polymer chemistry sem-6, dse-b3 molecular weight of

POLYMER CHEMISTRY

SEM-6, DSE-B3MOLECULAR WEIGHT OF POLYMERS

PART-2, PPT-23

Dr. Kalyan Kumar Mandal

Associate Professor

St. Paul’s C. M. College

Kolkata

Polymer ChemistryMolecular Weight of Polymers: Part-2

Contents

• The Number-Average Molecular Weight ( ഥ𝑴n)

• The Weight-Average Molecular Weight ( ഥ𝑴w)

• Viscosity-Average Molecular Weight ( ഥ𝑴v)

• Molecular Weight Averages

• Determination of the Number-Average Molecular Weight: Colligative Properties: Osmotic Pressure

Number-Average Molecular Weight

• Since typical polymers consist of mixtures of many molecular species of (nearly) identical

chemical structure but varying in chain length or molecular weight, molecular weight

methods always yield average values. The molecules produced in polymerization reactions

have lengths that are distributed in accordance with a probability function which is governed

by the mechanism of the reaction and by the conditions under which it has been carried out.

• The measurement of the colligative properties in effect counts the number of moles of solute

per unit weight of sample. This number is the sum over all molecular species of the number

of moles Ni of each species present:

𝒊

∞

𝑵𝒊

• The total weight w of the sample is similarly the sum of the weights of each molecular

species,

𝒘 =𝒊

∞

𝒘𝒊 =𝒊

∞

𝑵𝒊𝑴𝒊

This Lecture is prepared by Dr. K. K. Mandal, SPCMC, Kolkata

Number-Average Molecular Weight

• The average molecular weight given by these methods is known as the number average

molecular weight, 𝑀n. ഥ𝑴n is defined as the total weight w of all the molecules in a polymer

sample divided by the total number of moles present. Thus the number-average molecular

weight is expressed in Equation 1 where the summations are over all the different sizes of

polymer molecules from i = 1 to i = ∞ and Ni is the number of moles whose weight is Mi. By

the definition of molecular weight as weight of sample per mole,

𝑴𝒏 =σ𝒊𝒘𝒊

σ𝒊𝑵𝒊=σ𝒊𝑵𝒊𝑴𝒊

σ𝒊𝑵𝒊=

σ𝒊𝒘𝒊

σ𝒊(𝒘𝒊𝑴𝒊)−−−−𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏

• The number-average molecular weight ഥ𝑀n is determined by experimental methods that count

the number of polymer molecules in a sample of the polymer. The methods for measuring ഥ𝑀n

are those that measure the colligative properties of solutions-vapour pressure lowering

(vapour pressure osmometry), freezing point depression (cryoscopy), boiling point elevation

(ebulliometry), and osmotic pressure (membrane osmometry).


Number-Average Molecular Weight• The colligative properties are the same for small and large molecules when comparing

solutions at the same molal (or mole fraction) concentration. For example, a 1-molal solution

of a polymer of molecular weight 105 has the same vapour pressure, freezing point, boiling

point, and osmotic pressure as a 1-molal solution of a polymer of molecular weight 103 or a

1-molal solution of a small molecule such as hexane.

• Equation 1 can also be written as,

𝑴𝒏 =

𝒊

𝑵𝒊𝑴𝒊 −−− −𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐

where 𝑵𝒊 is the mole fraction (or the number-fraction) of molecules of size Mi.

• The most common methods for measuring ഥ𝑴𝒏 are membrane osmometry and vapour pressure

osmometry since reasonably reliable commercial instruments are available for those methods.

Vapour pressure osmometry, which measures vapour pressure indirectly by measuring the

change in temperature of a polymer solution on dilution by solvent vapour, is generally useful

for polymers with ഥ𝑴𝒏 below 10,000-15,000.


Number-Average Molecular Weight• Above that molecular weight limit, the quantity being measured becomes too small to detect

by the available instruments. Membrane osmometry is limited to polymers with ഥ𝑴𝒏 above

about 20,000-30,000 and below 500,000. The lower limit is a consequence of the partial

permeability of available membranes to smaller-sized polymer molecules. Above molecular

weights of 500,000, the osmotic pressure of a polymer solution becomes too small to measure

accurately.

• End-group analysis is also useful for measurements of ഥ𝑴n for certain polymers. For example,

the carboxyl end groups of a polyester can be analyzed by titration with base and carbon-

carbon double bond end groups can be analyzed by 1H NMR. Accurate end-group analysis

becomes difficult for polymers with ഥ𝑴𝒏values above 20,000-30,000.

