and B, passes

through the column. As it is filtered through the layer containing the adsorbed

components A and B, it will gradually be washed out by both of them, beginning

from the more weakly adsorbed component A. As a result, the components A and

B will be distributed among the different zones of the adsorbent. As the elution

continues, these zones move down the column, and, finally, components A and B

are carried out from the column in different portions of elute that are separated

from each other by pure elute.

164

The displacement method differs from the preceding by that the displacer

compound is used which is adsorbed more strongly than the components of the

mixture A and B. In other words the displacer has a higher affinity for the station-

ary phase than any of the feed components. The action of a displacer causes the

feed components to migrate through the column at velocities greater than that dic-

tated solely by their individual adsorption ability. This liquid therefore displaces

both components, and to a greater extent A, from the adsorbent. In some time we

shall have two zones with one component in each, separated by a zone containing

both of them. When these zones are removed from the column, we obtain each of

the components partly in its pure form, and partly in a mixture.

In the frontal method the original mixture passes through the column. Owing

to the fact that component A is more weakly adsorbed, in some time the distribu-

tion of the components sets in. As the process continues, first the pure component

A comes out of the column and then the component B appears. By means of this

method we are able to obtain a certain amount of weakly adsorbed component A. It

is therefore used to obtain small amounts of certain valuable substances (rare-earth

elements, proteins).

4. The character of the interactions on which these processes are based var-

ies in different processes, which can be divided in this respect into four main types:

Adsorption chromatography is based on the different surface (adsorption)

activity of different substances and described in the above-mentioned examples.

Solid adsorbents are used, such as activated carbon, silica gel, activated alumina.

In partition chromatography a liquid thin film on a solid support is formed

(stationary phase) and a favorable distribution of the desired component between

the origin mixture and the stationary liquid is achieved.

In ion exchange chromatography ion exchange is affected by using natural

or synthetic inorganic or organic materials (see adsorption of electrolytes). The

process of separating is due to the difference between the exchange constants of the

components. This method is widely used in the purification of biological materials.

Separation by this method is highly selective.

In precipitation chromatography the molecules that are immobilized on a

stationary phase (on a solid support) forms insoluble precipitates with some of the

components of the original mixture. Separation is usually based on the difference

in solubility of the compounds formed. For example, the immobilized molecule

may be an antibody to some specific protein. When original mixture of proteins is

165

passed by this molecule, only the specific protein is reacted to this antibody, bind-

ing it to the stationary phase.

As it was shown above adsorption process is applied not for division of the

mixtures into their components and for purification of the materials, but also for

analytical analysis - qualitative and quantitative. For this purpose the explored mix-

ture is placed into the chromatograph (apparatus used in chromatography), where

adsorption process oc-

curs and as a result we

obtain a trace of it in

the form of chromato-

gram. For example, let

us consider such an

analysis of the mixture

consisting of two sub-

stances, which can be

adsorbed (Fig.6.12.).

Since the sample is separated in the column, different peaks on the chromatogram

correspond to different components in the sample mixture. The chromatogram

above shows two peaks (3 and 4) of two adsorbed components. (Line 1 corre-

sponds to the elute outlet and peak 2 corresponds to the non-adsorbed component)

Each peak consists of front line and rear line. The peak is characterized by its

height (h) and width (μ). The relation μ/h is characterized the spreading of a peak.

There is the productivity of the division, which shows how well the division

of the components occurs. It is equal

21

l

For qualitative analysis the time of keeping is used, which is equal to the

distance between the initial and maximal points of the peak (l1 and l2 for the first

and the second component accordingly). Qualitative information about the sample

composition also could be obtained by comparing peak positions with thse of

standards. The quantitative analysis makes counting the area of the peak (hatching

part on the diagram).

Review questions:

1. What method is called chromatography and what phenomenon lies in the

bases of this method?

V

1

2

3 4

l2l1

l

fron

t

r ear

h

Fig.6.12. An analysis of chromatogram

166

2. What two obligatory phases take place in this method?

3. What are classifications principles of method of chromatography there are?

4. Describe the paper method. What are the stationary and the mobile phases

here?

5. How is qualitative analysis made in paper chromatography? What is the

retinue factor?

6. What is the difference between liquid adsorption and gas adsorption chro-

matography?

7. What is the stationary phase in liquid partition chromatography?

8. Is adsorption process involved in gel permeation chromatography? How

does this process occur?

9. Draw the example of chromatogram and explain how quantitative and

qualitative analyses are made.

§ 10. Applying of the surface phenomena in pharmacy

Besides the surface phenomena spreading in almost all processes in our life

and human activity, there is a large region of surface phenomena application in

medicine and pharmacy. We can list some of them:

– Chromatography: for drug purification, division and analysis.

– Heterogeneous catalyse: for some drugs synthesis.

– Purification of water for obtains demineralized water used in pharmacolo-

gy.

– Adsorption of poison substances in cases of intoxication of organism by

some toxins or overdosed of drugs.

– In choosing drug containers we must keep in mind the interactions between

drugs and container walls what could influence on drug activity.

– Adsorption participates in all the processes which drugs undergo in an or-

ganism.

167

CHAPTER 7

COLLOIDAL CHEMISTRY

INTRODUCTION

Between pure bulk materials and molecularly dispersed solutions lies a wide

variety of important systems in which one phase is dispersed in a second, but in

units which are much larger than the molecular unit (a classical sol) or in which the

molecular size of the dispersed material is significantly greater than that of the sol-

vent or continuous phase (a macromolecular or polymer solution). Such systems

are generally defined as colloids, although there may be accepted limitations on the

unit size of the dispersed phase, beyond which other terminology may be used.

Colloidal chemistry studies the properties of the heterogeneous high-dispersed sys-

tems, the rules of the processes occurring in them, and the processes proceeding on

the high-developed surface.

In the middle of the past century, T. Graham when investigating solubility

phenomena noted that some substances, which are insoluble in water, might never-

theless under certain conditions form apparently homogeneous solutions. Many

properties of such systems differ markedly from those of true solutions:

– The solute does not diffuse through the semi-penetrable membranes.

– Diffusion occurs very slowly in them.

– They give phenomena of the electrophoresis and diffraction of light.

– Osmotic pressure has a very low and changeable magnitude.

– They are unstable systems.

Due to these investigations Graham concluded, that all substances divides on

two groups: crystalloids, which formed true solutions and colloids formed colloid

solutions. For this reason, the science studying such systems was called colloidal

chemistry. Further investigations showed that the distinction between colloids and

crystalloids is conditional, since the same substance can exist as a crystalloid or a

colloid depending on the method by which it is obtained or separated. For instance,

sodium chloride, a typical crystalloid under ordinary conditions, forms a colloidal

solution in benzene and a true solution in water, while soap, which forms a colloi-

dal solution in water, acts like a crystalloid in alcohol. For this reason, the term col-

168

loid is now applied not to a given class of substances, but rather to a specific state

of matter.

Colloid systems are only one of the specific forms of disperse systems;

hence, it will be true to call this subject the chemistry of disperse systems.

There is an extremely great variety of disperse systems. They are widespread

in nature, and find application in many industrial processes. Many natural sub-

stances, such as milk, blood, egg albumin, smoke, gels, mayonnaise, numerous

vegetable and animal tissues are disperse systems. Other disperse systems are

clouds, fog, and many natural liquids. They are widely represented in great variety

in the mineral world. Opals, agates, and many other minerals are solid disperse sys-

tems. The coloring of many minerals and rocks is due to impurities present in a

dispersed state. The list of only some important products and processes involving

colloids is following: pharmaceuticals, cosmetics, inks, paints, foods, lubrication,

paper coating, catalysts, chromatography, membranes, wetting of powders, purifi-

cation, coating technology, powder flow, water purification, and so on.

As about of their importance in medicine and pharmacy, many drugs are

prepared in the form of disperse systems, such as emulsions, powders, liniments,

aerosols, etc. Knowing the disperse systems laws helps to understand the behavior

of the liquids of the organism.

Systems in which one substance is uniformly distributed throughout of an-

other one in the form of finely divided particles are called disperse systems. They

usually consist of two or more phases: the dispersed phase, consisting of the ag-

gregate of dispersed particles and the dispersion medium (or the continuous phase)

surrounding them. A great number of molecules form these aggregates. Therefore,

all disperse systems are heterogeneous systems with large summary interphase area

between particles and medium.

Disperse systems are characterized by two properties: degree of dispersion

and specific surface. The term degree of dispersion shows the degree of comminut-

ing of the substance forming the dispersed phase. Its equal the inverse of the parti-

cle size: D=1/a. The term specific surface area shows the surface of the unit vol-

ume or unit mass of material. A system containing very fine particles is called

highly disperse with highly specific surface. When particles are not so finely divid-

ed, we speak of a coarse dispersion. As the bulk phase is subdivided into finer and

finer particles, the relative ratio of surface to bulk molecules will increase until the

effect of specific surface properties will begin to become significant, or even dom-

inate the characteristics of a system.

169

§ 1. Classification of the disperse systems

There are different principals of disperse systems classification.

1. Depending on the size of disperse phase particles (dispersion degree) the

following types are they distinguished:

– Coarse disperse systems with particle size about 100 nm(1 nm = 1x10-9)

and more (suspensions, some emulsions, powders).

– Colloid disperse systems with particle size from 1 to 100 nm (colloid solu-

tions).

This classification is rather relative, because colloid systems usually contain

particles of different size, i.e., they are polydisperse systems.

2. Disperse systems can be classified according to the aggregation state of

both the dispersion medium and the dispersed phase, either of which can be in the

solid, liquid or gaseous state.

– When the dispersion medium is a gas, the system is called aerosol. De-

pending on the state of aggregation of the dispersed phase, aerosols are subdivided

into smokes (solid dispersed phase) and fogs (liquid dispersed phase).

– Disperse systems with liquid dispersion medium are called liosols or just

sols. They are subdivided into foams (with gaseous dispersed phase), emulsions

(liquid dispersed phase), suspensions and colloid solutions (solid dispersed phase).

– Disperse systems with solid dispersion medium are called solidosols. They

could be with gaseous dispersed phase (solid foams, silicagel, activated coals), with

liquid dispersed phase (wetted soil), and with solid dispersed phase (colored glass-

es, alloys).

3. Depending on the character and intensity of dispersed particles interaction

with the dispersion medium for liquid-medium systems they are divided into two

types:

– A lyophilic (solvent attracting) colloids is one particles of which have a

strong attraction for the molecules of the dispersion medium.

– In lyophobic (solvent repelling) colloids the particles do not interact so

strongly with the molecules of the surrounding medium.

In the particular case of aqueous colloid solutions one speaks, correspond-

ingly, of hydrophilic or hydrophobic systems.

Lyophobic colloids are heterogeneous highly disperse colloid systems such

as real soles. They are usually characterized as irreversible systems without ther-

modynamic stability. On the contrary, lyophilic colloids (such as solutions of high

170

molecular compounds) formed spontaneously, have a certain thermodynamic sta-

bility because of the solvent layers, which decreases an interphase tension.

4. According to the interaction intensity of dispersed phase particles disperse

systems are divided into two types:

– Free disperse systems in which there is no interaction between dispersed

phase particles and the latter can easily be moved through the medium.

– Linked disperse systems in which the particles are binding to each other,

restricting their motion.

§ 2. Preparation and purification of the disperse systems

With respect to particle size, colloid systems occupy an intermediate position

between molecules and macroscopic bulk phases. They can therefore be produced

by breaking down large pieces to the size required (dispersion methods) or, vice

versa, by bringing up the agglomeration (condensation) of molecules, ions or atoms

to particles of the required size (condensation methods).

In dispersion methods, a coarse material can be made into a colloid by dif-

ferent type of colloid mills in the presence of a stabilizer. It also could be done by

ultrasound or by acting of an electrical force (electrodispersion). Besides, there is

the method of peptization, in which fresh sediment is converted into a colloid solu-

tion by adding a special substance (peptizator). Moreover, at last, one can obtain a

colloid solution just by dilution of certain substances (such as starch, gelatin) in the

suitable solvent. It is reasonable to assume that the work required reducing a given

material to colloidal size; higher surface-energy materials require more work input.

In addition, the natural tendency of subdivided particles is to reduce the total sur-

face area by some aggregation process. This tendency could be reduced by the in-

troduction of an intervening medium, usually a liquid.

Condensation methods are based on purely physical processes, such as the

fast condensation of vapor or on different chemical reactions. A common feature of

both classes of methods is that the formation or separation of the colloid in the new

phase occurs at strong supersaturation, i.e. from a highly supersaturated vapor or a

highly supersaturated solution. Under such conditions, nuclei of the new phase

simultaneously are formed at many points. These nuclei become centers of conden-

sation or crystallization. Their further agglomeration is prevented by the addition of

a stabilizer. One of the simplest examples of the formation of a colloid system by

171

vapor condensation is the formation of atmospheric fog consists of tiny droplets of

water condensed when the water vapor cools in the atmosphere.

In chemical condensation methods, the colloidal substance is obtained by

means of some chemical reactions in which it is separated out in the colloidal state.

