Subject PHYSICAL CHEMISTRY Paper No and Title 6, …CHEMISTRY PAPER No.6: PHYSICAL CHEMISTRY-II...

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PAPER No.6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules)

MODULE No. 34: Method of determining molar mass - I (osmometry)

Subject PHYSICAL CHEMISTRY

Paper No and Title 6, PHYSICAL CHEMISTRY-II (Statistical

Thermodynamics, Chemical Dynamics, Electrochemistry

and Macromolecules)

Module No and Title 34, Method for determining molar mass - I (osmometry)

Module Tag CHE_P6_M34

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Table of Contents

1. Learning Outcomes

2. Introduction

3. Measurement of Molar Mass and Size

4. Colligative Property Measurement

4.1 Number Average Molar Mass

4.2 Osmosis and Osmotic Pressure

4.3Expression of Osmotic Pressure for a Polymer Solution

5. Summary

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1. Learning Outcomes

After studying this module you shall be able to:

Learn about the osmometry technique for determination of Molar Mass of polymers.

Understand the importance of second virial coefficient in case of polymer soultions.

Mathematically prove that osmosis technique gives Number Average Molar Mass of a

polymer.

2. INTRODUCTION

Macromolecules, also known as polymers, are formed by the covalent linkages between

many repeating small molecules known as monomers. In the preparation of the polymer from

monomer units, the polymerization reactions proceed through different extents of reaction.

Thus, this results in polydispersity in molar masses. As a result, macromolecules do not have

a unique molar mass. Depending upon the method by which the macromolecule is produced,

same molecule may have different molar mass. Hence, for macromolecules the concept of

average of molar masses is used. In order to describe the distribution of molar masses, the

following averages are commonly used:

1. Number Average Molar Mass (𝑀𝑛̅̅ ̅̅ ) :

𝑀𝑛̅̅ ̅̅ =

∑ 𝑁𝑖𝑀𝑖𝑖

∑ 𝑁𝑖𝑖 (1)

Where:

Ni is the number of molecules each having molar mass Mi

2. Mass Average Molar Mass:

𝑀𝑤̅̅ ̅̅̅ = ∑ 𝑤𝑖𝑀𝑖𝑖 (2)

Where :

wi = Mass fraction

Mi = Molar mass

3. Z – average molar mass:

𝑀𝑧̅̅ ̅̅ =

∑ 𝑁𝑖𝑖 𝑀𝑖3

∑ 𝑁𝑖𝑖 𝑀𝑖2 (3)

4. Viscosity Average molar mass:

𝑀𝑣̅̅ ̅̅ = (

∑ 𝑁𝑖𝑀𝑖𝑎+1

𝑖

∑ 𝑁𝑖𝑀𝑖𝑖)

1𝑎⁄

(4)

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3. MEASUREMENT OF MOLAR MASS AND SIZE

The molar mass of polymers can be determined by a variety of physical and chemical

methods such as – by functional group analysis, by measuring the colligative properties, by

light scattering method, by ultracentrifugation, by viscosity measurement of dilute solutions.

All these methods, except the viscosity method, are in principle, absolute: Molar masses can

be calculated without reference to calibration by some other method. However, dilute

solution viscosity measurement method does not involve direct estimation of molar mass. Its

value lies in the simplicity of the technique and the fact that its results can be related

empirically to molar masses for many systems. Thus, dilute solution viscosity method yields

Viscosity Average Molar Mass which is not an absolute value but a relative mass based on

prior calibration with known molar mass for the same polymer-solvent-temperature

conditions.

With the exception of some types of end-group analysis, all molar-mass methods require

solubility of the polymer, and all involve extrapolation to infinite dilution or operation in a 𝜃

(theta) solvent in which ideal-solution behavior is attained.

Table 1 lists the different molar masses described above along with the experimental methods

to determine the various average molar mass.

Average Definition Methods

𝑀𝑛̅̅ ̅̅ ∑ 𝑁𝑖𝑀𝑖𝑖

∑ 𝑁𝑖𝑖

Osmotic pressure and other colligative properties, End group analysis

𝑀𝑤̅̅ ̅̅̅ ∑ 𝑤𝑖𝑀𝑖

𝑖

Light Scattering, Sedimentation velocity

𝑀𝑧̅̅ ̅̅

∑ 𝑁𝑖𝑖 𝑀𝑖

3

∑ 𝑁𝑖𝑖 𝑀𝑖2

Sedimentation equilibrium

𝑀𝑣̅̅ ̅̅

(∑ 𝑁𝑖𝑀𝑖

𝑎+1𝑖

∑ 𝑁𝑖𝑀𝑖𝑖)

1𝑎⁄

Intrinsic viscosity

Table 1: Description of average molar masses.