• Evaluation of number average molecular weight is useful in understanding the

polymerization mechanism and kinetics. It also assumes prime importance in determining the

solution properties of the polymer, commonly known as the colligative properties. Polymer

molecules of lower molecular weight contribute equally and enjoy equal status with those of

higher molecular weight in determining these properties.


Weight -Average Molecular Weight• Light scattering by polymer solutions, unlike colligative properties, is greater for larger-sized

molecules than for smaller-sized molecules. The average molecular weight obtained from

light-scattering measurements is the weight-average molecular weight ഥ𝑴w defined as

𝑴𝒘 =σ𝒊𝑵𝒊𝑴𝒊

𝟐

σ𝒊𝑵𝒊𝑴𝒊=σ𝒊𝒘𝒊𝑴𝒊

σ𝒊𝒘𝒊−−−−𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟑

where wi is the weight fraction of molecules whose weight is Mi.

• ഥ𝑴w can also be defined as,

𝑴𝒘 =σ𝒊 𝒄𝒊𝑴𝒊

σ𝒊 𝒄𝒊=σ𝒊 𝒄𝒊𝑴𝒊

𝒄=σ𝒊𝑵𝒊𝑴𝒊

𝟐

σ𝒊𝑵𝒊𝑴𝒊−−−−𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟒

• where ci is the weight concentration of Mi molecules, c is the total weight concentration of all

the polymer molecules, and the following relationships hold:

𝒘𝒊 =𝒄𝒙

𝒄−−−−−𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟓

𝒄𝒊 = 𝑵𝒊𝑴𝒊 −−−−𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟔

𝒄 = σ𝒊 𝒄𝒙 = σ𝒊𝑵𝒊𝑴𝒊 −−−−𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟕

Weight -Average Molecular Weight

• Since the amount of light scattered by a polymer solution increases with molecular weight,

this method becomes more accurate for higher polymer molecular weights. There is no upper

limit to the molecular weight that can be accurately measured except the limit imposed by

insolubility of the polymer. The lower limit of ഥ𝑴w by the light scattering method is close to

5000-10,000. Below this molecular weight, the amount of scattered light is too small to

measure accurately.

• Weight average molecular weight is important in relation to bulk properties of polymers that

reflect their load bearing capacity. Softening, hot deformation, tensile and compressive

strength, modulus and elongation, toughness and impact resistance and some other related

bulk properties of polymer are better appreciated on the basis of weight average molecular

weight, keeping in mind, however, the influence of chemical nature of the repeat units,

degree of branching and cross-linking, thermal or thermomechanical history of the sample.


Viscosity-Average Molecular Weight

• Solution viscosity is also useful for molecular-weight measurements. Viscosity, like light

scattering, is greater for the larger-sized polymer molecules than for smaller ones. However,

solution viscosity does not measure ഥ𝑴w since the exact dependence of solution viscosity on

molecular weight is not exactly the same as light scattering.

• Solution viscosity measures the viscosity-average molecular weight ഥ𝑴v defined by

𝑴𝒗 = [

𝒊

𝒘𝒊𝑴𝒊𝒂]𝟏𝟐] = [

σ𝒊𝑵𝒊𝑴𝒊𝟏+𝒂

σ𝒊𝑵𝒊𝑴𝒊]𝟏𝒂−−−−𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟖

where a is a constant.

• The viscosity- and weight-average molecular weights are equal when a is unity. ഥ𝑴v is lessthan ഥ𝑴w for most polymers, since a is usually in the range 0.5-0.9. However, ഥ𝑴v is muchcloser to ഥ𝑴w than ഥ𝑴n, usually within 20% of ഥ𝑴w. The value of a is dependent on thehydrodynamic volume of the polymer, the effective volume of the solvated polymermolecule in solution, and varies with polymer, solvent, and temperature.


Molecular Weight Averages

• More than one average molecular weight is required to reasonably characterize a polymer

sample. There is no such need for a monodisperse product (i.e., one composed of molecules

whose molecular weights are all the same) for which all three average molecular weights are

the same. The situation is quite different for a polydisperse polymer where all three

molecular weights are different if the constant a in Equation 8 is less than unity, as is the

usual case. A careful consideration of Equations 1 through 2 shows that the number-,

viscosity-, and weight-average molecular weights, in that order, are increasingly biased

toward the higher-molecular-weight fractions in a polymer sample. For a polydisperse

polymer

𝑴𝒘 > 𝑴𝒗 > 𝑴𝒏

• with the differences between the various average molecular weights increasing as the

molecular-weight distribution broadens. A typical polymer sample will have the molecular

weight distribution shown in Figure 1 the approximate positions of the different average

molecular weights are indicated on this distribution curve.