Such methods are usually based on interactions in solution in which the substance

formed is insoluble. Originally formed in the molecular-dispersed state it tends to

precipitate from the solution. The conditions of the reaction must be so chosen that

aggregation of the molecules into larger particles stop at a certain stage, before co-

agulation begins. This is usually achieved by taking solutions of sufficiently low

concentration (with excess one of them) and slowly mixing them.

Condensation methods are commonly employed for the production of col-

loids and aerosols, and less commonly in the production of emulsions.

When colloid solution is formed, some quantities of the ionic material have

accompanied their formation. So, it is necessary to purify them. Colloids are often

purified by dialysis. In this method the semipermeable membranes are used, which

are pervious only for ions and small molecules, but impervious for colloidal parti-

cles. Dissolved ions or small molecules move

from the region of their higher concentration

to a region of lower concentration. For purifi-

cation the colloid solution containing a crys-

talloid (electrolytes) it is placed into the bag

made of dialyzing (semipermeable) membrane

and the bag is immersed into the water. Water

continuously circulates around the bag. The

molecules and ions move out of the bag into

the circulating water where they are removed.

Colloid particles cannot cross the membrane and therefore remains in the bag. A

simple dialyzer shows on the Fig.7.1.

Dialysis can be greatly accelerated by the simultaneous action of an electric

current. This technique is called electrodial-

ysis. A schematic view of an electrodialyzer

is given on Fig.7.2. When current is sent

through the solution, the electrolytes in it are

carried in the form of ions to the correspond-

ing electrodes and carried off by water.

Fig.7.1. Purification of colloids by

dialysis

Fig.7.2. A schematic view of an

electrodialyzer

172

Dialysis is often applied in industry and in medicine and biology, especially

for analysis and purification of biological liquids. The

principal of the dialysis is also used also in the appa-

ratus of the “artificial kidney” whereby separates

waste molecules of metabolism such as urea and ex-

cess ions from the blood for the people with kidney

failure.

For colloid purification the method of ultrafil-

tration is also applied, the scheme of which is shown

on Fig.7.3. The colloid solution purified by passing

through such a filter (3), from which only the mole-

cules and ions of the crystalloid pass through the

pores of the filter, while colloid particles are held

back by the membrane. The process is hastened by

pressure difference on the both sides of the filter.

MOLECULAR – KINETIC PROPERTIES OF COLLOIDS

Molecular-kinetic properties of colloids include Brownian motion, diffusion,

osmotic pressure and sedimentation. They subordinate to molecular-kinetic laws of

the true solutions, but are expressed much weakly (because of large size of parti-

cles).

§ 3. Brownian motion

This is the term applied to the random motion of particles suspended in a

liquid (colloid particles) that is due to the random impacts by the molecules of the

surrounding medium (liquid) in their thermal motion. It is called Brownian motion

after its discoverer, the botanist Robert Brown. If the fluid pressure on all the parts

of the colloidal particles were always the same, the particle would remain at rest.

However, tiny fluctuations in fluid pressures on the colloidal particles cause ran-

dom motion. If the particle is large, it experiences many millions of impacts on all

the sides at every instant, so that on average these impacts balance of each other.

However, the number of impacts on a small particle is much less, so that the prob-

ability of their being exactly balanced becomes small. Thus, a colloid particle usu-

ally experiences at any instant a resultant impact in one direction, and in the next


the method of ultrafiltra-

tion

173

instant a resultant impact in another direction, and

as a result executes a continual erratic motion.

Figure 7.4. shows the path of the colloidal particle

due to Brownian motion which is very disordered

and there is no ability to follow its movement ex-

actly. Because of it quantitatively Brownian mo-

tion is expressed by average square displacement

(∆2), which shows projection of the distance be-

tween the initial and final positions of the particle

in a unit of time on the abscise. Einstein averaged

over many colloidal particles and found the equation for ∆2:

trN3

RT

A

2

7.1

where t is the time of displacement, R is the gaseous constant, T is the absolute

temperature, η is the viscosity of the surrounding, r is the radius of the particle, NA

is the number of Avogadro.

§ 4. Diffusion

The solute always has a tendency to pass from points of higher concentration

to the ones of lower concentration. This spontaneous decrease in concentration

differences is called diffusion. It is occurred due to the difference of chemical po-

tentials of the substances in different parts of the system caused by Brownian mo-

tion. This process is characteristic to both true solutions and colloids, but the diffu-

sion rate is many times less in the latter.

Consider the simplest case of a solution containing a single solute. The so-

lute will be spontaneously diffused from a region of high chemical potential to one

of low chemical potential that is from a region of higher concentration to the one of

lower concentration, whilst the solvent molecules move in the reverse direction.

Although the driving force for diffusion is the gradient of chemical potential, it is

more usual to think of the diffusion process in terms of the concentration gradient

(the difference in concentration between two layers of a system). The expression,

which relates the flow of material to the concentration gradient, is referred to as

Fick’s first law equation:

dx

dCDS

dt

dm 7.2

Fig.7.4. The path of the col-

loidal particle due to Browni-

an motion

174

where dm is the amount of matter passing through a cross section S in a time inter-

val dt when the concentration changes by dC through a distance dx (dC/dx is called

concentration gradient).

The proportionality constant D is called the diffusion coefficient. It is numer-

ically equal to the quantity of matter that diffuses in unit time (1s) through a cross

section of 1 sq cm when the concentration gradient is unity, i.e., when the concen-

tration decreases by one unit per cm of path.

Fick’s first law equation describes the diffusion process under conditions of

steady state; that is, the concentration gradient does not change with time. In many

of the experimental methods used to study diffusion, however, the variation of C

with both time and distance is of interest. In such cases equation 7.2 may be con-

verted into an express of the Fick’s second law equation

2

2

dx

CdD

dt

dC

Fick’s second law states that the rate of concentration change in a volume

element within the diffusion field is proportional to the rate of change in the con-

centration gradient at that point in the field, the proportionality “constant” being

the diffusion coefficient or diffusivity.

The relationship between the radius (r) of the drug molecule (or other parti-

cle) and its diffusion coefficient is given by the Stokes-Einstein equation as

ArN

RTD

6 7.3

This relation shows that the rate of diffusion increases in direct proportion to

the temperature, and in inverse proportion to the viscosity of the medium and the

size of the particles. By means of this relation, it is possible to determine the “mo-

lecular weight” of colloid particles from measurements of the rate of diffusion.

There is a connection between the diffusion coefficient and the average

square displacement, which is expressed by equation

∆2 = 2Dt 7.4

The solution of the diffusion equation can be used to predict the concentra-

tion of particles at any location. We can also use them to calculate the net distance

through which the particles diffuse in a given time. It is very important also to de-

termine drug diffusion rate through the organism tissues and liquids as well in dif-

ferent diffusion processes in organism (for example, diffusion of neurotransmitters

is used to transmit signals from one nerve cell to another).

175

§ 5. Osmotic pressure

The osmotic pressure of colloid solutions as well as true solutions is directly

proportional to the number of colloid particles in a unit volume of the solution:

RTV

n

Since the size and mass of colloid particles are many times greater then those

of ordinary molecules, then obviously the number of solute molecules in, say, a 1%

true solution will be the corresponding number of times greater than the number of

colloid particles in an equal volume of a 1% colloid solution. Consequently, the

osmotic pressure of colloid solutions is much lower than that of true solutions. Be-

sides that, the osmotic pressure for the same colloid solution measured in different

time under the same conditions has a different value. It is due to the aggregating

and disaggregating processes occurred in the colloid solutions, hence the number of

the colloid particles changes, so changes the osmotic pressure.

For these reasons, the measurement of the osmotic pressure is not applied for

solute concentration determination in colloid systems.

§ 6. Sedimentation

If the size and density of particles suspend-

ed in a liquid are sufficiently more than those of

the liquid they settle under the force of gravity.

This process is called sedimentation. The rate of

sedimentation is expressed by Stock’s equation:

grV 0

2

3

2 7.5

where r is the radius of the particle, ρ and ρo are

the solution and the solvent densities accordingly,

g is the acceleration of gravity, η is the viscosity of

the medium. It is clear that the more particles size

and the less the viscosity of the medium the more the rate of sedimentation is.

This equation is used in the sedimentation analysis, from which one can de-

termine the size of the colloid particles by determining the rate of the sedimenta-

tion. However, sedimentation process run very slow and for accelerating it applies

the gravitation field by a centrifugal force. Centrifugation, of course, is basically


ultracentrifuge

176

the same as sedimentation except that the force of gravity is replaiced by artificial

forces of greater strength. The latter can be achieved in an ultracentrifuge, which is

essentially a cylinder that can be rotated at high speed about its axis with a sample

in a cell near its periphery (Fig.7.5). Modern ultracentrifuges can produce accelera-

tions equivalent to about 105 that of gravity. Initially the sample is uniform, put the

“top” (innermost) boundary of the solute moves outwards as sedimentation pro-

ceeds. Ultracentrifuges are widely applied in biology to divide biological liquids

into fractions (depends on their size and weight). An example is the work of a milk

separator, where the cream is separated in a far shorter time than if left to the action

of the natural force of gravity.

The particles in the colloid solutions are not only under gravitation force, but

also under diffusion force. With large particles, the force of gravity is the decisive

factors, and such particles eventually, depending on their size, settle on the bottom

of the vessel. However, with smaller particles the rate of diffusion is sufficient to

prevent their settling. Hence, for small particles equilibrium between gravitation

and diffusion force could be states, which is known as sedimentation equilibrium.

It is characterized by a gradual decrease in concentration from the bottom of the

vessel to the upper layers of the solution. This phenomenon, for example, causes

the decrease in atmospheric pressure (i.e., the concentration of molecules of air),

with altitude.

Because of colloid systems being polydisperse the concentration gradient of

the larger particles is greater than that of the smaller particles under sedimentation

equilibrium, hence, in equilibrium the average particle size is less in the upper part

of the suspension than in the lower.

The quantitative study of sedimentation phenomena gives important infor-

mation of the colloids, in particular, of the size of its particles. On basis of this

phenomena Perrin determined Avogadro’s number in 1908.

OPTICAL PROPERTIES OF COLLOIDS

Optical properties of colloids differ from those of true solutions due to dif-

ference in optical densities of particles and medium. It depends on particle amount

and size, as well as on the wavelength of the light. To put it more precisely optical

properties depend on the ratio of the diameter of the particles of dispersed phase

(2r) to the length of the light (λ) passing through dispersed system as well as on the

distance between particles.

177

When the beam of the light passes throughout the colloid solution the fol-

lowing optical phenomena could occur: scattering, absorption, deflection and re-

flection of the light. If 2r>>λ, the reflection, deflection and absorption of light hap-

pen. That is why rude dispersed systems are feculent in both passing light and

when light passes sidely.

§ 7. Light scattering

If particle size of colloid dispersed system is about or less then λ, the diffrac-

tion light scattering takes place. Each colloid particle becomes secondary source of

light, causing an opalescention. This is a process, when the colors of sole under the

passing light and light passing from a side are different. It has been observed that

the path of a beam of light through a perfectly transparent colloidal solution of gold

becomes outlined when observed laterally against a dark background. This is

known as the Tyndall effect and is due to the scattering of light by the colloidal par-

ticles. A similar phenomenon is familiar to anyone who has observed a narrow

beam of light in a dark room (for instance, in a cinema). The beam is visible from

the side only when it passes through numerous fine particles of dust or fog that

scattered the light. This phenomenon of light scattering describes by Reley’s equa-

tion:

4

2

VKI 7.6

where I is the intensity of scattering light, ν is the number of the particles in the

unit volume, V is the volume of a particle (its size), λ is the length of falling light

wave, and K is the optical coefficient. It is clear that the intensity of scattering light

depends on the amount of the particles, quadrant of volume of colloid particle and

reversely depends on the fourth degree of the wavelength of falling light. It is not

difficult to notice that the shorter the wavelength the larger the intensity of light

spread. This is the reason of blue color of the sky; the blue component of white

sunlight more intense scattered by the molecules of the atmosphere.

§ 8. Light absorption

In colloidal systems, besides scattering absorption of the certain wavelength

of white sunlight by colloidal particle might occur. It depends on the nature of par-

178

ticles, their size and shape. Light absorption is described by Lambert-Buger equa-

tion:

klon eII 7.7

where Io is the incident intensity of the light, I – transmitted intensity, l is the thick-

ness of the sample, k is the absorption coefficient. Then by Beer-Lambert was

found that k is proportional of the molar concentration: k=εc, where ε is the molar

absorption coefficient. Hence, the equation obtains the form

clon eII

and after finding the logarithm we obtain:

clI

I

n

o ln 7.8

n

o

I

Iln is called optical density of the sample and used in spectrophotometry for

concentration determination.

Optical properties of colloid systems are used to investigate such systems.

Thus, the ultramicroscope is based on the scattering phenomenon. In opposite to an

optical microscope, in this one a beam of light illuminate the sample from the side

and we can see colloidal particles as sparkling points on the dark background. In

such conditions, the number of colloidal particles in a unit of volume (by determi-

nation the number of shining points in certain visible volume) can be counted. Us-

ing the data one can determine the concentration of the compound and, moreover,

knowing the mass of the compound, the density of the dispersed phase, the radius

of colloid particle. One can determine also the form of colloid particle by ultrami-

croscope. When the intensity of spread light is constant, the particles are spherical,

when the shine of particles is changeable (flickering shine), the particles have dif-

fereht shapes.

