Chapter0 Fundamentals

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Chem 4615 Physical Chemistry for the Life Sciences

Chem 4615

Physical Chemistry for the Life Sciences

Diego Troya

Department of Chemistry

Virginia Tech

Lecture notes
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Chapter 0. Fundamentals

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F.0 Units

-We will be using the SI system of units most of the time (but

not always!).

-We will be using 5 out of the 7 Base SI units:

The unit of length is meter (m)

The unit of mass is kilogram (kg)

The unit of time is second (s)

The unit of temperature is kelvin (K)The unit of amount of substance is mole (mol)

-Resource section 2 gives a great review of units.

-Make sure that you know the prefixes of SI and other metric

units Table 3, p 558
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F.1 The constituents of matter

- As we learned in General Chemistry, matter is composed of

atoms.

- Atoms are characterized by their atomic number Z, which

denotes the number of protons.- Isotopes are atoms with the same number of protons, but

different mass number A (they have a different number of

neutrons).

- In neutral atoms, a number of electrons (which carry negative

charge) identical to the number of protons (positively charged)surround the nucleus.

- Ions are formed by the removal or addition of electrons to a

neutral atom.

- Molecules are formed by the aggregation of atoms.
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- In most biological molecules, the bonding between the atoms

is covalent, and emerges by the sharing of electrons among

atoms. We will study this type of bonding in detail in CHEM

4616.

- Other substances in nature (mostly solids) have ionic ormetallic bonding, but they are less relevant here.

- Two molecules may interact via non-bonding interactions (also

called intermolecular interactions). These interactions might

also take place between two parts of a large molecule. When aprotein folds, sections of the protein come in contact with one

another and interact via non-bonding interactions. Hydrogen

bond is the quintessential example of a non-bonding

interaction, but there are others, like dipole-dipole, dispersion,

etc. (More on this in CHEM 4616).
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A recurring theme in large biological polymers is that they are

composed of simple building blocks.

- Aproteinis composed by aminoacids

-Nucleic acidsare composed by bases, sugars, and

phosphate units-Polysaccharidesare complex molecules formed by sugars

-Lipidsare formed by long hydrocarbon units and polar

headgroups

- There are fourlevels of structurethat are commonly

discussed. For the specific case of a protein, the primarystructure is simply the aminoacid sequence. The secondary

structure is the local spatial arrangement of aminoacids. The

tertiary structure is the global 3D shape of the protein. Finally,

the quaternary structure refers to the way that aggregation of

protein subunits takes place in large proteins(e.g.hemoglobin).

C C S
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F.2. Bulk matter: States of matter

In the end, atoms and molecules aggregate to form bulk matter,

which is what we study in CHEM 4615. A difference has to be

made between thestate of matter and thephysical state.

- We will consider three states of matter: gas, liquid and solid.

Macroscopically:A gas fully occupies a container and conforms to its shape

A liquid does not occupy a container but conforms to its shape

A solid neither occupies a container nor conforms to its shape

At themicroscopiclevel, these states of matter can be

distinguished by molecular motion and separation between

particles. Gas molecules are far away from one another and

travel mostly freely. Liquid molecules are close to one another,

and can move past each other with some restriction. Particles

in a solid are also close to one another but cant move pasteach other.

Ch 4615 Ph i l Ch i t f th Lif S i
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F.2 Bulk matter: Physical states

The physical state of a sample refers to the conditions that

sample is in. To define a physical state completely, one needs

to specify:

i) state of matter

ii) amountiii) volume

iv) pressure

v) temperature

Amountmight be specified either by using moles (e.g. 1.5 mol

of hydrogen), or simply mass (e.g. 16.5 g of oxygen). Volume

is also something intuitive. Notice that the SI units of volume

are m3, but liters (L) are more convenient to use. Please, make

sure that you dominate this unit transformation (1000 L= 1 m3).