Molecular Weight Averages• For most practical purposes, one usually characterizes the molecular weight of a polymer

sample by measuring 𝑴𝒏 and either 𝑴𝒘 or 𝑴𝒗 . 𝑴𝒗 is commonly used as a close

approximation of 𝑴𝒘, since the two are usually quite close (within 10–20%). Thus in most

instances, one is concerned with the 𝑴𝒏 and 𝑴𝒘 of a polymer sample. The former is biased

toward the lower-molecular-weight fractions, while the latter is biased toward the higher-

molecular weight fractions.

• The ratio of the two average molecular weights

𝑴𝒘/𝑴𝒏 depends on the breadth of the distribution

curve (Figure 1) and is often useful as a measure

of the polydispersity in a polymer. The value of

𝑴𝒘 / 𝑴𝒏 would be unity for a perfectly

monodisperse polymer. The ratio is greater than

unity for all actual polymers and increases with

increasing polydispersity.



• The characterization of a polymer by 𝑴𝒏 alone, without regard to the polydispersity, can

be extremely misleading, since most polymer properties such as strength and melt viscosity

are determined primarily by the size of the molecules that make up the bulk of the sample by

weight.

• Polymer properties are much more dependent on the larger-sized molecules in a sample than

on the smaller ones. Thus, for example, let us consider a hypothetical mixture containing

95% by weight of molecules of molecular weight 10,000, and 5% of molecules of molecular

weight 100 (The low-molecular-weight fraction might be monomer, a low-molecular weight

polymer, or simply some impurity.).

• The 𝑴𝒏 and 𝑴𝒘 are calculated, using the appropriate equations, as 1680 and 9505,

respectively. The use of the 𝑴𝒏 value of 1680 gives an inaccurate indication of the properties

of this polymer.


Molecular Weight Averages• The properties of the polymer are determined primarily by the 10,000-molecular-weight

molecules that make up 95% of the weight of the mixture. The weight-average molecular

weight is a much better indicator of the properties to be expected in a polymer. The utility of

𝑴𝒏 resides primarily in its use to obtain an indication of polydispersity in a sample by

measuring the ratio 𝑴𝒘/𝑴𝒏.

• In addition to the different average molecular weights of a polymer sample, it is frequently

desirable and necessary to know the exact distribution of molecular weights. As indicated

previously, there is usually a molecular weight range for which any given polymer property

will be optimum for a particular application. The polymer sample containing the

greatest percentage of polymer molecules of that size is the one that will have the optimum

value of the desired property. Since samples with the same average molecular weight may

possess different molecular weight distributions, information regarding the distribution

allows the proper choice of a polymer for optimum performance.


Molecular Weight Averages• The methods that are used in the past to determine the molecular weight distribution of a

polymer sample, e.g., fractional extraction and fractional precipitation are laborious and

determinations of molecular weight distributions were not routinely performed. However, the

development of size exclusion chromatography (SEC), also referred to as gel permeation

chromatography (GPC) and the availability of automated commercial instruments have

changed the situation. Molecular-weight distributions are now frequently performed using

SEC.

• Size exclusion chromatography involves the permeation of a polymer solution through a

column packed with microporous beads of crosslinked polystyrene. The packing contains

beads of different-sized pore diameters. Molecules pass through the column by a

combination of transport into and through the beads and through the interstitial volume (the

volume between beads). Molecules that penetrate the beads are slowed down more in moving

through the column than molecules that do not penetrate the beads; in other words, transport

through the interstitial volume is faster than through the pores.



• The smaller-sized polymer molecules penetrate all the beads in the column since their

molecular size (actually their hydrodynamic volume) is smaller than the pore size of the

beads with the smallest-sized pores. A larger-sized polymer molecule does not penetrate all

the beads since its molecular size is larger than the pore size of some of the beads. The larger

the polymer molecular weight, the fewer beads that are penetrated and the greater is the

extent of transport through the interstitial volume. The time for passage of polymer molecules

through the column decreases with increasing molecular weight.

• The use of an appropriate detector (refractive index, viscosity, light scattering) measures the

amount of polymer passing through the column as a function of time. This information and a

calibration of the column with standard polymer samples of known molecular weight allow

one to obtain the molecular weight distribution in the form of a plot such as that in Figure 1.

Not only does SEC yield the molecular weight distribution, but 𝑴𝒏 and 𝑴𝒘 (and also 𝑴𝒗 if a

is known) are also calculated automatically. SEC is now the method of choice for

measurement of 𝑴𝒏 and 𝑴𝒘 since the SEC instrument is far easier to use compared to

methods such as osmometry and light scattering.