The electron microscope, which uses electron beams instead of light beams,

allows us to achieve magnification of 100 – 150 thousand, so that it is now possible

to see colloidal particles, or rather their projections on a screen.

Review questions:

1. What systems are called disperse? By what properties colloid solutions

are distinguished from true solutions?

2. By what principles do classifications of disperse systems occur?

3. Why is classification of disperse systems by particles size relative?

179

4. What does it mean liophobic or liophilic colloids? Give a few examples

of such colloids.

5. By what reason are there two methods of colloids obtaining. What are

they?

6. Explain the method of purification of colloids by dialysis.

7. What is Brownian motion explained by and what is its quantitative char-

acteristic?

8. Define the process of diffusion. Why does it occur (from the point of

view of thermodynamics)?

9. Write the equation of Fick’s first law and explain it? What does the rate

of diffusion show? What is the concentration gradient?

10. Which are two peculiarities of osmotic pressure in colloids? Explain

their reason.

11. What is sedimentation process in colloids? Write the equation of sedi-

mentation rate of colloid particles and denote the factors it depends on?

12. What is sedimentation analysis applied for?

13. Under which forces are colloid particles in the solution. When is sedi-

mentation equilibrium stated on?

14. What are the peculiarities in optical properties there in the colloid solu-

tions and why do they occur?

ELECTRICAL PROPERTIES OF COLLOIDS

Most solid surfaces in contact with water or an aqueous solution will be

found to develop some type of electrical charge. In macroscopic systems it may

often be overlooked. However, in the microscopic world of colloids and interfaces,

the presence or absence of even a small surface charge can have a great importance

in terms of stability, sensitivity to environment, electrokinetic properties, and other

factors.

Thus, in the colloid solutions on the boundary surface between particles of

dispersed phase and dispersion media a double electrical layer is formed, due to

colloid particles owing the electric charge, which has a great significance for col-

loid system properties. An important consequence of the existence of electrical

charges at interfaces, whether they are colloids, porous materials, or some other

system, is that they will exhibit certain phenomena under the influence of an ap-

180

plied electric field related to movement of some part of their electrical double lay-

er. Those phenomena are collectively defined as electrokinetic phenomena and

include the following kinds:

1. Electrophoresis is the movement of a charged interface (usually colloidal

particles or macromolecules) with its electrical double layer together relative to a

stationary liquid under the influence of an ap-

plied electrical field. Electrophoresis was dis-

covered in 1808 by F. Reiss by the following

experiment (Fig.7.6). Two glass tubes full of

water were immersed into the wet clay and were

being under the action of constant electricity

current. Under the condition water in anode part

is feculent because the particles of clay migrated

to the positive electrode, and, hence, are nega-

tively charged.

2. Electroosmosis is the movement of a liquid relative to a stationary

charged interface under the influence of an applied electrical field. As it shown on

the Fig.7.7, the level of water in cathode part is raised, i.e. the solvent of the colloid

solution is moved through the pores of a diaphragm of very fine particles under the

influence of an applied potential gradient.

Particle electrophoresis has provide to be

very useful in many areas of theoretical and prac-

tical interface and colloid science, including

“model” polymer latex, water purification, deter-

gency, emulsion science, the characterization of

bacterial surfaces, blood cells, viruses, and so on.

It allows for the separation and identification of

components that would be extremely difficult or

impossible to separate using other techniques. It is

especially applicable to biological systems where

sample availability may be a problem.

§ 9. The structure of a double layer

Electric charges appear on the colloid particles as a result of adsorption pro-

cess, when the particles electoral (selectively) adsorb ions of a given species from

Fig.7.7. The experiment of

electroosmosis

Fig.7.6. The experiment of

electrophoresis

181

the solution, and due to the particle acquiring a charge suitable of a given ion.

What kind of ions is adsorbed determined by Panet’s rule, which is formulated as

following: ions that are preferentially adsorbed on the surface of the particle and

that determine the sign of the charge on the particle usually have an element in

common with the particles itself. In other words, the surface of solid substance ad-

sorb those ions which exist in the structure of later, i.e. those ions which can con-

tinue the crystalic net of the solid substance. For example, if the colloid particles

consists of the numbers of AgCl molecules, and there are Ag+ and NO3– ions in the

solution, so the Ag+ ions are selectively adsorbed on the surface of the particles,

because such species there are in the particles. These ions, which formed the first

rank of the double electrical layer, are called potential-causing ions. Because the

particle is acquired a charge suitable of the potential-causing ions, oppositely

charged ions being in the solution, called counter ions, are attracted to them and

form the second film of the double electrical layer. So, the ions of the first rank are

connected with the surface of hard phase by adsorption forces and the ions of the

second layer are bound due to adsorption and electrostatic forces. The double elec-

trical layer is a neutral system, because amounts of opposite charges in it are equal.

Double electrical layer can occur not only because of ion adsorption, but also

because of the electrolytic dissociation of the hard phase surface layer’s molecules

as well. For instance, in case of H2SiO3 sol, the surface molecules are dissociated

and obtained SiO3-2 ions form the first rank of the double electrical layer (potential-

caused ions) and H+ ions – the second one (counter ions).

There are a few theories about the arrangement of ions in the double electri-

cal layer, i.e., about its structure.

According the Helmholtz model the solvated

ions range themselves along the surface of the particle

but are held away from it by their hydration spheres

(Fig.7.8.). This theory considers a double electrical

layer as consisting of two parallel planes of opposite

charges as illustrated in Fig.7.9.a, hence it is assumed

that the electrical potential in the solution surrounding

the surface falls off exponentially with distance from

the surface. This model ignores the disrupting effect of

thermal motion, which tends to break up and disperse

the rigid outer plane of charge.

In the Gouy-Chapman model is taken into ac-

Fig.7.8.The Helmholtz

model of a double electri-

cal layer

182

count the disrupting effect of thermal motion due to counter ions of the second film

of the double electrical layer is arrangement not as the plane film but formed diffu-

sion layer and the drop in potential occurs not linearly (Fig. 7.9.b).

Neither the Helmholtz nor the Gouy-Chapman model is a very good repre-

sentation of the structure of the double layer.

Nowadays the Stern model, in which the double layer consists of two films,

is assumed. The closest to the particle surface film is called adsorption layer and

contain potential-causing ions and some of the counter ions. The ions in the liquid

phase are under the influence of two forces in opposite directions – electrostatic

and diffusion due to which a diffuse atmosphere creates nearby the adsorption lay-

er. The outer film layer is called diffusion layer and contains the rest amount of the

counter ions (Fig.7.9.c). A common amount of counter ions in both adsorption and

diffusion layers are equal to the amount of potential determining ions.

§ 10. The electrical potentials at the double layer

Because of such structure of the double electrical layer, two kinds of poten-

tial difference must be distinguished. One of them is the total potential difference

between the surface of the particle and the bulk of the solution. It is called thermo-

dynamic or φ potential and formed due to the potential-causing ions and, hence,

has the same charge. The other called the electrokinetic or zeta potential, ξ, which

is formed due to two regions of charge in the double electrical layer. To understand

j

x

– +– +– +– +– +– +– +– +– +– +

j

x

– – +– – – +– – – – –

++

++

+ +

++

j

x

– – +– +– – +– – +– +– – +

++

+

+

a cb

solution

par

tic

le

I IIIII

–– +– +– – +–– +– +– – +

++

+

+

Fig.7.9. The models of a double electrical layer by Helmholtz (a), Gouy-

Chapman (b) and Stern (c)

183

it let us come back to the Stern model of the double

electrical layer structure (Fig.7.10). The same number

of the potential-causing ions (1) compensates the

numbers of counter ions found at the adsorption layer

(2). The rest amount of the potential-causing ions re-

mains uncompensated; hence, between them and the

rest counter ions at the diffusion layer (3) the potential

difference or zeta potential is formed. The adsorption

layer (immobile layer of ions) adheres tightly to the

surface of the colloidal particles and moves with them when displacement of col-

loidal particles takes place. We can say that this layer belongs to the solid phase

(colloidal particle). Hence, the movement of hard and liquid phases (or adsorption

and diffusion layers) relatively each other occurs along the so-called sliding sur-

face and the potential difference (zeta potential) appears at that time. For this rea-

son, zeta potential is also called the sliding surface potential. It appears only when

a movement of charged particles takes place in the system, i.e. in electrical field.

The zeta potential is always less then the φ potential because it’s a part of it,

and has the same charge of the potential determined ions. The value of zeta poten-

tial is defined by thickness of the diffuse layer. The thicker diffusion layer the larg-

er value of zeta potential (because the more numbers of counter ions in the diffu-

sion layer). One can say that moving of the particle occurs along the adsorption and

diffusion layers boundary due to the zeta potential appears there.

Presence and value of zeta potential plays a very important role in properties

and stability of colloidal systems, therefore let us consider the factors influencing

on zeta potential.

First of all it is effect of electrolytes. When an electrolyte is added to the col-

loidal solution increasing the concentration of the elec-

trolyte leads to a decrease in the zeta potential. It oc-

curs due to pressure effect of electrolyte ions (but such

ions the charge of which is the same with the counter

ions) on the ions of the diffusion layer and part of

them transfers through the sliding surface into the ad-

sorption layer (Fig.7.11). Therefore, the thickness of

diffusion layer becomes less, which also affects the

decrease in the zeta potential. If this process is contin-

ued it may lead all the counter ions passing to the ad-

Fig.7.10. Adsorption and

diffusion layers of DEL

Fig.7.11. The effect of

the electrolytes on the

zeta potential

–I

–I

–I

–I

–I

–I

+K +

K

+K

+K

+K +K

+K

{ {

+K+K

+K

AgI

ads.l. dif.layer

–I

184

sorption layer, hence to complete neutralization of the potential determined ions

and the zeta potential falls to zero. It said that a colloid to be in an isoelectric state.

If the addition of an electrolyte to a colloidal solution accompany selective

adsorption of one species of ion (in cases of polycharged ions) by the colloid parti-

cles, the process may not merely lead to a

decrease in the charge of the particles, but

even a change in the sign of the charge.

These ions are replaced by the potential-

causing ions from the surface and them-

selves becomes potential-causing ions.

Previous potential-causing ions become as

counter ions; re-charging of colloid parti-

cle occurs (Fig.7.12).

The influence of the ions on the zeta potential depends on a charge of the

ion, its size and hydration degree (we will speak about below).

The temperature also affects on the value of zeta potential. With temperature

increase the value of zeta potential is increased too. It is due to thermal motion ac-

tivation, which leads to the passage of a number of counter ions from the adsorp-

tion layer to the diffusion layer. The number of compensated ions in adsorption

layer becomes less, so the value of zeta potential is increased. However, if very

high increase in temperature occurs, the value of zeta potential may decreases due

to desorption of the potential determined ions from the particle surface. Thanks to

this the value φ potential is decreased, hence the value of zeta potential as well.

Dilution of the colloidal solution also leads to the increase of zeta potential

value. It occurs because under this condition diffusion forces become more inten-

sive, counter ions leave a diffusion layer and pass to the bulk of a solution, so a

new number of counter ions pass from the adsorption layer to the diffusion layer

and zeta potential value is increased. In infinite dilution the value of a zeta poten-

tial decreases due to desorption of the potential determined ions from the particle

surface.

Zeta potential value depends on the pH of the medium too. Its influence de-

pends on the charge of the colloid particle. For example, in case of the positive sol

of H2SiO3 with H+ as counter ions adding of H+ ions in the solution leads to the

zeta potential decrease (due to bonding them with SiO3-2 potential-causing ions).

§ 11. Methods of zeta-potential determination

Fig.7.12. Re-charged of colloid

particle

185

These methods are based on the ability of colloid particles movement in

electrical field according their value of zeta-potential.

Electroforetic method, in which measures the rate of particles (U) in electric

field and determine zeta potential by Helmholtz-Smoluchovski equation:

l

EU

4 7.9

where ε is the dielectric constant of the solution, E is the potential difference of

electric field, l is the distance between electrodes, η is the viscosity of the medium.

Electroosmotic method measures the volume rate of the solution (V) and de-

termined zeta potential by the equation:

4

iV 7.10

where χ is the conductivity of the solution, i is the density of the electrical current.

§ 12. Micellar theory of colloid particle structure

According of this theory colloidal solutions (it relates only to lyophobic sols)

consist of the micelles and the intermicellar liquid (solvent with dissolved inorgan-

ic species). Micelle is

the colloidal particles

with double electrical

layer. It has the fol-

lowing structure: the

centre of the micelle,

nucleus, is the origi-

nal particle com-

posed of a great

number of molecules

of a certain species.