4. Colligative Property Measurement

The relations between the colligative properties and molar masses, for infinitely dilute

solutions, lay upon the fact that the activity of the solute becomes equal to its mole fraction as

the solution concentration becomes sufficiently small. The activity of the solvent must equal

to its mole fraction under these conditions, and follows that the depression of the activity of

the solvent by a solute is equal to mole fraction of the solute. The colligative property

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methods used are: lowering of vapour pressure, elevation of boiling point (ebulliometry),

depression in freezing-point (cryoscopy) and the osmotic pressure measurement

(osmometry). A colligative property measures the number of particles in solution. Therefore,

colligative property measurement of a polymer yields number average molar mass.

4.1 Number Average molar mass

Number average molar mass can be calculated by using a dilute solution of a polymer

through any of the methods from ebulliometry, cryoscopy and osmometry. Direct

measurement of lowering of vapour pressure for dilute polymer solutions, is not precise and

gives uncertain results. However, Vapour-phase osmometry, is an indirect method based

Clapeyron equation, using which the vapour pressure lowering of a polymer solution, at

equilibrium, can be related to a temperature difference that is comparable to or of the same

order of magnitude as those observed in cryoscopy and ebulliomtry. These methods require

calibration with low molar mass standards and they may produce reliable results for polymer

with molar masses < 30,000. The working equations for ebulliometric, cryoscopic and

osmometric measurements are as follows:

2b

x 0 f n

T RT 1lim

c H M

(5)

2f

x 0 f n

T RT 1lim

c H M

(6)

x 0 n

RTlim

c M

(7)

Where:

∆𝑇𝑏 = elevation in boiling-point

∆𝑇𝑓 = depression in freezing-point

= osmotic pressure

= density of the solvent

c = solute concentration (in g/mL or g/cm3)

vH and fH are the enthalpies of vaporization and fusion, respectively, of the solvent per

gram

The number-average molar mass nM has been introduced in equations (5) , (6) and (7) in

order to make theses equations applicable to poly-disperse solutes.

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Majorly, osmotic pressure measurements are used for studying macromolecules because

osmotic changes are larger than the changes in other colligative properties. This is due to the

fact that even a very small concentration of the solution produces a fairly large magnitude of

osmotic pressure while at this small concentration, the values of other colligative properties

such as boiling point elevation and freezing point depression are too small to be

experimentally determined. Thus, osmometry technique is more useful and thus, is more

widely used than other colligative techniques for polymer systems.

4.2 Osmosis and Osmotic pressure

When a solution and pure solvent are separated with the help of a semi-permeable membrane,

diffusion of solvent molecules take place from the side of pure solvent to solution side. This

flow of solvent molecules, from a region of their higher concentration (i.e. pure solvent) to a

region of their lower concentration (i.e solution) is known as Osmosis.

As a result of osmosis, the meniscus of the solution tube rises, whereas that of the pure

solvent falls, until equilibrium is reached, when the meniscus of the solution tube does not

rise further. At this stage, the excessive hydrostatic pressure created on the solution side

exactly balances the tendency of the solvent molecules to pass through the membrane (fig1).

This excessive hydrostatic pressure is known as the osmotic pressure of the solution and is

represented by the symbol Π. Thus, osmotic pressure arises because of the equilibrium

between the solvent in solution and the pure liquid solvent.