Determination of the Number-Average Molecular Weight

• The number-average molecular weight, 𝑴𝒏, involves a count of the number of molecules of

each species, NiMi, summed over i, divided by the total number of molecules as shown in

Equation 1. It is the simple average that most people think about. For many simple single-

peaked distributions, 𝑴𝒏 is near the peak (Figure 1).

𝑴𝒏 =σ𝒊𝑵𝒊𝑴𝒊

σ𝒊𝑵𝒊=

σ𝒊𝒘𝒊

σ𝒊(𝒘𝒊𝑴𝒊)−−− −𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏

• There are two important groups of methods for determining 𝑴𝒏.

1. End-Group Analyses; 2. Colligative Properties

• End-Group Analyses: The first group of methods involves end-group analyses. Many types

of syntheses leave a special group on one or both ends of the molecule, such as hydroxyl and

carboxyl. These can be titrated or analyzed instrumentally by such methods as infrared. For

molecular weights above about 25,000 g mol-1, however, the method becomes insensitive

because the end groups are present in too low a concentration.


Colligative Property Measurement

• The second group of methods makes use of the colligative properties of solutions.

Colligative properties depend on the number of molecules in a solution, and not their

chemical constitution. The relations between the colligative properties and molecular weight

for infinitely dilute solutions rest upon the fact that the activity of the solute in a solution

becomes equal to its mole fraction as the solute concentration becomes sufficiently small.

• Colligative properties reflect the chemical potential of the solvent in solution. Alternatively, a

colligative property is a measure of the depression of the activity of the solvent in solution,

compared to the pure state. The activity of the solvent must equal its mole fraction under

these conditions, and it follows that the depression of the activity of the solvent by a solute is

equal to the mole fraction of' the solute.

• The relative applicability of the colligative methods to polymer solutions is demonstrated in

Table 1.



• The colligative properties include boiling point elevation (ebulliometry), freezing-point

depression (cryoscopy), vapour pressure lowering, and osmotic pressure (osmometry). The

basic equations for boiling point elevation, and melting point depression may be represented

as the following.

𝐥𝐢𝐦𝒄→𝟎

∆𝑻𝒃𝒄

=𝑹𝑻𝟐

𝝆∆𝑯𝑽

𝟏

𝑴𝒏

−−−−𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟗


∆𝑻𝒇

𝒄= −

𝑹𝑻𝟐

𝝆∆𝑯𝒇

𝟏

𝑴𝒏

−−−−𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏𝟎

• where ∆Tb, and ∆Tf are the boiling point elevation, and freezing point depression,

respectively. ρ is the solvent density. ∆Tb and ∆Tf are the latent heats of vaporization and

fusion per gram of solvent, and c is the solute concentration in grams per cubic centimeter.

• The number-average molecular weight ഥ𝑴n, defined below, has been inserted to make the

equations applicable to polydisperse solutes.

𝑴𝒏 =σ𝒊𝑵𝒊𝑴𝒊

σ𝒊𝑵𝒊=

σ𝒊𝒘𝒊

σ𝒊(𝒘𝒊𝑴𝒊)−−− −𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏


• If the vapour pressure of the solute is small, and the solvent follows Raoult’s vapour

pressure law,𝑷𝟏𝟎 − 𝑷𝟏

𝑷𝟏𝟎 = 𝒙𝟐 −−−−𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏𝟏

• where 𝑷𝟏𝟎 is the vapour pressure of the pure solvent, P1 is that of the solution, and x2 is the

mole fraction of the solute. The osmotic pressure π depends on the molecular weight as

follows:


𝝅

𝒄=𝑹𝑻

𝑴𝒏

−−−−𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏𝟐

• Taking membrane osmometry as an example, Equation 10 can be written separately for

each species i in the mixture present. Omitting the limit sign for convenience and

transposing, one obtains

𝝅𝒊 = 𝑹𝑻𝒄𝒊𝑴𝒊

−−−−𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏𝟑



• Summing over i, σ𝝅𝒊 may be replaced by π, since each solute molecule contributes

independently to the osmotic pressure, and it is to be noted that c = σ𝒄𝒊. The mixture is

then described by an unspecified average molecular weight 𝑀:

𝝅 = 𝑹𝑻𝒄𝒊𝑴𝒊

= 𝑹𝑻𝒄

𝑴−−−−𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏𝟒

• Solving, one gets

𝑴 =𝒄

σ𝒄𝒊𝑴𝒊

−−−−𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏𝟓

• For unit volume the c’s may be replaced by w’s. Noting that wi = MiNi shows that 𝑴 = 𝑴𝒏.

