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Chapter 5
Colligative Properties
5.1 Introduction
Properties of solutions that depend on the number of molecules present and not on the kind of molecules are called colligative properties. These properties include boiling point elevation, freezing point depression, and osmotic pressure. Historically, colligative properties have been one means for determining the molecular weight of unknown compounds. In this chapter we discuss using colligative properties to measure the molecular weight of polymers. Because colligative properties depend on the number of molecules, we expect, and will show, that colligative property experiments give a number average molecular weight.
5.2 Boiling Point Elevation
Figure 5.1 shows the vapor pressure of a liquid for pure liquid and for a solution with that liquid as the solvent. In an ideal solution, the vapor pressure of the solvent, PA, is reduced from the vapor pressure of a pure liquid, P (^) A◦, to XAP (^) A◦ where XA is the mole fraction of liquid A. This reduction is reflected in a shift to the right of the vapor-pressure curves in Fig. 5.1. By definition, boiling point is the temperature at which the vapor pressure of the liquid reaches 1 atm. Thus, the right-shift caused by the dissolution of component B in solvent A causes the boiling point to increase. This increase, ∆Tb, is the boiling point elevation effect. A well known result from introductory chemistry is that the boiling point elevation is propor- tional to the molar concentration of solute particles
∆Tb = Kbm (5.1)
where m is the molality of solute molecules and Kb is the boiling point elevation coefficient that is a function of only the solvent. Molality is the number of moles of component B per 1000 grams of solvent. If we prepare a solution of an unknown compound of molecular weight B at a concentration
65
66 CHAPTER 5. COLLIGATIVE PROPERTIES
Vapor Pressure
Vapor Pressure
Solution
Temperature
Pure Solvent
1 atm
Tb
∆T
Figure 5.1: Boiling point elevation effect is a consequence of the effect of solute molecules on the vapor pressure of the solvent.
c in g/cm^3 , then m =^1000 M c B ρ^
where ρ is the density of the solvent (in g/cm^3 ). Substituting into the expression for ∆Tb gives
MB =^1000 ρ∆KTbc b
or ∆Tb c =
1000 Kb ρMB^ (5.4) For a given solvent (e.g., water where Kb = 0.52 and ρ = 1.00) and concentration (c), all terms in Eq. (5.4) are known except for MB. Thus, measuring ∆Tb can be used to determine the molecular weight MB. We can also express boiling point elevation in terms of mole fraction. Mole fraction is
XB = M^ cV B ρV MA +^ McVB
≈ cM ρMA B
where V is total volume and MA is molecular weight of the solvent. The boiling point elevation becomes ∆Tb =^1000 MKb A
XB (5.6)
To apply boiling point elevation to polymers, we begin by using solution thermodynamics to derive an expression for ∆Tb. At equilibrium, the chemical potential of the vapor is equal to the chemical potential of the liquid
μvapA = μliqA = μ◦ A + RT ln XA or μ
vapA − μ◦ A RT = ln^ XA^ (5.7)
68 CHAPTER 5. COLLIGATIVE PROPERTIES
where Xi is the mole fraction of polymer with molecular weight Mi. We more conveniently rewrite Xi in terms of concentration:
Xi =
cMiV i ρV MA +^
i cMiVi
≈ c ρMiMA i
where ci is the concentration in weight/unit volume (e.g., g/cm^3 ) of polymer with molecular weight i. The approximation in this expression is valid for dilute solutions in which the number of moles of solvent is much greater than the total number of moles of polymer. Summing the mole fractions, Xi, results in ∑ i
Xi = cM ρA
∑^ i^ Mcii i ci
= cM ρA
i
wi Mi^ =^
cMA ρMN^ (5.17)
where c =
i
ci (5.18)
The final expression for the boiling point elevation becomes
∆Tb c =^
MART (^) b^2 ρ∆HvapMN
It is common to express the boiling point elevation in terms of the latent heat of vaporization, lvap, defined as energy or vaporization per unit weight or
lvap = ∆ MHvap A = Jg//molemole = heat of vaporization in J/g (5.20)
The boiling point elevation becomes
∆Tb c =^
RT (^) b^2 ρlvapMN^ (5.21) Except for incorporation of polydispersity, there is nothing new about the boiling point eleva- tion expression for polymer solutions vs. the comparable expression for small molecule solutions. In polymers, however, the solution is more likely to be non-ideal. For this equation to apply we will probably need to use very low concentrations or techniques to extrapolate to very low concen- trations. For an example, let’s consider a solution of polystyrene in benzene. For benzene ρ = 0.8787 g/cm^3 , Tb = 55◦C, and lvap = 104 cal/g. We assume a relatively concentrated solution of c = 1 g/cm^3 of a polymer with molecular weight MN = 20, 000. The change in the boiling point elevation for this solution is ∆Tb = 1. 4 × 10 −^3 ◦C. This boiling point elevation is very small. It is probably beyond the accuracy of most temperature measuring equipment. The small change arises despite relatively ideal conditions of a fairly concentrated solution and a low molecular weight polymer. More dilute solutions or higher molecular weight polymers would give an even smaller ∆Tb. The problem with polymer solutions is that for a given weight of material, the polymer solution will
5.3. FREEZING POINT DEPRESSION 69
have many less molecules than the comparable small molecule solution. When there are a small number of molecules, the change in boiling point (a colligative property) is small. The problem with the boiling point elevation method applied to polymer solutions is that it is not sensitive enough. It has found some use with polymers but it is limited to polymers with relatively low molecular weights. (e.g., MN less than 20,000 g/mol).