Then nucleus with

adsorption layer is formed granule. In addition, granule with diffusion layer is

called micelle. Let us consider formation of the AgI micelle by the reaction AgNO3

+ KI = AgI + KNO3 potassium iodide and silver nitrate are in equivalent concen-

trations so the sediment of AgI is formed. However, if the solution contains a slight

excess of silver nitrate then the following colloidal solution is formed (Fig.7.13).

mAgI

NO3–

NO3–

NO3–

NO3–

NO3–

NO3–

movin

g

bounda

ry

gra

nule

nu

cle

us

mic

ell

e

Ag+

Ag+

Ag+

Ag+

Ag+

Ag+

Ag+

NO3–

Ag+

Ag+

NO3–

NO3–

Fig.7.13. The structure of the micelle of AgI sol

186

The nucleus of each particle of this sol is composed of a great number of molecules

of the AgI. Let m be the average number of such molecules. The nucleus has a

crystalline structure. It is composed of fine crystallites that preferentially adsorb

from a surrounding solution the same ions as are contained in the lattice of the

crystallites and capable to complete its structure. Hence, when the surrounding so-

lution contains Ag+, K+ and NO3– ions, silver iodide particles preferentially adsorbs

Ag+ ions (potential-causing ions). Let n be the average number of such ions. The

counter ions will be NO3–, which average number also is n. But some of them are

located in the diffusion layer (x) and the rest amount in the adsorption layer (n–x).

The structure of micelles is expressed by micellar formulas, which for considered

case have the following form:

{m(AgI) nAg+ (n–x)NO3–}x+ xNO3

–

In the other case, if the solution contains a slight excess of potassium iodide,

so the surrounding solution contains K+, NO3– and I– ions. Hence, silver iodide par-

ticles preferentially are adsorbed I– ions, and the micelle will have the following

form:

{m(AgI) nI– (n–x)K+}x– xK+

We can see that in the case of the same different micelles with different

charge in dependence of the conditions colloid may be formed.

In case of sol of BaSO4 obtained by reaction BaCl2 + Na2SO4 = BaSO4 +

2NaCl when BaCl2 is in excess amount the micelle has the following form:

{m(BaSO4) nBa2+ 2(n–x)Cl–}2x+ 2xCl–

Due to such structure of a micelle, one can explain the phenomenon of an

electrophoresis. In the figured braked there is a granule with accordingly charge,

due to a granule (consists of the original particle, adsorption layer and molecules of

hydrates water) is moved in the electrical field toward to the corresponding elec-

trode.

§ 13. Stability and coagulation of colloids

Stability of colloid systems means the stability of their properties for a pre-

cise time. It is related to a degree of dispersion and uniform distribution of particles

through the solution, i.e., there mustn’t be increase in size of particles and sediment

formation. In terms of colloids stability, we can say the following: interfacial ener-

gy consideration dictate that the “position” of lowest energy for a given system will

be that in which there is a minimum in the interfacial area of contact between phas-

es. Put another way, in the absence of another factors, colloids should be unstable

187

and rapidly revert to a state of complete phase separation. However, Nature has

designed things in such a way that we can impose barriers of various types between

matastable and stable states so that useful (and vital) colloidal systems can be made

to exist and persist for enough time so that tyhey can serve a useful function (like

make up a significant part of our biological systems!)

Colloid systems widely differ in stability. Some can be preserved unchanged

for a long period of time (lyophilic sols), which are formed spontaneously and exist

without stabilizers, such as solutions of surfactants and polymers. Others are com-

paratively unstable (lyophobic sols), being more sensitive to various influence,

such as colloid solutions, emulsions, suspensions, etc. This instability can be ex-

pressed thermodynamically by noting that, because of the large summary surface

area in these systems. Gibbs’s energy of surface also has a great value. From ther-

modynamics, we know that systems with great value of free energy aspire to de-

crease it by some spontaneous process. And because ∆G=σ∆S, consequently

Gibbs’s energy could be decreased by the way of either σ or S decreasing. Really,

decreasing of σ occurs by adsorption process, and decreasing of S occurs by colloid

particles size increasing (they spontaneously aspire to joins with more big particles

formation).

There are two kinds of processes leads to the disintegration of colloid system

and that under certain conditions can take place spontaneously. These are sedimen-

tation and coagulation processes. In sedimentation processes, the particles of the

dispersed phase settle out or rise to the surface of the system, depending on the ra-

tio of the densities of the dispersed particles and the dispersion medium. In coagu-

lation processes, the particles of the dispersed phase adhere to one another increas-

ing in size.

There are the concepts of kinetic and aggregate stability of colloid systems,

which characterize their stability according to sedimentation processes and to a

change in particle size.

The ability of dispersed phase to stay equally diffused in dispersed medium

is called the kinetic stability of the system. Two conflicting processes determine

kinetic stability: sedimentation and thermal motion of the particles of the dispersed

phase. The particles doing movement tend to spread equality in the whole volume

of the system. If the diffusion forces are larger than the gravitation forces of the

Earth, the system is kinetically stable. Kinetic stability depends on the particles

size; the smaller particles size, the more stability. The molecules of the dispersion

188

medium continually collide with the dispersed particles, which in view of their

small size are thereby kept in a state of suspension.

Aggregate stability is a measure of the ability of a colloid system to preserve

its degree of dispersion. It is because the particles of the dispersed phase are elec-

trically charged and are surrounded by a solvate (or in case of water – hydrate)

shell. In comparatively stable colloid systems, the particles of the dispersed phase

usually acquire an electric charge of the same sign for all the particles in the given

system. This is easily demonstrated by applying a steady electric field to the colloid

system, upon which all the particles of the dispersed phase are displaced toward

one of the electrodes. So, as all the particles of the dispersed phase acquire an elec-

tric charge of the same sign, they repulsion each other and prevent from their asso-

ciation. It is the main factor of the colloid systems stability. Solvation (hydration)

of the particles (more precisely to say – of the ions surrounding particles) is the

second factor of the stability. When these solvate layers of particles approach due

to their flexible properties repulsion each other occurs and prevent from their asso-

ciation.

The stability of the colloid systems one could provide with addition to the

colloid system the certain species, which are called stabilizers (such us surfac-

tants). They are adsorbed on the surface of the particles and the result will be a sys-

tem with a free-swimming “tail” projecting into the solution providing the protec-

tive action. In addition, with an increase in lyophilic properties of the system, the

role of the charge on the particles becomes relatively less important, while that of

solvation becomes more important.

If stability of colloid system is got broken coagulation process occurs; the

particles of the colloid system combine into larger particles. It occurs spontaneous-

ly due to the great Gibbs free energy; the state of a system with a highly developed

surface is always less stable than a state with a smaller surface. The rate of coagu-

lation will be controlled entirely by diffusion kinetics, analogous to a diffusion-

controlled bimolecular reaction.

Coagulation may occur under condition, which is broken the stability of the

colloid system: temperature changing, mixing of the system, influencing of the

electricity or lighting. However, the most important way of coagulating lyophobic

sols is by the addition of an electrolyte. It occurs, as we consider above, due to in-

fluencing on the zeta potential, value of which becomes less and the main factor of

colloids stability gets broken.

189

Investigations of the coagulation of lyophobic sols by electrolytes have led

to the following conclusions:

1. All electrolytes, in sufficient concentration, are capable of coagulating

lyophobic sols.

2. The coagulating action of an electrolyte is due to the ion whose charge is

opposite in sign to that of the colloid particle.

3. The effect of coagulation may be visible to the naked eye (apparently by

turbidity of a solution) and is known as the stage of visible coagulation. In contrast,

in the stage of invisible coagulation the process cannot be detected from the ap-

pearance of the colloid system (in this case, coagulation begins but the size of the

particles doesn’t reach the visible size).

4. Visible coagulation begins when the concentration of the electrolyte (mol

electrolyte/dm3 of sol) exceeds a certain minimum value of the electrolyte, called

the coagulation threshold (mol electrolyte/dm3 of sol). Thus, one may characterize

the stability of a charge-stabilized colloidal system by its critical coagulation con-

centration (ccc), the concentration of electrolyte necessary to bring the system into

the regime of rapid coagulation.

The effect of coagulation depends on the coagulating ion size and valence.

According Schulze-Hardy’s rule the larger the valence of the coagulating ion the

stronger effect of coagulation and the smaller coagulation threshold. For example,

the coagulation effect (coagulation threshold) of ions of different valence K+, Ba+

and Al+ is accordingly 540:7,4:1.

For the univalent ions, the coagulation effect is non-equal because of their

different size. Thus, the cations of the alkali metals form the series in the order of

decreasing coagulating ability:

Cs+>Rb+>K+>Na+>Li+

Such series are called lyotropic.

Coagulation rate dependence on the amount of an electrolyte is apparent on

the Fig.7.14. Segment OS corresponds to invisible coagulation. During this period

the aggregation of particles takes place and a decrease of the dispersion degree, but

there are not any visible changes. This period is called hidden period of coagula-

tion (invisible stage). Point S is the coagulation threshold point, after which the

visible coagulation occurs. During the segment OK the rate of the coagulation in-

creases with the electrolyte concentration increasing. Point K is an isoelectric state

of the colloid system, when zeta potential is equal to zero. After this point, coagu-

lation occurs independently on the presence of an electrolyte.

190

Coagulation may occur not only by act-

ing of one electrolyte but also by a mixture of

electrolytes. In such cases, three types of co-

agulation could be observed:

1. Additive coagulation occurs when

electrolytes are close in nature, such as NaCl

and KCl. In this case the coagulation threshold

is intermediate in value between the coagula-

tion thresholds of these electrolytes when used

separately.

2. Much more frequently, the coagulat-

ing power of a given electrolyte decreases

when it is used with another electrolyte together. This phenomenon is called an-

tagonist coagulation.

3. In synergist coagulation the coagulating power of a given electrolyte in-

creases when another electrolyte is added to it.

Coagulation of a sol can be brought about by the addition of another sol

whose particles are oppositely charged (due to attraction between oppositely

charged particles). This is known as the mutual coagulation of lyophobic colloids.

Example: Coagulation of 10 ml sol of Fe(OH)3 begins from 2ml of 0.00125

mol/dm3 Na2SO4 solution. Calculate the coagulation threshold point of electrolyte.

Solution: In 2ml of 0.00125 mol/dm3 Na2SO4 solution contains the following

amount of electrolyte: 20.00125/1000 = 2.510–6 mol. If the unit of coagulation

threshold is mol electrolyte/dm3 of sol, hence for 1000ml of sol needs 2.510–

6100 = 2.510–4.

§ 14. Protection of colloids

For greater stability of colloids some special substances, which are called

protective species or stabilizers, are added to colloids. Usual-

ly it is high-molecular compounds (proteins, gelatin, albu-

min, polysaccharide), which are adsorbed on the surface of

the colloid particles forming protective films that stabilizes

the interface and cannot be penetrated when two particles touch each other.

One must be careful in the precise amount of the stabilizer. When a protec-

tive species is applied in an amount too small for protection it may, on the contrary,

C

N

S

K

slow fast

invisible visible

V

0

Fig.7.14. Coagulation rate de-

pendence on the amount of an

adding electrolyte

191

reduce the stability of the sol. This increase in sensitivity of the sol is called sensi-

tization. It occurs as the small amount of the polymer of high molecu-

lar weight as stabilizer is insufficient for all particles surfac-

es adsorption and one molecule of the stabilizer could simul-

taneously adsorb on a few particles surfaces promoting faster coagula-

tion.

§ 15. Peptization

Peptization (resolution) is the disintegration of the coagulation product

(fresh-forming sediment) into a sol under the action of compounds called pep-

tisate)rs. Their main function is to decrease significantly the intermolecular adhe-

sion forces. Peptization is divided into two groups: adsorption and ablution. For

example, a deposit of ferric hydroxide can be peptizated by treatment with very

small quantities of a ferric chloride solution after preliminary removing of the co-

agulating substances. In this case, peptization is due to adsorption of Fe3+ ions on

the surface of Fe(OH)3 particle, due to which all of them acquire the positive

charge, repulsion from each other, and a deposit is transformed to a sol. Thus, pep-

tization occurs if there are no electrolytes in the surrounding medium for zeta po-

tential formation and peptizators in such cases are usually electrolytes.

In other cases (ablution peptization), when there was an excess of electro-

lytes in the solution, as a result zeta potential falls to zero, and a deposit is formed.

The excess of electrolytes is washed with water to remove them and form diffusion

layer.

Review questions:

1. What indicates electrokinetical phenomena in colloids? What do these

phenomena mean?

2. Define the process of electrophoresis and electroosmosis.

3. Explain the mechanism of double electric layer formation.

4. Under which forces the potential-causing ions and counter ions are at-

tracted to the solid surface.

5. Explain the structure of double layer by Stern theory.

6. Which ions are located in the adsorption and diffusion layer?

7. What kinds of potentials are formed in double layer?

192

8. What potential is called thermodynamic, where is it formed, which

charge does it carry and from what does its value depend on?

9. What potential is called electrokinetic, on which boundary is it formed,

which charge does it carry and from what does its value depend on?

10. How and why does the thickness of diffusion layer influence on the val-

ue of ξ potential?

11. Why is electrokinetic potential also called the sliding surface potential?

12. How does addition of electrolytes influences on the value of ξ potential?

Explain its mechanism of action. What ion of added electrolyte influences on ξ po-

tential?

13. What position do the ions of double layer assume in isoelectric state of

colloid?

14. How does re-charging of colloid particle happen?

15. How do the following factors influence on the value of ξ potential: tem-

perature, dilution, pH? Explain the action mechanism.

16. Explain the structure of micelle according the micellar theory of colloids.

17. What charge do granule and micelle carry? How can the phenomenon of

electrophoresis by micellar structure of colloids be explained?