End-group methods as well as colligative methods give the number-average molecular

weight. The number average is very sensitive to changes in the weight fractions of low-

molecular-weight species, and relatively insensitive to similar changes for high-molecular-

weight species.



• Though the osmotic pressure is inversely proportional to solute molecular weight, the

relatively large, measurable values of osmotic pressure (π) obtained even for dilute solutions

make osmotic pressure measurements valuable in determining molecular weights of

substances with high molecular weights like polymers.

• Typical values for the colligative properties for a polymer having a molecular weight of

20,000 g/mol are shown in Table 1. Only osmotic pressure is large enough for fruitful

studies at this molecular weight or higher.

Table 1: Comparison of the Colligative Solution properties of a

1% polymer solution with M = 20000 g/mol

Property Value

Vapour pressure lowering 4.0 x 10-3 mm Hg

Boiling point elevation 1.3 x 10-3 °C

Freezing point depression 2.5 x 10-3 °C

Osmotic pressure 15 cm solvent


Osmotic Pressure: Thermodynamic Basis

• Polymer solutions exhibit osmotic pressures because the chemical potentials of the pure

solvent and the solvent in the solution are unequal. Because of this inequality, there is a net

flow of solvent, through a connecting membrane, from the pure solvent side to the solution

side. When sufficient pressure is built up on the solution side of the membrane, so that the

two sides have the same activity, equilibrium will be restored.

• While the discussion above gives an exact thermodynamic interpretation of the

phenomenon, a consideration in terms of the number of solute molecules per unit volume is

useful for practical calculations. An analogy exists between Equation 12 and the ideal gas

law:𝑷𝑽 = 𝒏𝑹𝑻 −−− −𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏𝟔

• Where n is in moles, as usual. The quantity n/V is equal to c/M, yielding

𝑷 =𝒄

𝑴𝑹𝑻 −−− −𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏𝟕

• Setting the gas pressure equal to the osmotic pressure, P = π, and rearranging, Equation 12

is obtained.


Instrumentation

• A typical static osmometer design includes a membrane that permeates only the solvent, a

capillary, and a reference capillary. Typical membranes are made from regenerated cellulose

or other microporous materials. Such an instrument usually requires 24 h to reach

equilibrium.

• There are now several types of automatic osmometers that operate with essentially zero flow

and that reach equilibrium very rapidly, usually within minutes. Osmotic equilibrium depends

on an equal and opposite pressure being developed. The critical part of their design relates to

the method of automatic adjustment of the osmotic pressure of the solution side so that the

activity of the two sides is equal.

• Since several concentrations usually need to be run, the time required to determine a

molecular weight by osmometry has been reduced from a week to a few hours by these

automatic instruments.


The Flory θ-Temperature

• The basis for determining the molecular weight by osmometry has been given in Equation

12. At finite concentrations, interactions between the solvent and the solute result in the virial

coefficients A2, A3, and so on. The full equation may be written𝝅

𝒄= 𝑹𝑻

𝟏

𝑴𝒏+ 𝑨𝟐𝒄 + 𝑨𝟑𝒄

𝟐 +⋯ −−−−𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏𝟖

• Interactions between one polymer molecule and the solvent result in the second virial

coefficient, A2. Multiple polymer-solvent interactions produce higher virial coefficients, A3,

A4, and so on. A further interpretation of the virial coefficients is considered along with

solution theory. For medium molecular weights, the slope is substantially linear below about

1% solute concentration. Of course, 𝑨𝟏 =𝟏

𝑴𝒏.

• The quantity A2 depends on both the temperature and the solvent, for a given polymer. A

unique and much desired state arises when A2 equals zero. The temperature at which this

condition holds is called the Flory θ-temperature.


The Flory θ-Temperature

• In this state, π/c is independent of concentration, so that only one concentration need be

studied to determine 𝑀𝑛. Since it is also the state where an infinite molecular weight polymer

just precipitates and χ1= 0.5, considerable care must be taken to keep the polymer in solution.

Table 2 presents a selected list of polymers and their theta solvents.

Table 2: Polymers and their θ-solvents

Polymer Solvent Temperature (°C)

cis-Polybutadiene n-Heptane -1

Polyethylene Biphenyl 125

Poly(n-butyl acrylate) Benzene/methanol (52/48) 25

Polystyrene Cyclohexane 34

Poly(oxytetramethylene) Chlorobenzene 25

• It must be emphasized that molecular weights determined by any of the colligative

properties, osmometry in particular, are absolute molecular weights; that is, the values are

determined by theory and not by prior calibration. The practical limit of osmometry is about

500,000 g/mol because the pressures become too small.


polymer chemistry sem-6, dse-b3 molecular weight of

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