Fig1: The phenomenon of osmosis and generation of osmotic pressure

The osmotic pressure of a solution depends on its concentration i.e larger the concentration

larger will be the osmotic pressure. The relationship of osmotic pressure with the

concentration of the solution can be derived thermodynamically. In this module we will not

consider the entire derivation; we will only use the following results:

Π𝑉𝑙,𝑚∗ = −𝑅𝑇 ln 𝑥1 (8)

Where:

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𝑉𝑙,𝑚∗ = molar volume of the solvent

𝑥1 = amount fraction of solvent in the solution

We will simplify equation (8) for a dilute solution using the following approximations:

1. Since the solution is dilute 𝑥2 (𝑎𝑚𝑜𝑢𝑛𝑡 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑜𝑙𝑢𝑡𝑒) ≪ 1 , therefore the

term ln 𝑥1 can be approximated as:

ln 𝑥1 = ln(1 − 𝑥2) ≃ − 𝑥2 (9)

2.The amount fraction of solute is given by:

𝑥2= 𝑛2

𝑛1+𝑛2=

𝑛2𝑛1

(10)

3.For a solution , the total volume V can be written in terms of the partial molar volumes

following the additivity rule:

𝑉 = 𝑛1𝑉1 + 𝑛2𝑉2

Where :

V1 = partial molar volume of the solvent

V2 = Partial volume of the solute.

Thus , for a dilute solution, we make the following assumptions:

i) Partial molar volume of the solvent in the solution is same as that of the pure solvent i.e.:

V1 = 𝑉𝑙,𝑚∗

ii) The factor n2V2 << n1V1 since n2<<n1 and can be neglected.

Thus, we have

V≃ 𝑛1𝑉1 = 𝑛1𝑉𝑙,𝑚∗ (11)

Subsituting equations 9, 10 and 11 in equation 8 we get:

𝑉

𝑛1Π = −𝑅𝑇(−𝑥2) = 𝑅𝑇𝑥2

Or

𝑉

𝑛1Π = 𝑅𝑇

𝑛2

𝑛1

or

𝑉Π = 𝑛2𝑅𝑇

or

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Π = (𝑛2

𝑉) 𝑅𝑇 = 𝑐𝑅𝑇 (12)

Where:

Π = Osmotic pressure

c = molar concentration of solute in the solvent

R = Gas constant

T = Temperature

Equation 12 is known as the van’t Hoff equation for determining osmotic pressure of the

solution. This equation is exactly identical to the ideal gas equation i.e

pV = nRT (13)

Except that Π replaces the gas pressure p. This resemblance indicates that the solute

molecules are dispersed in the solvent in the similar way as the gas molecules are dispersed in

the empty space. Thus, the solute is analogous to the gas molecules and the solvent is

analogous to the empty space between the gas molecules.

4.3 Expression of Osmotic pressure for a polymer solution

We have already studied that osmotic pressure measurement is a method primarily used in the

determination of the molar masses of substances which have either very low solubility or

very high molar masses, such as proteins or polymers.

The van’t hoff equation (12) is valid either only at infinite dilution or to a very dilute solution

where the solute-solute interaction is negligible. Thus, this equation might not be applicable

to a substance of high molar mass at concentrations which are employed for experimental

methods. Moreover a polymer solution does not behave as an ideal solution because there is a

large difference in the molar volumes of polymeric solute and that of low molar mass solvent.

Therefore we take this fact into consideration and apply following conditions to equation (8).

i.e.

Firstly, we perform the expansion of logarithm in expression (9):

ln 𝑥1 = ln(1 − 𝑥2)

= − (𝑥2 + 𝑥2

2

2+

𝑥23

3+ … . ) (14)

And thus, expression 8 can be expressed as:

Π𝑉𝑙,𝑚∗ = 𝑅𝑇 (𝑥2 +

𝐵′𝑥22

2+

𝐶′𝑥23

3+ … . ) (15)

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Where :

B′ = second virial coefficient

C′ = Third virial coefficient

If the concentration if solution is expressed in mass of solute per unit volume of solution,

then:

(16)

Also,

𝑛2 = 𝑚2

𝑀2 and 𝑛1 =

𝑉

𝑉𝑙,𝑚∗ (17)

Using equation 17 in equation 11 we have:

(18)

Substituting equation 18 in equation 15 we get:

(19)

The higher terms are insignificant. Thus retaining only the first two terms on the right hand

side, we get:

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(20)

Equation (20) , predicts that for a non-ideal solution , the plot of Π/𝑐2RT and 𝑐2 is linear:

The slope obtained from this graph is equal to the second virial coefficient B′ and its intercept

is equal to the reciprocal of the molar mass M2 of the solute.