PressureandTemperaturerequire more attention, and are

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F.2 Bulk matter. Pressure

Pressure Pressure = ForceArea

P= FAThe unit of pressure is the pascal (Pa). 1 Pa= 1 N m2

There are other more popular units of pressure, such as

atmosphere (atm), bar (bar), torr (torr), and millimeters of

mercury (mmHg).1 atm = 101325 Pa =1.01325105 Pa

1 atm = 1.01325 bar

1 atm = 760 torr = 760 mmHg

The atmospheric pressure we feel ( 1 atm) is due to collisionsof gas molecules with our body. Atop the Everest, there is less

gas, so the pressure is lower. Something that will be important

during this course is the concept of mechanical equilibrium

(Fig. F.4) in a cylinder with a movable piston, which occurs

when the gas inside the cylinder exerts the same pressure on

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Hydrostatic pressure

Under water, we are subject tohydrostatic pressure. This is

the same effect as what was used in the original barometer by

Torricelli (www.ems.psu.edu/ nese/f4 2.gif)The equation of hydrostatic pressure in that barometer can be

derived from the definition of pressure:

pressure= force

area =

mass acceleration

area

Multiplying and dividing by the height of the column, and noting

that the acceleration is due to gravity:

pressure=mass gravity height

area height = densitygravityheight

The resulting equation,p= g hdetermines the atmosphericpressure in the barometer, and will be used when we talk about

osmotic pressure.

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F.2. Bulk matter. Temperature

Temperatureis hard to define. In thermodynamics,

temperature helps determine the direction in which energy will

flow when two objects that are not isothermal are brought in

contact. Energy will flow from the body with higher temperature

to the body with lower temperature until the temperaturesbecome equal. At that point we will have reachedthermal

equilibrium. See Fig. F5.

As you have known for some time, there are three main

temperature scales: Kelvin, Celsius and Fahrenheit. Kelvin is

the SI unit of temperature. Here are some useful

transformations:

oK=o C+ 273.15

oF=o C 95+ 32

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F2. Bulk matter. Equations of state

We have said that, aside from the state of matter, V, p, T, and n

are the four quantities that are needed to specify the physical

stateof a system. Well, it turns out that these 4 quantities are

not independent of each other. This gives rise to the equations

of state, which relate the four quantities.

The easiest equation of state is that for an ideal gas (termed

perfect gas in your book), which was established some 200

years ago by combining Boyles, Charless, and Avogadros

laws:

pV= nRT R = 0.08206 atm L mol1K1 = 8.314 J mol1K1

where R is the ideal-gas constant, and can be expressed in

multiple units (please, give attention to TableF2).

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F2. Bulk matter. Equations of state

Something that emerges from the ideal-gas equation of state is

that themolar volume(volume occupied by 1 mol of gas) is

identical for all gases for a given set of temperature and

pressure.

pV= nRT Vn = Vm= RTp

Atstandard ambient temperature and pressure (SATP)

(p=1 bar, T=298.15 K), the molar volume of any gas that

behaves ideally is 24.8 L/mol.

Atstandard temperature and pressure (STP) conditions(p=1 atm, T=273.15 K), the molar volume of any gas that

behaves ideally is 22.4 L/mol.

The SATP and STP conditions will appear quite often during

the rest of the semester.

Thestandard state(Table F.3) will also beimportant.

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y y


Equations of state for non-perfect gases

-ThepV= nRTis an equation of state only followed by gasesthat behave ideally. There are two main conditions for ideal

behavior:

i) There are no interactions between the gas particles

ii) The size of the gas particles is negligible compared to theseparation between the gas particles.

-These conditions are satisfied by most gases at ambient

temperature and pressures below ca. 10 atm. Deviations from

ideality are studied by thecompressibility factor(this is not in

the book):

Z= pV

nRT =

pVm

RT

-For a gas that behaves ideally, the compressibility factor

should be 1. Z=1.

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y y



Deviations from ideal behavior (not in the book):

If Z>1 the gases are more difficult to compress than ifideal (there are repulsions, the non-negligible volume of

the molecules starts to factor in)

If Z


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1.- Corrections to the pressure term (Pideal= (Preal+ an2/V2)):

Pressure depends on both the number of collisions of the gas

molecules with the walls of the container and the momentum of

the particles colliding. If there are many collisions with the walls

of the gas container and the impinging particles have highmomentum, the pressure will be high. Attractive interactions

among gas molecules reduce both the number of collisions and

their momentum, thus reducing the pressure. Both types of

reductions in the pressure are proportional to the number of

molecules present, orn/V(density of the gas). ais a measureof the strength of the attractions.

2.- Corrections to the volume term: Videal= (Vreal nb)Molecules have a finite volume and this should be taken into

account in a proper description. Notice the minus sign:

Vreal>Videal if molecules occupy a non-negligiblevolume.

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F.3 Energy

Force Force = mass acceleration F= m aThe unit of force is the newton (N). One newton is the force

required to push a 1 kg mass to an acceleration of 1 m s2.