18. What is writing order of micellar formule? Write an example of micellar

formule in case of BaSO4 colloid if it was obtained by reaction BaCl2 + H2SO4 =

BaSO4 + 2HCl when BaCl2 is in excess amount.

19. Are colloidal systems stable or not? Explain it. What properties do we

mean when we speak about the stability of a system?

20. Define the kinetic and aggregate stability of colloids.

21. What does the main factor of colloids stability mean? Explain its role for

stability.

22. What does the second factor of colloids stability mean and how is it af-

fected?

23. What is the role of stabilizers for stability? What kind of substances are

they?

24. What is coagulation? Why does it spontaneously occur in colloids?

25. Explain the mechanism of coagulation by electrolyte addition. What ion

of electrolyte affects coagulation? What does it mean visible and invisible coagula-

tion?

26. Give formulation of coagulation threshold. In which units is it ex-

pressed? What properties of coagulating ion does it depend on?

193

27. Draw a diagram of coagulation rate dependence on concentration of add-

ed electrolyte. What occurs in points S and K?

28. How is a protection of colloids made? Explain the mechanism of stabi-

lizers acting.

29. Define a process of peptization. Explain the mechanism of adsorption

and ablution peptization.

OTHER TYPES OF DISPERSE SYSTEMS

§ 16. Suspensions

Suspensions are widely applied in different areas of industry and medicine,

particularly in pharmacy in the form of different drugs.

Suspensions, like colloid solutions are the systems with solid dispersed

phase and liquid dispersion medium but with larger size of particles. Due to it, mo-

lecular-kinetic and optical properties of suspensions vigorously differ from those of

colloid solutions. The process of osmosis and diffusion is not peculiar to suspen-

sions. The beam passing throughout a suspension is not accompanied by opalescen-

tion; only turbidity occurs due to light reflection.

Suspensions to some extent possess an aggregate stability because of double

electric layer presence. The kinetic stability is absence here; due to a large size of

particles, sedimentation of them takes place rapidly. As suspensions are instable

systems, the application of stabilizers must accompany to obtain them.

In suspensions of high concentration paste is formed. In paste because of a

large amount of dispersed phase almost all dispersion medium is bonded with par-

ticles and only thin films of solvent divides particles from each other. Due to it

such systems possess of high degree of viscosity and any mechanical properties.

When in suspension of high concentration the links formed between the par-

ticles of dispersed phase the gels are formed. Gels are capable to reverse transfor-

mation to sols under action of external forces, which destroy the links into the gel.

If the liquid of gel is removed, the gel holds the shape of essential gel and is called

xerogel (dry gel). Xerogels have a high degree of perforation, hence the high value

of specific surface, which allows applying them as adsorbents (silicagel, alu-

minagel etc.).

194

§ 17. Emulsions

Emulsions are the systems in which the dispersion medium and the dispersed

phase are liquids. Emulsions are lyophobic colloids.They are heterogeneous mix-

tures of at least one immiscible liquid dispersed in onother in the form of droplets,

the diameter of which are, in general, greater than 0.1 μm.

Emulsions are widely spread among various naturally occurring materials

and industrial products (milk, yolk, cream, margarine, milky sap of plants, petrol,

latex, etc.). Many food products as well as pharmaceutical preparations and cos-

metics (salves, ointments, cold creams) are also emulsions.

Formation of emulsions supposes the comminuting of liquid phase in liquid

media and is generally referred to as “emulsification”. These include spontaneous

emulsification, electroemulsification and spontaneous microemulsion formation.

The preparation of an emulsion requires the formation of a very large amount of

interfacial area between two immiscible liquids.

Emulsions are classified according two principles: the liquids polarity and

the concentration.

By polarity, the liquids in most emulsions are water and oil, where “oil” de-

notes an organic liquid essentially immiscible with water. Such emulsions are clas-

sified as either oil-in-water (O/W) emulsions (the straight emulsions) or the first

type, in which water is the continuous phase and oil is present as tiny droplets. The

second type or reverse emulsions are water-in-oil (W/O), in which oil is the contin-

uous phase.

In the later years has been published about more complex systems generally

referred to as multiple emulsions. Multiple emulsions, as the name emplied, are

composed of droplets of one liquid dispersed in larger droplets of a second liquid,

which is then dispersed in a final continuous phase. Typically, the internal droplet

phase will be miscible to the final continuous phase. Such systems may be W/O/W

emulsions as indicated in Figure, where the internal and external phases are aque-

ous; or O/W/O, which have the reverse composition. Although known for almost a

century, such systems have only recently become of practical interest for possible

use in cosmetics, pharmaceuticals, controlled drug delivery, etc.

According to the concentration, three types of emulsions are distinguishes:

1. Dilute emulsions, in which the concentration of a dispersed phase is not

more 0,1% of the volume. These systems are stable and might be obtained without

emulsifying (stabilization) agents. The particles (drops of a dispersed phase) in

195

such solutions have a spherical form and uniformly distribute among a dispersion

medium.

2. In concentrated emulsions, the concentration of a dispersed phase is run

up to 74%. The particles keep their spherical form, though they achieve a high de-

gree of packing. To obtain such emulsions the presence of emulsifying agent is ob-

ligatory. When the difference between the densities of the phases is great, the parti-

cles may rise to the surface or settle on the bottom of the vessel. Even so they re-

tain the structure of emulsion, except that the concentration is higher (as in the

cream or milk).

3. In highly concentrated emulsions, the dispersion medium sometimes re-

mains only as a thin film between the particles of the emulsion, which prevents

coalescence (junction) of the particles. In such systems, the emulsion particles are

deformed; the contact area between neighboring particles becomes flat, and the

particles take the form of irregular polygons separated by a thin film of the disper-

sion medium.

For an emulsion to be stable, two liquids must be practically insoluble in

each other or only very slightly soluble. Besides, an emulsion is made more stable

when certain substances are introduced into the system are adsorbed on the inter-

face and reduce the interfacial tension. Such substances are called emulsifying

agents or emulsifiers. Usually they are surfactants, polymers or powders. Protein,

gelatin, casein are good emulsifying agents in an aqueous medium. For instance,

fatty emulsion of cow’s milk (O/W emulsion) is stabilized by casein; in natural

latex (the milky sap of rubber plants) proteins are again the emulsifying agents.

The cleansing action of soaps and other detergents results in part from their acting

as emulsifying agents to keep tiny droplets of grease suspended in water.

The stabilizing effect of such agents may be explained by different mecha-

nisms:

1. Nonionic surfactants are adsorbed on the interface and reduce the interfa-

cial tension. The way of adsorption depends on the nature (polarity) of the liquids.

In the O/W emulsions, hydrocarbon tails of

the surfactants direct to the oil particles

(Fig.7.20.a), and in the reverse emulsions –

vice versa (Fig.7.20.b).

2. Ionic surfactants cause the emul-

sion particles to acquire electric charges of

the similar sign imposing a slight electro-

a b

Fig.7.20. Different arrangement of

the surfuctunts on the interface of

emultions of different types

196

static barrier between approaching drops.

3. The interface becomes covered with a compact film of the emulsifying

agent having a certain mechanical strength. Such films prevent the particles of the

emulsion from merging (coalescing) upon contact. For this reason, soaps and other

substances that form strong films used for stabilization, especially in concentrated

emulsions, where emulsifying agents that only charge the particles electrically are

not enough to stabilize the system.

4. Certain solid powdered materials can also act as emulsifying agents. Their

effect is usually because they are wetted selectively by one of the phases (the dis-

persion medium), as a result of which, upon adsorption on the interface, they are

also wetted by the dispersed phase and envelop the emulsion particles with a film.

In general, they while not directly affecting interfacial tension, can stabilize an

emulsion by forming a physical barrier between drops, thereby retarding or pre-

venting drop coaleascence. Hydrophilic materials (such as chalk, gypsum, glass)

are used to stabilize O/W emulsions, and hydrophobic (soot, lead) for W/O emul-

sions.

Without stabilizing agent coalescence occur in emulsions. Coalescence re-

fers to the joining of two (or more) drops to form a single drop of greater volume,

but smaller interfacial area. Such a process is obviously energetically favorable in

all cases in which there exists a positive interfacial tension.

In emulsions the process might occur, which is called reversal of emulsions.

In this process the dispersion medium and the dispersed phase change roles; for

instance, the transformation of a W/O emulsion to an O/W. An example of such a

transformation occurs when cream is shacked up to give for butter. Since some

emulsifying agents are more active for W/O emulsions and others for O/W ones,

the transition of the emulsifying agent (in the emulsion) from one type to another

facilitates the reversal of the emulsions. For example, O/W emulsions that are sta-

bilized by a sodium soap can be transformed to W/O type emulsions by introducing

a calcium salt; this leads to the formation of a calcium soap, which, owing to its

oleophilic properties, is a better stabilizer of W/O type emulsions.

§ 18. Foams

Foam is a colloidal system in which gas bubbles are dispersed in a liquid or

solid medium. Although the diameters of the bubbles usually exceed 104Ǻ, the dis-

tance between bubbles is usually less than 104Ǻ, so foams are classified as colloi-

197

dal systems. Their peculiarity is the excess of the volume of dispersed phase in

comparison with dispersion medium many times.

Like other colloidal systems, foams may be formed either by dispersion or

condensation processes. The formation of the “head” on a glass of beer is a classic

example of foam formation by condensation. In such a system, when the bottle are

opened, carbon dioxide produced by fermentation in the container and solubilized

under pressure is liberated. The solution becomes supersaturated, and the excess

gas forms a dispersed phase which rises to the top and forms the head. Many indus-

trial processes for the formation of solid foams employ a similar process in which a

“blowing agent” is added to the polymerizing system creating the foam. Foam for-

mation involves a large increase in the surface of the liquid, which requires that

work be done; this process takes place more easily, the lower the surface tension of

the liquid. Hence, the addition of surfactants in general should increase the ability

of a liquid for foam formation. To be true, the formation of stable foams is also

promoted by an increase in the viscosity of the liquid, a decrease in its volatility,

and by the mechanical strength of the foam.

A primary characteristic of foams is that they have very low densities. Relat-

ed to this foams will have a large surface area for a given weight of foam.

Like almost all systems containing two or more immiscible phases, foams

involve thermodynamic conditions in which the primary driving force is to reduce

the total interfacial area between the phases – that is, they are thermodynamically

unstable. In spite of their ultimate tendency to collapse, however, foams can be

prepared that have a lifetime (persistence) of minutes, days, or even months. Due

to it stabilizers (which are called “foam makers”) are required for their formation.

They are usually surfactants, which are adsorbed on the interphase boundaries

forming the jelly-like film which does not allow the particles junction.

The presence of foam in an industrial product or process may or may not be

desirable. Foams have wide technical importance, as such in the field of fire

fighting, polymeric foamed insulation, in flotation processes, in detergents proper-

ties improvement, in food and cosmetic industry. Some of pharmaceutical prepara-

tions are also foams (for skin treatment, as tourniquet, etc.).

198

§ 19. Aerosols

Dispersions, emulsions and foams are the most commonly treated and in-

tensely studied examples of colloidal systems. There exists one another class of

true, lyophobic colloids – the aerosols – which, although seemingly less importent

in a theoretical or applied sense, are of great practical important. Aerosols are sys-

tems in which there exists a condensed phase of one material (solid or liquid) that

is dispersed in a gaseous phase and that has dimensions that fall into the colloidal

range. There are two subclasses of aerosols depending on whether the dispersed

phase is a liquid or a solid. Where the dispersed phase is a liquid, the system is

commonly referred to as a “mist” or a “fog”. For solid aerosols, one can commonly

refer to a “dust” or “smoke”. One of the examples of the natural aerosol is a cloud

that is a large collection of water droplets or ice crystals moving through the at-

mosphere.

Aerosols, both liquid mists and solid smokes, have a great deal of technolog-

ical and natural importance. They are usefully employed in coating operations, fire-

fighting, medical treatments (allergy and asthma sprays), chemical production pro-

cesses, spray drying, and other procedures. On the opposite side of the ledger, we

have the smoke, smog and haze from industry and automobiles, forest fires, chemi-

cal and biological weapons, and so on. Today, probably the most visible aerosols

are those resulting from air pollution. The effects of pollution on human health, on

vegetation, and on atmosphere itself are more apparent and frightening ever day.

Preparation of aerosols is the same for such of colloids – by dispersion and

condensation methods. The majority of industrial aerosols (both wanted and un-

wanted) are produced by processes of dispersion in which small particles are

formed from larger solid masses. Condensation aerosols, on the other hand, are less

common but include those formed by the chemical reaction of one or more gaseous

materials or by oxidation. The condensation methods can be divided into two clas-

ses: chemical and physical condensation processes. A typical physical method may

involve the heating of a material of relatively low volatility sufficiently to produce

a high degree of supersaturation and passing the vapor into a stream of cold gas,

rapidly condensing the vapor into a solid aerosol.

A “chemical” method for the production of liquid aerosol involves the direct

condensation of drops or particles in the air or other gaseous environment. In order

for a vapor to condense, certain conditions must be fulfilled. If the vapor contains

no foreign substances that may act as nucleation sites for condensation, the for-

199

mation of aerosol drops will be controlled by the degree of saturation of the vapor.