5.3 Freezing Point Depression
A similar analysis (but with sign changes) can be applied to the freezing point depression of a polymer solution. The final result is ∆Tf c =^
RT (^) f^2 ρlf MN
where Tf is the freezing point of the solvent and lf is the latent heat of fusion. We consider the same example of polystyrene in benzene with Tf = 5. 5 ◦C, lf = 30.45 cal/g for the freezing point of benzene. For a c = 1 g/cm^3 solution of polystyrene with molecular weight MN = 20, 000, the change in the freezing point of the solution is ∆Tf = 2. 9 × 10 −^3 ◦C. Like the boiling point elevation effect, the freezing point depression effect is too small. The technique is insensitive and only useful for low molecular weight polymer (e.g., MN less than 20,000 g/mol).
5.4 Osmotic Pressure
Another colligative property is osmotic pressure. Figure 5.2 illustrates the osmotic pressure ef- fect. Imagine a pure solvent and a solution separated by a semipermeable membrane. An ideal semipermeable membrane will allow the solvent molecules to pass but prevent the solute molecules (polymer molecules) from passing. The different concentrations on the two sides of the membrane will cause an initial difference in chemical potential. At equilibrium, this difference in potential will be counteracted by an effective pressure across the membrane. As shown in Fig. 5.2, it can be imagined that solvent molecules pass from the pure solvent side to the solution side. The excess height in the column of liquid above the solution side is related to the osmotic pressure by π = ρgh. Here π is the osmotic pressure, ρ is the density of the solution, g is the acceleration of gravity (9.81 m/sec^2 ) and h is the height of the column of liquid. We begin with a thermodynamic analysis of osmotic pressure. At equilibrium the chemical potential in the solution will be equal to the chemical potential in the pure solvent:
μsolventA = μsolutionA = μ◦ A + RT ln aA (5.23)
where μsolventA is the chemical potential of the pure liquid or
μsolventA = μ◦ A (5.24)
5.5. PRACTICAL ASPECTS OF OSMOTIC PRESSURE 71
But, MA/ρ is the grams per mole of solvent divided by the grams per cm^3 of solvent. The grams cancel and we have cm^3 per mole of the solvent or the partial molar volume of component A — VA. Substituting into the osmotic pressure equation thus gives:
π c =^
RT
MN
Rewriting the osmotic pressure equation gives a result that is similar to the ideal gas law
cRT = π
∑^ i^ NiMi i Ni
or
i NiMi V RT^ =^ π
∑^ i^ NiMi i Ni
which simplifies to πV =
i
NiRT (i.e. P V = nRT ) (5.33)
For an example, let’s consider the solution of polystyrene in benzene that was used for examples of boiling point elevation and freezing point depression; i.e., a solution of polystyrene in benzene with MN = 20, 000 and a concentration of c = 1 g/cm^3. For the correct units we use R =
- 3143 × 107 ergs/K/mol and calculate π = 1. 24 × 104 dynes/cm^2. This pressure will be measured by a difference in heights of liquids in columns. The height difference comes from π = ρgh or
h = 1.^24 ×^10
(^4) dynes/cm^2 0 .8787 g/cm^3 981 cm/sec^2 = 14.3 cm (5.34)
This height difference is large and is an easily measurable quantity. In fact we expect to be able to measure distances at least 100 times smaller than this result. Thus osmotic pressure can, in principle, be used to determine molecular weights in polymers with MN up to 2,000,000 g/mol.