1 N= 1 k g m s 2

Work Work = Force distance W= F dThe unit of work is the Joule (J). One joule is the amount of

work that is done when one newton of force is applied to push

an object over a distance of 1 meter. 1 J= 1 kg m2 s2 = 1 Nm

Energy.Energy is the ability to do work, and it has the same units as

work (J).

There are two types of energy, kinetic (due to motion) and

potential (due to position).

Kinetic energyalways isEK= 12 mv2.

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F.3 Energy

Potential energycan come in a variety of ways.

-The potential energy of interaction between two charged

particles in a vacuum separated by a distanceris described by

Coulombs law: Ep= Q1Q240rwhere the Qs are the charges, and

0 is the permittivity of vacuum. Coulombic energy plays a bigrole in a variety of biomolecules, including DNA.

- Another example of potential energy is gravitational energy.

The potential energy of an object above the ground is:Ep= mghwhere m is the mass of the object, g the gravitationalconstant (9.81m s2), andhis the height above ground.

- Total energy is conserved, and is the sum of kinetic and

potential energies: E= EK+ EP

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F.3 Energy. The Boltzmann distribution

-We will learn in CHEM 4616 that counter to what we

experience in our daily life, energy is not continuous, but

discrete. This becomes more noticeable at the microscopic

level.

-Molecules can have various types of energies: translational,rotational, vibrational, and electronic. Since energy is discrete

at the atomic level, molecules can only have given amounts of

energy, called energy states (see Fig. F8).

-At a given temperature, matter has thermal energy, which

allows the system to have various energy states.-The relative population between two energy states N1 andN2at a given temperature is provided by the Boltzmann

distribution:

N2N1 = e

(E2E

1)/RT

or

N2N1 = e

(E2E

1)/k

BT

kB= R

NA

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F3. Kinetic Theory of Gases

Kinetic Theory of Gases

The ideal gas equation of state is a relation between properties

that define the state of a gas (temperature, pressure, volume,

amount).

Although this law is very useful, it does not provide any insight

into the behavior of gases at the atomic level.

The kinetic theory of gases tries to go beyond the

macroscopic description of gases provided by the gas laws by

introducing simple models that attempt to characterize thebehavior of gases at themicroscopiclevel.

For instance, we are going to try to define pressure and

temperature through investigation of the behavior of individual

gas molecules.

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Thekinetic theory of gasesis based on three main

approximations:

1) The particles of the gas are in random motion

2) There are no interactions between the gas particles until

they collide. In addition, the collisions are elastic (no energy islost to the surroundings)

3) The size of the particles is negligible compared to the

interparticle separation

With these assumptions, we will be able to calculate the

pressure exerted by a gas based solely on the mass andvelocity of the particles making up the gas:

p=2

3

N

V

whereNis the number of gas particles,Vis the volume, and

< 12 mv2 >is the average kinetic energy ofeachgasparticle.

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Derivation of pressure from the Kinetic Theory of Gases

(This derivation is easier than what you have in the book p267))

- Intuitively, we can think that the pressure of a gas is related to

the force exerted by gas molecules on the walls of the container

that keeps the gas (think about a pressure cooker). P= FA- Lets try to calculate the force exerted by a molecule traveling

perpendicularly to a flat wall of a cubic container at speed vx.

force= mass acceleration= massdistance

time2

force=momentum

time- So the force is the change in momentum per unit time.

-The momentum of the gas molecule before a collision with the

wall ismvx, and mvxright after it (the particle is moving only

in the x direction, and the collision is elastic). Thus the total

change in momentum is 2mvx.
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- It follows that:

P= force

area = Nmv2

xl

1

l2 =Nmv2

xV

- This is the pressure exerted byNmolecules traveling in only

one direction at the same speed vx.

- However,molecules travel randomlyin the three directions

of space (x,y,z). In addition, at a given temperature, themolecules can have abroad range of velocities. To keep

things simple, we are going to do the calculations taking into

accountroot-mean-squaredvelocities in each direction

(,,,=v2x,1+v2x,2+v2x,3+

N ).- The average total squared velocity of the gas particles is:

= + +

- If we have many molecules, ==, thus

= 3< v2x> or = 3


Ch t 0 F d t l
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- Now that we have included the fact that the molecules move

randomly in all three directions of space, and that they might

have a broad range of velocities, we can redefine pressure as:

P= Nm

3V

- Notice that we are using a statisticaltreatment. Use ofaverage velocities is only justified if we have many molecules.