The formation of a new phase involves first the formation of small clusters of mol-

ecules which than may disperse or grow in size until some critical size is reached,

at which point the cluster becomes recognizable as a liquid drop.

One or two processes, depending on whether the dispersed system begins as

a liquid or undergoes a phase change from vapor to liquid during the formation

process, may form liquid aerosols. Liquid aerosols formation by spraying widely

applies in many industrial processes, and particularly in medical nose and throat

sprays preparation.

While aerosols are typical colloids in that they respond to the same forces al-

ready introduced, but the special conditions that prevail in terms of the intervening

gaseous medium results in an apparent qualitative difference from colloids in liquid

media.

Optical properties of aerosols, although the fundamental rules remain the

same, are more expressed than in liosols due to great difference in refraction coef-

ficient of dispersion medium and dispersed phase particles.

Molecular-kinetic properties of aerosols also obey to the fundamental rules

but diffusion and sedimentation are more expressed here. It is because in aerosols

the density and viscosity of the continuous phase (medium) is always significantly

less than that of the dispersed particle. Even under ideal conditions, the dynamic

flow behavior of aerosols in contrast to other colloids can be markedly different.

For example, in water, with a viscosity approximately 50 times that of air, mineral

particles would have sedimentation rates on the order of 1.6 m day–1 and 23 m day–

1, respectively. Such calculations are important for modeling problems of sediment

accumulation in different reservoirs.

Small temperature differences or mechanical movements may be translated

over large distances in gases and produce a much greater effect in aerosols. The

movement of particles in the line of the low temperature is called thermophoresis.

The movement of particles due to different lightening of aerosol is called photo-

phoresis. Besides, a gaseous medium, because of its very different unit dencity,

dielectric constant, and other properties, is very ineffective at screening the forces

acting between colloidal particles and between particles and surfaces of the materi-

als involved. The later is called thermoprecipitation. Such attractive interactions

can be particularly important in situations where the presence of even a few extra-

neous particles on a surface can be highly detrimental, as in the production of mi-

crochips for the electronic industry.

200

In aerosols one have to take into consideration the free path, λ, that is the av-

erage distance a particle will travel before colliding with another particle and a cor-

rection factor, Cc (the Cunningham correction factor) can be incorporated into the

Stokes equation to give

F=6πηrV/ Cc

Obviously, the correction given in this equation becomes more important for

aerosols of smaller particle radius, or in conditions of lower gas pressures.

Electric properties of aerosols. As a practical matter, almost all aerosol

particles will rapidly acquire an electric charge leading to electrostatic interactions.

The mechanisms for acquiring charge in aerosols are basically the same as those

described for liosols, although direct ionization by dissociation will be of minor

importance because of the lack of ionizing solvent. Perhaps most important are

charge acquisition due to friction, electron gain or loss due to collision with ioniz-

ing radiation, and adsorption of ions from the air. It is estimate that a cubic meter

of “normal” city air will contain 108 ions (both positive and negative) producing a

charged surface. The larger the aerosol particle, the more charges it can accumu-

late. In fact, nevertheless, that the coulombic interactions between charged aerosol

particles has little influence on particle–particle interactions at distances much

greater than those at surface contact do do.

The presence of charge on aerosol particles does have its important implica-

tions, however. Many practical applications of aerosols depend on the presence of

charge, as does one of the most important processes for destroying unwanted aero-

sols. It takes a special significance in the natural aerosols of a large size as a cloud.

A cloud is a large collection of water droplets or ice crystals moving through the

atmosphere. Due to mechanisms described above the upper and lower layers of a

cloud acquires an opposite charges that becomes a reason of stormy discharge.

§ 20. Powders

Powders can be regarded as aerosols with a great part of solid dispersed

phase. Solid particles in powders (sand, dust) are in contact to each other and a sur-

face of contact is an important characteristic of such systems. It could be determine

by the following equation: y = hnc/d, where h is the height of contact between two

particles, nc is the number of contacts in the system, d is the diameter of a particle.

The less the size of a particle the greater a surface of contact correspondingly the

201

less a specific surface, which is important for adsorption properties of powder ad-

sorbents.

There are any properties of powders having a great importance in powders

technology, preparation and applying processes. They are the following:

Adherence of particles to each other or to certain surfaces. This property is

important when medical (drug) powders are applied for surface of wound treat-

ment. Particularly important are pesticide dusts that may be applied to wide areas

by airplane or surface dispersal techniques.

Filling volume denotes the mass of powder be placed in a vessel of a cer-

tain volume.

Fluidity shows the ability of powder flow under external force affection.

Scattering in the surrounding that is important in aspects of air pollution

and technological safety.

Wettability of powders is important for keeping their physical and chemi-

cal properties in safety.

Granulation of powders is a process of their transformation in form of

granules under weak forces affecting. It is applied for a transportation powders

without quantitative losses, and besides granulation is the intermediate stage in

pills preparation.

§ 21. Solutions of surfactants

As pointed above, a colloid is characterized by a particle size range of about

10–7 to 10–3 cm. Using that size rang as the only criterion, we can define two types

of colloids: the lyophobic (solvent-hating) and the lyophylic (solvent-loving) col-

loids. The lyophobic colloids, as previously pointed out, are normally formed by

the comminution of coarse particles to achieve a desired particle size or by con-

trolled growth from solutions of small molecules or ions. The lyophylic colloids,

on the other hand, are composed of either solutions of large molecules or reversible

associated or aggregated structures formed spontaneously in solutions of certain

types of molecules. The large molecules usually with molecular weight ranging

from about 5000 to seversl million) are, of course, macromolecules or polymers.

The best known association colloids are aqueous solutions of surfactants, although

certain dyes and drugs may also form association structures.

Some substances, such as surfactants, depending on concentration, might

form both real and colloidal solutions (lyophilic colloids). Their formation does not

202

result from the input of energy such as in commutation or emulsification; it is spon-

taneous association process resulting from the energetic of interaction between the

individual units and the solvent medium.

It is generally accepted that most surfactant molecules in aqueous solution

can aggregate to form micellar structures with an average of 30–200 monomers in

such a way that the hydrophobic portions of the mole-

cules are associated and mutually protected from ex-

tensive contact with the bulk of the water phase. In

certain concentration, which is called critical micelle

concentration (CMC), molecules of surfactant cluster

together, formed colloid-sized clusters of molecules as

micelles. In polar solvent, such as water, their hydro-

phobic tails tend to congregate, and their hydrophilic

heads provide protection (Fig.7.15). For instance, in soap micelle formation in

aqueous solution the hydrocarbon part of each monomer an-

ion is directed toward the center (forming hydrophobic me-

dium), and the polar COO– group is on the outside (in the

hydrophilic medium). The hydrocarbon tails are mobile, but

slightly more restricted than in the bulk.

In a nonpolar solvent reverse micelles are formed with

reverse arrangement of the surfactant molecules (Fig.7.16).

Formation of micelles and their destruction occurs

spontaneously and depends on the concentration and temper-

ature. Lyophilic colloids are thermodynamically more stable than the two-phase

system of dispersion medium and bulk colloid material.

Fig.7.15.

Formation of micelle in

polar solvent

Fig.7.16. For-

mation of micelle

in non polar sol-

vent

Fig.7.17. Construction of different types of micelles

203

Surfactants in dependence on their nature might be nonionic and ionic.

Nonionic surfactant molecules may cluster together in clumps of 1000 or more, but

ionic species tend to be limited to groups of less than about 100.

The micelle population is often polydisperse and the shapes of the individual

micelles vary with concentration. Spherical micelles do occur close to the CMC.

Some micelles at concentrations well above the CMC form extended parallel

sheets, called lamellar micelles, of two-molecule thickness (Fig.7.17). The individ-

ual molecules lie perpendicular to the sheets, with hydrophilic groups outside in

aqueous solution and inside in nonpolar media.

Such lamellar micelles show a close resemblance to biological membranes,

and are often a useful model on which investigations of biological structures are

based.

The investigation of such solutions properties is very important to determine

CMC. Micelle formation could change dramatically the properties of the solution

and CMC is detected by noting discontinuity in

physical properties of the solution at this concen-

tration, such as molar conductivity, surface tension

and osmotic pressure (Fig.7.18). The sudden

change n a measured property is interpreted as

indicating a significant change in the nature of the

solute species affecting the measured quantity.

The hydrocarbon interior of a micelle is like

an oil droplet. Due to it many organic species (hy-

drophobic), which cannot be solved in water, are

solved in the surfactants micellar solutions. This

phenomenon is called solubilization. It occurs due

to intestinal absorption of organic molecules (in

particular, fats) in the inner part of micelles

(Fig.7.19). For this reason,

micellar systems are used as

detergents and in water insoluble drug carrier, and for organ-

ic synthesis, petroleum recovery. It is very important also in

biology; for instance, solubilization of cholesterol in such

micelles aids in excretion of cholesterol from the body.

In general, extensive interest in the self-association

phenomenon of surface-active species is evident in such


CMC by changing in physical

properties of a solution

surfactant

solute

Fig.7.19. The

phenomenon of

solubilization

204

wide-ranging chemical and technological areas as organic and physical chemistry,

biochemistry, polymer chemistry, pharmaceuticals, cosmetics, food science, etc.

An interesting and important phenomenon related to the micellar solubiliza-

tion is that of the digestion and absorption of fatty nutrients by the body. Since fat

is a major source of energy, its transport through the digestive track and ultimate

transfer through the intestinal walls is obviously of great importance. While the

body produces several natural surfactants, the most important in terms of digestion

are the bile salts derived from cholesterol and lecithin. The bile salts are synthe-

sized in the liver and secreted into the upper small intestine in response to stimuli

indicating the presence of nutrients. When no nutrients are present, the acids are

normally concentrated and stored in the gallbladder. As fatty materials, usually in

the form of triglycerides, enters the intestine it is emulsified by muscular action of

the duodenal wall, which also causes the secretion of pancreatic enzymes that hy-

drolyze the triglycerides to produce fatty acids and 2-monoglycerides. At the same

time, the gallbladder releases bile acids into the intestine that, in combination with

lecithin already present and the 2-monoglycerides produced by hydrolysis form

mixed micelles that solubilize the essentially insoluble fatty acids.

Once incorporated into the micelles, the fatty acids are transported through

the small intestine during which time random motion and diffusion bring the mi-

celles into contact with the intestinal wall. Because the solubilized lipid/micelle

system is a dynamic structure, the fatty acids can be released and transferred

through the membranes of the epithelial cells where the are reesterified to produce

triglycerides again. The triglycerides then continue their journey to produce energy

(good) or be processed and deposited for future use (usually not so good).

When functioning normally, the human body uses the equivalent of approx-

imately 30 g of bile acid per day.

SOLUTIONS OF HIGH MOLECULAR SUBSTANCES

High molecular substances or macromolecules are those molecules of which

consist of a great number of simple structural units – monomers. There are macro-

molecules everywhere, inside and outside us. Life in all its forms, from its intrinsic

nature to its technological interaction with its environment, is the chemistry of

macromolecules.

205

Some of macromolecules are natural: they include polysaccharides such as

cellulose, polypeptides such as enzymes, and nucleic acids such as DNA as well as

proteins, carbohydrates, gums and other biocolloids. They are monodisperse,

meaning that it has a single, definite molar mass. Others are synthetic: they include

polymers such as nylon, polyethylene, and polystyrene. Synthetic polymers are

polydisperse in the sense that a sample is a mixture of molecules with various chain

lengths and molar masses. They are characterized by number-average molar mass.

Macromolecules formed true solutions, i.e., molecular-disperse homogene-

ous systems, and not colloids. But the characteristic properties of these systems,

unlike other groups of true solutions, are due mainly to the great difference in size

of the solvent and solute particles and to the structure of these particles, which are

in form of long flexible molecules (chains). Owing to the large size of the mole-

cules, solutions of such substances are in many respects similar to colloid systems

and like to the lyophilic colloids; they form a separate class known as high molecu-

lar solutions.

§ 22. Structure of polymers

.The properties of polymers depend on their intrinsic structure and primarily

on the type of the repeating unit, degree of polymerization, chain structure, and on

the nature and intensity of interaction of the chains.

According to their primary structure, polymers are classified on three types:

Linear or chain polymers consist of long hydrocarbon chains, every of

which contain hundreds or even thousands of units (Fig.7.21.a).

Branched or two-dimensional polymers consist of long hydrocarbon chains

with branches on it (Fig.7.21.b). The branches occur more or less regularly along

the chain.

S

S

b

dc

a

Fig.7.21. Linear (a), branched (b) and network (c,d) structure of

polymers

206

Three-dimensional or network polymers in which the existence of strong

chemical bonds between the chains leads to the formation of a single network

(Fig.7.21.c,d).

The first step of polymer structure is a chain of identical units that are inca-

pable of forming hydrogen bonds or any

other type of specific bond. The simplest

model is a freely jointed chain, in which

any bond is free to make any angle with

respect to the preceding one (Fig.7.22).

Therefore, in a given polymer chain the

carbon atoms are not in a straight line, but

in random arrangement.