Fig 2: plot of 𝚷/𝒄𝟐 (reduced osmotic pressure) versus 𝒄𝟐

We can also plot Π/𝑐2 (reduced osmotic pressure) versus 𝑐2 (Fig 3). From this graph, which

when extrapolated to 𝑐2= 0 gives:

Slope= B′RT

Intercept= RT/M2

i.e lim𝐶2→0

(Π

𝑐2) = 𝑅𝑇/𝑀2

4.3.1 Discussion of Second Virial Coefficient (𝑩′).

The virial coefficients are determined empirically for a given solute-solvent system. The

second virial coefficient i.e 𝐵′ represents the interaction of a single solute particle with the

solvent while the higher order virial coefficients are associated with correspondingly larger

number of solute particle clusters interacting with the solvent. Thus, second virial coefficient

is a measure of inter-particle interactions and hence, is temperature dependent.

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Fig 3: Plot of reduced osmotic pressure 𝚷/𝒄𝟐 versus 𝒄𝟐 at different temperatures

The temperature at which, for a given polymer-solvent pair, the polymer exists in an

unperturbed state is termed as theta (θ) temperature. This is also the temperature at which the

second virial coefficient becomes zero. In Fig 2, T2 is θ temperature.

This temperature is also the critical solution temperature for a polymer of infinite molar mass.

Since 𝐵′ is temperature dependent therefore, for each polymer-solvent pair, there exists a

unique temperature at which 𝐵′= 0. At theta temperature, there exists no interactions between

different polymer chains; the polymer solution behaves as an ideal solution at this

temperature.

Above theta temperature, i.e T1 (Fig 2) there is positive deviation and a positive value

of 𝐵′ indicates that the polymer is insoluble in the solvent, while below theta temperature,

there is negative deviation (T3) (Fig 2) and the polymer chain segments will attract one other

and eventually phase separation will occur. Moreover, a negative value of 𝐵′ is indicative of

a ‘good solvent’ (i.e. the polymer will be highly soluble in the solvent due to favourable

polymer-solvent intermolecular interactions).

4.3.2 Mathematical proof: Osmotic pressure gives number average molar mass.

The osmotic pressure measurement leads to number-average molar mass because osmotic

pressure is a colligative property which depends upon the number of molecules of the

polymer and not on their masses. All the molecules, whether heavy or light, make equal

contribution.

Mathematically:

Since osmotic pressure (Π) consists additively of the partial osmotic pressures Π1, Π2, Π3,…

for the molecules of different degrees of polymerization 1,2,3 , etc. Thus from equation 12:

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Π = Π1 + Π2 + Π3 +……..

= 𝑐1𝑅𝑇

𝑀1 +

𝑐2𝑅𝑇

𝑀2 +

𝑐3𝑅𝑇

𝑀3 +…..

The same holds for the total concentration:

c = c1+ c2 +c3 +….

Thus, the osmotic-average molar mass can be written as:

�̅�osmotic =(𝑐1 +𝑐2 +𝑐3 +⋯ )𝑅𝑇

Π1+Π2+Π3+…

=(𝑐1 +𝑐2 +𝑐3 +⋯ )𝑅𝑇

𝑅𝑇(𝑐1

𝑀1⁄ +

𝑐2𝑀2

⁄ +𝑐3

𝑀3⁄ +⋯ )

=∑ 𝑐𝑖𝑖

∑ 𝑐𝑖𝑖∑ 𝑀𝑖𝑖

⁄ = �̅�n

Hence the expression proves that osmotic pressure yields number average molar mass (�̅�n).

5. SUMMARY

In this module, we have learnt:

Osmometry technique for determination of molar mass of polymers.

Osmotic pressure of a solution is given by the van’t hoff equation i.e.

Π = (𝑛2

𝑉) 𝑅𝑇 = 𝑐𝑅𝑇

Polymer solution is a non – ideal solution. Thus osmotic pressure for a polymer

solution is given by:

Where:

𝐵′ = second virial coefficient.

𝑀2= Molar mass of the polymer.

Osmometry technique provides number average molar mass i.e �̅�n

𝑀𝑛̅̅ ̅̅ =

∑ 𝑁𝑖𝑀𝑖𝑖

∑ 𝑁𝑖𝑖

Subject PHYSICAL CHEMISTRY Paper No and Title 6, …CHEMISTRY PAPER No.6: PHYSICAL CHEMISTRY-II...

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