- A more common expression of this equation takes into

account the definition of thekinetic energyof a particle:

= 1

2 m

- With this, the formulation of pressure from a microscopic point

of view is:

P= 2

3

N

V


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Temperature and velocity (kinetic energy)

- We have just defined pressure in terms of molecularproperties. Now we are going to define temperature also at the

microscopic level.

- Intuitively, temperature and molecular motion seem related

(hot molecules move faster). To prove this point, lets insert the

microscopic definition of pressure in the ideal-gas equation:

pV= nRT 2

3

N

V V=

N

NART

- If we solve for< EK>

=3

2

R

NAT=

3

2kBT

whereNA is Avogadros number, andkBis the Boltzmann

constant (R/NA= 1.381 1023

J K1

)


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Important observations:

Temperature and kinetic energy are related. Hence, we

commonly refer to the motion of a gas as thermal motion

Different gases at the same temperature have the same

average kinetic energy, regardless of their mass.

Since we know that< EK>= 12 m, we can find the

velocityof a gas at a given temperature:

1

2m=

3

2kBT

= vrms=

3kBT

m =

3RT

M

- Thatvelocity vrmsis called theroot-mean-squarevelocity ofa gas, and it depends on mass.

- The average kinetic energy of a gas does not depend on

mass (i.e. all gases have the same average kinetic energy at

the same temperature), but the rms velocity of the gas

molecules depends on the mass.


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Maxwell distribution of velocitiesIn the foregoing discussion,

average (more precisely root-mean-squared) velocities have

been considered. However, in a real gas, not all of the

molecules move at the same velocity. For an ideal gas, Maxwell

deduced that the distribution of velocities can be obtained fromthe following probability distribution:

f= F(s)s with F(s) = 4

M

2RT

3/2(s2)

e

Ms2

2RT

wherefis the probability of finding a particle moving at a speed

betweensands+s.The two most important components of the distribution are the

exponential and the (s2) terms.


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- The exponential term is aGaussianfunction (see page 14),

which has a bell shape. A Gaussian function has a maximum

around a central point and decays at high or low values of the

velocitys. This means that the probability to find particles that

are moving very fast or very slow is small.- The exponential depends on the mass Min such a way that

the largerM is, the faster the function decays. I.e., the larger

the mass, the lower the probability to find particles moving at

high speeds (see Fig. F11)

- The opposite occurs with the dependence on T. SinceT isdividing in the exponent, the largerTis, the smaller the

exponent is, and the slower the decay. At high T, the molecules

will be able to move faster. (See Fig. F10).

- Finally, the (s2) term becomes small at small s values,

introducing asymmetry in the distribution (SeeFigs.F10,F11)


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Extensive and intensive properties

Throughout the semester, you will sometimes hear the termsextensive and intensive properties.

Extensive properties are those that depend on the size of the

system. Examples of extensive properties are: mass, volume,

energy, heat capacity, ...

Intensive properties are those that do not depend on the size of

the system. Example of intensive properties are: temperature,

pressure, density, and all molar quantities (molar energy, molar

heat capacity, ...)It is generally more convenient to work with intensive

properties. For instance, if you want to report how much energy

is relased by combusting glucose, you can either say: 2808 kJ

for a sample that weighs 180.1 grams, or 2808 kJ/mol. Working

with intensive properties avoids having to specify theamount.


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p

Summary of equations

- Force: F= m a(mass x acceleration)- Work: W= F d(force x distance)

- Kinetic energy: EK= 1

2mv2

- Gravitational potential energyEP= mgh

- Pressure: P=F

A(force over area)

- Hydrostatic pressure: P= g h(density x gravity x height)- Equation of state for an ideal gas: pV= nRT

- Compressibility: Z= pVRT

= VmRT

- Van der Waals equation of state for a real gas:

(P+ an2

V2)(V nb) = nRT


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p

Summary of equations

- Pressure from kinetic theory of gases: p=2

3

N

V

- Temperature and kinetic energy of a gas: EK=3

2kBT or

EK= 32

RT (molar quantity)

- Root-mean-square velocity of a gas: vrms=

3RT

M- Maxwell speed distribution:

f= F(s)s with F(s) = 4

M

2RT

3/2(s2)

e

Ms2

2RT

(this last equation will be provided in a test if needed)
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