Due to their structure polymers, es-

pecially linear polymers possess a chain

flexibility (mobility of the various segments) because a chain consist of a large

number –CH2– groups, every of which be able to move around of C–C bond. Flex-

ibility becomes less with branches, with polar groups increases; it also depends on

the temperature, on the medium, on the intermolecular attractions, etc. In polymers

containing hydroxyl groups, imino groups NH and some other groups, the chain

may be linked by hydrogen or other chemical bonds. All this corresponds to the

transition to the network polymer and increases its hardness.

Starting from their physical state polymers (linear) can be in three states de-

pending on the temperature. At relatively low temperatures they are in the elastic-

hard (glassy) state; with a rise in temperature they pass over into the high-elastic

(retarded elastic) rubbery state, and on the further heating they acquire fluidity,

transforming into the plastic (viscofluid) state.

§ 23. Dissolution of polymers

Polymers, like other substances, can dissolve in some low-molecular liq-

uids. Depending on the chemical composition and internal structure, the solubility

of the polymers with respect to a given liquid and different liquids differs greatly.

Linear polymers are most soluble when they have no appreciable interchain bond-

ing. When such bonds are formed, the solubility diminishes; a polymer with a rigid

three-dimensional network is insoluble.

Fig.7.22. A model of a

freely joined chain

207

The dissolution process (when it occurs) differs greatly from ordinary solu-

tions of low-molecular substances. When low-molecular substances are brought

into contact with a low-molecular liquid, due to the solvent and solute molecules

have a comparable molecular size, the particles of each substance permeate the

medium of each other with almost the same rate (reciprocal diffusion). For a solu-

tion of macromolecules in which the solute molecule is much larger than that of the

solvent, molecules of polymer move more slowly, and in the first stages it is the

low-molecular molecules of a liquid that predominantly penetrate (diffusion) into

the polymer medium and fill up spaces between polymer chains (Fig.7.23.a,b).

This process is called swelling. Then the chains of macromolecules move away

from each other, they also penetrate the liquid (water) and two layers of solutions

are formed; water in polymer and polymer in water (Fig.7.23c). At least polymer

acquires fluidity and gradually is

passed into solution in the given sol-

vent and dilute solution is formed

(Fig.7.23.d).

The swelling of a polymer is

accompanied by an increase in its vol-

ume. Its quantitative characteristic is

the so-called degree of swelling

o

o

m

mm

where mo is the weight of the polymer

before swelling, m – after swelling.

Hence, it shows amount of water

swallowing up by unit gram of dray polymer. In consequence of increasing the

volume the swelling pressure is formed which acts on the walls of the vessel.

When the gradual swelling of the polymer is not confined by any limits and

the solution is formed it is called unlimited swelling. In some cases (for polymers

with stronger bonding), there is a limit to the amount of swelling and the swollen

polymer may be in a state of dynamic equilibrium with the given solvent; for-

mation of the solution does not occur and it is called limited swelling. Thus, gelatin

has a limited swelling capacity in cold water.

In spite of the swelling of a polymer, it is accompanied by an increase in its

volume the whole volume of the system (polymer + solvent) decreases due to in-

creasing of the packing density. This phenomenon is called contraction.

Fig.7.23. Stages of polymer dilution

208

§ 24. Stability of polymer solutions

Solutions of polymers are thermodynamically stable systems, but in case of a

large amount electrolyte addition the process occurred, which is called desalting;

polymer is separated from solution and two-phase system is formed. The reason of

this phenomenon is due to hydration of electrolyte; it is takes away water from the

polymer and the solubility of the latter decreases.

In concentric solutions of polymers at changing of pH or temperature the co-

acervation may occur at which molecules of a polymer are attracted to each other

and form associates. They might be so large that they are separates from a solution

as big droplets.

Most polymer solutions under certain conditions form a stiff semiliquid-

simirigid precipitate, called a gel in which both components (polymer and liquid)

extend continuously throughout the system. Setting of a gel may be induced by

widely different factors, such as electrolytes, temperature changes, etc. Some of

them, such as gelatin, set to a gel at low temperature and liquefy at a high tempera-

ture, while others behave in the opposite way. Setting of a gel can be explained in

the general case as the result of mutual binding of the particles into unstable coagu-

lation networks loops of which retain the intermicellar liquid. The forces binding

the particles together may be of a different nature: van der Waals forces, hydrogen

bonds, covalent bonds, etc.

It is interesting to consider dehydration processes in hydrogels, which occu-

py an important part in many phenomena such as the formation of minerals from

colloidal deposits, the aging of cement, the staling of bread, and industrial drying

processes. The dehydration and hydration curves show the pressure of water vapor

over the gel do not coincide. Due to it the temperatures of gel formation and gel

liquefying is not the same (as in the case of the crystals). This phenomenon is

called hysteresis.

As dehydration proceeds, the gel is gradually transformed from a soft jelly-

like mass to a hard stone-like body with a great strength. Dried gels are called

xerogels. They are systems with great degree of a porous, and for this reason they

are applied as adsorbents.

The structure of gels may change with time. This process is known as aging

of gels.

209

Some gels have the property of reversibly liquefying when subjected to me-

chanical forces (shaking, mixing, vibration, etc.), i.e., the gel turns into a sol, which

on standing again becomes a gel. Such transformation can be repeated many times.

This phenomenon is called thixotropy. Thixotropy is characteristic of gels in which

the forces holding the particles together are weaker than valence forces.

The spontaneous exudation of liquid from a gel is called syneresis (weep-

ing). This phenomenon occurs, in particular, in natural conditions when water or a

solution is exuded from some colloidal deposits (silt, silica hydrogel), in the sepa-

ration of the serum upon the coagulation of blood or of whey from curdled milk.

§ 25. Osmotic pressure of polymer solutions

Highly dilute solutions of polymers obey to the Vant-Hoff’s law, i.e., the

usual relation characterizing the dependence of the

osmotic pressure and other properties of solutions

on their concentration are valid. However, with

increasing the concentration osmotic pressure in-

creases much more than it is due to the law

(Fig.7.24). The reason of it is the exceptional flex-

ibility of the polymer chain. The mobility of the

separate units of the chain makes possible the ex-

istence of a large number of conformations of the

macromolecules every of which come out as a

separate molecule. As a result, osmotic pressure

does not correspond to the real amount of macromolecules. For taking into account

this deviation Galer offered the following equation:

π=CRT/M + bc2 7.11

where b is the coefficient in which considered deviation from osmotic pressure. For

linear dependence obtained the equation write

down in the form

π/C=RT/M + bc 7.12

It allows determine the value of the b and the mo-

lar weight of the polymer by diagram (Fig.7.25).

In general, osmotic pressure measurements

widely used for molar mass determination of high

molecular compounds (osmometry). In this case,

Fig.7.24. Changing the os-

motic pressure of polymers

with change of concentration


constancies of the Galer

equation

210

one has to take into account of Donnan membrane equilibrium; the equilibrium

distribution of ions in two compartments in contact through a semipermeable

membrane, in one of which there is a polyelectrolyte. It relates to the case when in

the system there are electrolyte and membrane, which is permeable to ions but not

for macromolecules. This arrangement is one that actually in living systems, where

osmosis is an important feature of cell operation. The presence of the salt affects

the osmotic pressure because the anions and cations cannot migrate through the

membrane to an arbitrary extent. Apart from small imbalances of charge close to

the membrane and which give rise to transmembrane potentials, electrical neutrali-

ty must be preserving in the bulk on both sides of the membrane; if an anion mi-

grates, a cation must accompany it. Therefore, for precise measurement of osmotic

pressure we have to take into account a presence of electrolytes in solution. For

this, let us suppose the solution of the polyelectrolyte NaR (in concentration C1)

which is in contact through a semipermeable membrane with NaCl solution (in

concentration C2). The essential arrangement will be:

21

21

CCl CNa

CNa CR

-

The anion of polyelectrolyte is not able passing through a semipermeable

membrane and the condition of equilibrium is that the Gibbs energy of NaCl in

solution is the same on both sides of the membrane, so a net flow of Na+ and Cl–

ions occurs until equilibrium states (amount of flowing ions are denoted x). The

arrangement of ions in this state is the following:

xCl

xCCl xCNa

CNa CR

-

-

21

21

In equilibrium state equality is occurs:

x(C1 + x) = (C2 – x)2

from which

21

22

2CC

Cx

7.13

In cases when C1<<C2, i.e., an amount of polyelectrolyte is very small

the equation becomes in form x=C2/2. It means that the real value of the os-

motic pressure will be equal to the half of measured value. When C1>>C2, i.e.,

it is very little electrolyte be contained in the system, so the value of x is

closed to zero. It means that electrolyte almost do not flow through the mem-

211

brane. Therefore, if we measure the osmotic pressure in the presence of high

concentration of salt, the molar mass may be obtained unambiguously.

§ 26. Viscosity of polymer solutions

The property that characterizes a fluid’s resistance to flow is its viscosity

(η). The bulk flow of fluids occurs under a pressure gradient, and the speed of flow

through a tube is inversely proportional to the viscosity. The flowing of liquid

might be laminar and turbulent.

In case of laminar (or streamline) flowing layers of fluid slip of each other

without mixing. Adjacent horizontal layers of fluid flow at different speeds and

“slide over” one another. As two adjacent layers slip past each other, each exerts a

frictional resistive force on the other, and this internal friction gives rise to viscosi-

ty. Experiments on fluid flow show that the frictional force (F) is proportional to

the surface area of layers contact (S) and to the gradient of flow speed (dv/dx):

dx

dvSF 7.14

This equation is Newton’s law of viscosity. The proportionality constant (η)

is the fluid’s viscosity. The units of η are N s m–2.

At high rates of flow, this equation does not hold, layers of flowing liquid

mixed, and the flow called turbulent.

Newton’s law of viscosity allows the rate of flow of a fluid through a tube to

be determined. For laminar (nonturbulent) flow of a liquid in a tube of radius r, the

flow rate is

l

Pr

t

V

8

4 7.15

where V is the volume of liquid that pass a cross section of the tube with length l in

time t, and pressure difference at the ends of the tube is ∆P. This equation is

Poiseuille’s law. French physician Poiseuille was interested in blood flow in capil-

laries and measured flow rates of liquids in narrow glass tubes. Take into account

the very strong dependence of flow rate on tube radius and the inverse dependence

on fluid viscosity. (A vasodilator drug such as nitroglycerin increases the radius of

blood vessels, thereby reducing the resistance to flow and the loading on the heart.

This relieves the pain of angina pectoris).

A Newtonian fluid is one for which η is independent of dv/dx. For a non-

Newtonian fluid, η changes as dv/dx changes. Most pure nonpolymeric liquids are

212

Newtonian. Polymer solutions, liquid polymers, and colloidal suspensions are often

non-Newtonian. An increase in flow rate and in dv/dx may change the shape of

flexible polymer molecules, facilitating flow and reducing η. The viscosity of liq-

uids generally decreases rapidly with increasing temperature, because the higher

translational kinetic energy allows intermolecular attractions to be overcome more

easily. The viscosity of liquids increases with increasing pressure. Liquids of high

viscosity have high boiling points and high heats of vaporization.

The presence of a macromolecular solute increases the viscosity of a solu-

tion. The effect is large even at low concentra-

tions, because big molecules affect the fluid flow

over an extensive region surrounding them. At low

concentrations, the viscosity of the solution is re-

lated to the viscosity of the pure solvent and to the

concentration by Einstein equation:

j k 10 7.16

where φ is the volume part of the polymer, and k is

the coefficient depending on the macromolecules

shape. With increase in concentration the equation

is not hold; the viscosity increases much more that

the concentration (Fig.7.26). It is due to intermolecular attractions in a liquid that

hinder flow and make viscosity greater. The viscosity of a polymer solution depend

on size and shape (and hence on the molecular weight and the degree of compact-

ness) of polymer molecules in the solution. The molecular shape influences the

viscosities of liquids. Long-chain liquid polymers are highly viscous, because the

chains become tangled with one another, hindering flow. Besides, a molecule of a

long-chain synthetic polymer usually exists in solution as a random coil. There is

nearly free rotation about the single bonds of the chain, so we can crudely picture

the polymer as composed of a large number of links with random orientation be-

tween adjacent links. In part, it influence on the degree of compactness, and, hence,

on the viscosity.

Depending on concentration and the above-mentioned factors a few types of

viscosity are distinguishes.

A relative viscosity of a polymer solution is defined as

ηr = η/ηo 7.17

where η and ηo are the viscosities of the solution and the pure solvent.

A specific viscosity shows how viscosity changes with addition of a polymer:

Fig.7.26. Dependence viscos-

ity of the polymer solution on

the concentration of polymer

213

o

os

7.18

A specific viscosity has a linear dependence on the polymer molar mass, which

was showed by Shtaudinger equation:

ηs = kMC 7.19

If we write it in the form

ηs /C = kM 7.20

thus the relation ηs /C is called adjusted viscosity. A diagram of adjusted viscosity

against concentration shows on the Fig.7.27. As we

can see, an adjusted viscosity grows with concentra-

tion increasing due to intermolecular attractions.

The section on the ordinate axis is called in-

trinsic viscosity . It is also called limiting viscosi-

ty number, because it is equal to:

C

s

OC

lim 7.21

Experimental data show that for a given kind of syn-

thetic polymer in a given solvent, the following rela-

tion is well obeyed at fixed temperature:

kM 7.22

where M is the molar mass of the polymer, K and α are empirical constants.

Measurements of the viscosity of a solution are widely applied for polymer

molecular weight determination. To apply the equation 7.22, one must first deter-

mine K and α for the polymer and the solvent using polymer samples molecular

weights of which have been found out by some other methods (such as osmotic

pressure measurements). Once K and α are known, the molar mass of a given sam-

ple of the polymer can be found out by viscosity measurements.

In high concentric polymer solutions due to intermolecular attractions and

links formed betweenits, the network generates and, hence, the viscosity of a solu-

tion arises. Such type of viscosity is called plastic or structural viscosity. It de-

scribes by Bingam equation:

dx

dVP 7.23

where P is the applied pressure (for liquid flow), θ is the limited effort of dis-

placement (an effort of the network links above which they are destroyed), is

Fig.7.27. A diagram of

adjusted viscosity against

concentration

214

the structural viscosity, and dV/dx is the rate gradient. Such system flow only under

condition when P>θ, i.e., when the applied pressure destroys the network links.

§ 27. Polyelectrolytes

Polyelectrolytes are such polymers, which might be ionization in the solu-

tion. Some of them are strings of acid groups, as in poly(acrylic acid),

— (CH2CHCOOH)n —, and called polyanions. Others are strings of bases, as in

nylon, — [NH(CH2)6NHCO(CH2)4CO] —, and called polycations. A macromole-

cule with mixed cation and anion character is known as a polyampholyte.

Among polyelectrolytes we are interested in proteins, which are amphoteric, and

which possess a net charge character (anionic or cationic) that

depends on the pH of the aqueous solution. In general, pro-

teins can be classified as fibrous and globular. In a fibrous

protein, the chain is coiled into a helix (Fig.7.28). The helix is

stabilized by hydrogen bonds between one turn and the next.

Hair and muscle proteins and collagen are fibrous. Fibrous

proteins are generally insoluble.

In a globular protein, some portions of the chain are

coiled into hydrogen-bond-stabilized helical segments. Oth-

er portions of the chain are nearly fully extended and are

hydrogen-bonded to adjacent parallel (or antiparallel) por-

tions of the chain to form what is called a β sheet. Different

portions of the chain are held together by S–S covalent

bonds, hydrogen bonds, and van der Waals forces. Globular

protein molecules contain many polar groups on their outer

surface and are generally water-soluble. Most enzymes are

globular proteins.

Protein molecule contains both acid and basic groups: NH2–R–COOH. In isoelectric

point, the number of acid and basic groups is equal and molecule is no charge. Because of acid

groups being a little more that basic groups (so, the proteins is a weak acid), the isoelectric point

of proteins lies in weakly acidic medium; acidic medium depresses the excess ionization of acid

groups and isoelectric point established. In isoelectric point, the ionization of the protein occurs

by the following:

RNH2COOH + H2O → OH– + NH3+–R–COO– + H+

Fig.7.28. The helix

chain of a fibrous

protein

215

In this point, molecules of a protein have a ball shape because of positive and negative side’s

attractions.

When the medium of a protein solution is strong acidic (pH<7), the ionization of acidic

groups depresses and the protein molecule acquires a positive charge and basic properties:

RNH2COOH + H+ → RNH3+COOH

The same charged molecules pushes off from each other, molecule becomes a helix

(more straight) form (from a ball shape); moreover, because of a charge enquiring protein mole-

cule moves in the electrical field toward the negative electrode (electrophoresis occurs).

The same processes take place in a strong basic medium in difference, that polymer

molecule acquires a negative charge and moves in the electrical field toward the positive elec-

trode.

The charge of a protein depends on the pH, and hence the rate of migration varies with

pH. This apparent difficulty can be used to distinguish proteins. For example, at a given pH the

rate of migration of haemoglobin from people with sickle-

cell anemia is different from that of a sample taken from

people without the disease. This difference is an indication

that there is different charge on the protein molecule,

which in its turn, is ascribed to the presence of a different

amino acid residue in the polypeptide chain.

Example: The drift speed of bovine serum albumin (BSA)

under the influence of an electric field in aqueous solution

was monitored at several values of pH, and the data are

listened below (opposite signes indicate opposite direction

of travel). What is the isoelectric point of the protein?

pH 4.20 4.56 5.20 5.65 6.30

7.00

Speed/(μm s–1) +0.50 +0.18 –0.25 –0.65 –0.90 –1.25

Solution: The data are plotted in diagram. The drift speed is zero at pH = 4.8; hence pH =

4.8 is the isoelectric point.

216

APPENDICES

RELATIONS BETWEEN SELECTED QUANTITIES AND NUMERICAL

VALUES OF CONSTANTS

R (gas constant) = 8,3143 J = 8,3143.107 erg = 1,98725 cal = 0,082057 latm =

62,36 mm Hg

NA (Avogadro constant) = 6,021023 mol–1

F(Faraday constant) = 96500 (96484) C mol–1

1J = 107 erg = 0,239 cal = 9,867.10–3 latm

1cal = 4,1840 J

1 atm = 760 mm Hg = 1,01325.10–5 Pa =760 Torr = 1,013 bar

1 mm Hg = 133,32 Pa = 1Torr

1 bar = 105 Pa = 0,986923 atm = 750,062 Torr

1 latm = 101,327 J = 24,218 cal

1 kg = 9,8067 J = 2,342 cal = 9,67610–2 latm

1m3 = 1,0.103 dm3 = 1,0.106 cm3

1 l = 1 dm3 = 1103 cm 3 = 1.10–3 m3

1 ml = 1,0.10–6 m3

1 g/cm3 = 1000 kg/m3

1 ml/g = 1,0.10–3 m3/kg

1 year = 3,1557.107 sec

1 cm = 108 hour

t0C = T0K–273,150

217

Table 1

Enthalpy of formation of selected inorganic compounds in standard

conditions

Compound

Mm H0 kJ/mol Compound Mm H0 kJ/mol

H2O /gas/ 18.02 –241.82 N2O4 /g/ 92.012 +9.16

H2O /liq/ 18.02 –285.83 SO2 /g/ 64.063 –296.8

H2O2 /liq/ 34.015 –187.8 H2S/g/ 34.080 –20.6

NH3 /gas/ 17.031 –46.11 H2S/l/ 34.080 –39.3

HNO3 /liq/ 63.013 –174.1 HCI /g/ 36.461 –92.31

NH4CI /sol/ 53.492 –314.4 HBr /g/ 80.917 –36.4

H2SO4 /liq/ 98.078 –811.3 H /g/ 127.912 +26.5

H2SO4 /liq/ 98.078 –907.5 CO2 /g/ 44.010 –393.51

NaCI /sol/ 58.443 –412.1 AI2O3 /s/ 101.96 1669.8

NaOH /sol/ 39.997 –425.6 SiO2 /s/ 60.085 –859.4

KCI /sol/ 74.555 –435.6 FeS /s/ 87.91 –95.1

NO /gas/ 30.000 +90.25 FeS2 /s/ 119.98 –117.9

NO2 /gas/ 46.006 +33.2

218

Table 2

Enthalpy of formation and combustion of selected organic compounds

in standard conditions

Compound Mm H0form kJ/mol

H0comb kJ/mol

CH4 /g/ 16.043 –74.81 890.4

C2H2 /g/ 26.038 +226.8 1300

C2H4 /g/ 28.054 +52.30 1411

C2H6 /g/ 30.070 –84.64 1560

C2H6 /l/ 78.115 +48.99 3268

CH3OH /l/ 32.042 –239.0 726.1

CH3CHO /g/ 44.054 –116.4 1193

CH3CH2OH /l/ 46.070 –277.0 1368

CH3COOH /l/ 60.053 –484.2 874

CH3COOC2H5 /l/ 88.107 486.6 2231

C6H5OH /s/ 94.114 –165.0 3054

Urea/s/ 60.056 333.0 632

Table 3

Entropy value of selected compounds in standard conditions

Compound

S0 J/K mol Compound S0 J/K mol

AgBr /s/ 107.1 C6H6 /g/ 296.20

AgCI /s/ 96.11 CH3COOH /l/ 159.8

BaCI2 /l/ 121 CO /g/ 197.91

Ba(OH)2 /l/ –8 CO2 /g/ 213.64

C /diamond/ 2.43 Ca(OH)2 /l/ –76.2

C /graphite/ 5.69 CI2 /g/ 222.97

CH3CI /g/ 234.2 H2O /g/ 188.74

CCI4 /g/ 309.4 H2O /l/ 69.96

CH4 /g/ 186.19 CH3OH /g/ 237.6

NaCI /l/ 115.48 CH3OH /l/ 126.8

219

Table 4

Gibbs,s energy value of selected compounds in standard conditions

G0 (kJ/mol)

Solid compounds

NaCI –384.0 CaO –604.2 SiO2 –805.0

NH4CI –203.89 CaCO3 -1128.8 FeS –97.6

KCI 408.32 AI2O3 -1576.4 FeS2 –166.7

KOH –374.5 C/³ÉÙ³ëï/ 2.87 AgCI –109.7

Liquids

H2O –237.19 CH3CH2OH -174.76 C6H6 –172.80

H2O2 –113.97 HNO3 –79.91 H2SO4(l) –741.99

CH3OH –166.31 CS2 63.6 H2S (l) –27.36

Gases

NH3 –16.63 N2O4 104.2 HCN 120.1

NO 86.69 CH4 –50.79 HCI –95.26

NO2 51.84 C2H2 209.2 HBr –53.22

O3 163.43 C2H4 68.12 H2S –33.02

CO –137.27 C2H6 –32.89 N3H 328.0

CO2 –394.38 C4H10 –15.71

Table 5

Heat capacity value of selected compounds in standard conditions

Compound

Cv,mJ/K mol Cp,mJ/K mol

He, Ne, Ar, Kr,Xe 12.47 20.78

H2 20.50 28.81

O2 21.01 29.33

N2 20.83 29.14

CO2 28.83 37.14

NH3 27.17 35.48

CH4 27.43 35.74

220

Table 6

Standard electrode potentials value of selected electrodes

Electrode Electrode reaction 0, V

Li+/ Li Li++ e Li –3.024

K+ / K K+ + e K –2.924

Ba2+ / Ba Ba2++2e Ba –2.90

Ca2+ / Ca Ca2+ + 2e Ca –2.87

Na+ /Na Na+ + e Na –2.714

Al3+ /Al Al3+ + 3e Al –1.67

Mn2+ /Mn Mn2+ + 2e Mn –1.05

Zn2+ /Zn Zn2+ + 2e Zn –0.762

Cr3+ /Cr Cr3+ + 3e Cr –0.71

Fe2+ /Fe Fe2+ + 2e Fe –0.441

Cd2+ /Cd Cd2+ + 2e Cd –0.402

Co2+ /Co Co2+ + 2e Co –0.277

Ni 2+ /Ni Ni2+ + 2e Ni –0.250

S n2+ /Sn Sn2+ + 2e Sn –0.140

Pb2+ /Pb Pb2+ + 2e Pb –0.136

H+/H2(Pt) H+ + e 1/2H2 0.000

Cu2+ /Cu Cu2+ + 2e Cu +0.345

Hg22+ /Hg Hg2

2+ + 2e 2Hg +0.789

Ag+ /Ag Ag+ + e Ag +0.799

Pt2+ /Pt Pt2+ + 2e Pt +1.2

Au 3+ /Au Au3+ + 3e Au +1.5

221

Table 7

Molar conductivity of selected ions at limit dilution in standard condi-

tions

Cation ¥ -1 m2 mol-1 Anion ¥ -1 m2 mol-1

H+ 349.8 OH– 197.6

Ag+ 61.9 Br– 78.14

K+ 73.5 CI– 76.35

Na+ 50.1 I– 76.85

NH4+ 73.7 NO3

– 71.4

1/2 Ba2+ 63.6 HCO3– 44.5

1/2 Ca2+ 59.5 1/2 CO32– 69.3

1/2 Cu2+ 55.0 1/2 SO42– 80.0

1/2 Zn2+ 54.0 1/2 CrO42– 83.0

HCOO– 54.6

CH3COO– 40.9

C2H5COO– 35.8

C6H5COO– 32.3

222

Table 8

Saturated vapor pressure of water at different temperatures

mm Hg

Temperature, 0C P0

15 12. 783

16 13.634

17 14.530

18 15.477

19 16.477

20 17.535

21 18.650

22 19.827

23 21.068

24 22.377

25 23.76

26 25.21

Table 9

Cryoscopic constants of selected solvents

Solvent Kcryos. Tfreez, K

Benzol 5.07 278.9

Water 1.86 273.2

Dioxan 4.71 145.8

Camphor 40 451.2

Phenol 7.8 313.2

Nitrobenzol 6.9 278.8

223

R E F E R E N C E S

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3. Äîáû÷ Ä.Ï. è äð. Ôèзè÷åñêàÿ è êîëëîèäíàÿ õèìèÿ.Ì."Ïðîñâåùåíèå".1986.

4. Åâñòðàòîâà Ê.È. Êóïèíà Í.À. Ìàëàõîâà Å.Å. Ôèõè÷åñêàÿ è êîëëîèäíàÿ

õèìèÿ. Ì. "Âûñøàÿ øêîëà". 1990.

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