5.5 Practical Aspects of Osmotic Pressure
Osmotic pressure measurements appear to be a suitable method for measuring number average molecular weights in polymers. It is therefore worthwhile considering practical aspects of polymer characterization by osmotic pressure. The first practical consideration is that we expect polymer solutions to deviate from ideal behavior and thus the osmotic pressure expression will need to be corrected. In the limit of zero concentration, the solution will eventually become ideal. We can therefore take a series of measurements and extrapolate back to zero concentration to get the ideal result. In other words lim c→ 0 πc = (^) MRT N
The question which remains is “how do we extrapolate?” A common approach in thermodynamics is to use a virial expansion. We thus write πc as a sum of many terms:
π c =^
RT
MN
- RT A 2 c + RT A 3 c^2 + · · · (5.36)
72 CHAPTER 5. COLLIGATIVE PROPERTIES
or π c =^
RT
MN
1 + Γ 2 c + Γ 3 c^2 + · · ·
Here A 2 , A 3 , ... and the related Γ 2 , Γ 3 , ... are called the virial coefficients. If we include enough virial coefficients we will always be able fit experimental data. But, how many of these terms do we need? Furthermore, how do we analyze experimental data when virial expansion terms are required? We consider two approaches to this problem. In the first approach, we assume that only the second virial coefficient — A 2 or Γ 2 — will be needed. Then π/c is predicted to be linear in concentration:
π c =^
RT
MN
A set of data for π/c vs. c can be plotted. If the results are linear, the assumption in the first approach is valid. When the data is linear, the intercept of the data at zero concentration will be RT /MN and thus can be used to determine MN. Besides an intercept, we can measure the slope which is equal to RT A 2. In other words the slope of the π/c vs. c plot is proportional to the second virial coefficient — A 2. We can make use of the Flory-Huggins theory to get a physical interpretation of the second virial coefficient. The Flory-Huggins theory includes non-ideal interactions through the Flory interaction parameter, χ. Let’s use the Flory-Huggins theory to develop an osmotic pressure theory for nonideal solutions. We begin with an early osmotic pressure formula:
π = − RT V^ ln^ aA A = − μA^ −^ μ
◦ A VA^ (5.39)
The term μA − μ◦ A is found by differentiating the free energy of mixing
d∆Gmix dnA^ =^ μA^ −^ μ
◦ A (5.40)
To use the Flory-Huggins theory we differentiate the ∆Gmix from that theory. In performing the integration we must realize that vA and vB also depend on nA. The work is left as an exercise to the reader. The result is
μA − μ◦ A = RT
[
ln vA +
1 − (^) x^1
vB + χv^2 B
]
Substituting into the osmotic pressure formula and at the same time using the approximation ln vA = ln(1 − vB ) ≈ −vB + v^2 B /2 (Note that in this approximation to ln(1 − vB ) we keep one more term than we have used in the past. The reason for the extra term is that the μA − μ◦ A expression already includes terms with v^2 B ), the osmotic pressure becomes:
π = RTV A
[ (^) v B x +
2 −^ χ
v^2 B + · · ·
]
74 CHAPTER 5. COLLIGATIVE PROPERTIES
We now have two parameters — Γ 2 and g. Deriving two parameters for osmotic pressure data will be more complicated than deriving the slope and intercept of simple linear fits. It requires more advanced curve-fitting techniques. We can simplify the process by introducing some theoretical calculations about g. For hard spheres, g can be calculated to be g = 5/8. For polymer molecules, g has been estimated to be g = 0.25 to 0.28. The actual value of g depends on various properties such as the expansion coefficient α, the characteristic ratio, etc.. Fortunately, however, g is restricted to a relatively narrow range for most polymers. Because g must be positive and a polymer cannot be more impenetrable than hard spheres, g must be between 0 and 5/8. If we pick a value for g than we are left with only one parameter (Γ 2 ) and we calculate MN and Γ 2 by simpler curve fitting analyses. Fortunately it has been found that the results are not very sensitive to the exact value of g. Because polymers have g’s calculated to be near 0.25, we will assume g = 0.25. The choice of g = 0.25 is desirable because it completes the square and the data analysis can again be done by linear fits (Scientists, especially scientists that worked before computers, like linear theories):
π c =^
RT
MN
1 + Γ 2 c +^14 Γ^22 c^2
or
π c =
RT
MN
1 + Γ 22 c
When g can be assumed to be 0.25, a plot of
π/c vs. c should be linear. The slope will give Γ 2 and the intercept will give
RT /MN.
The advantage of setting g = 0.25 is that the data can be analyzed with a simple linear fit. This advantage was important before computers were readily available. Now we can easily treat g as a second parameter and do a two parameter fit to the data. You will try this type of analysis in one of the class labs and be able to discuss whether the added complexity improves or weakens the interpretation of the results.
5.6 Experimental Aspects of Membrane Osmometry
A simple type of osmometer is illustrated in Fig. 5.3. The solution is placed in a cell with membranes on either side (one or two membranes, but two gives more area and faster equilibration). The entire assembly is then immersed in pure solvent. The heights of the liquids in the capillaries are read and the height difference gives the osmotic pressure. This apparatus is called a block type osmometer. It is the type of osmometer used to get the data that will be given to you in a lab. This osmometer uses a small cell and a large membrane. The membrane is supported by stainless steel plates with holes. By supporting the membrane, the membrane can be made larger; with larger membrane area equilibrium will be reached sooner. Block osmometers are called static osmometers because they wait for the natural development of equilibrium. The problem with static osmometers is that it can take hours (12-24 hrs) to reach
5.6. EXPERIMENTAL ASPECTS OF MEMBRANE OSMOMETRY 75
SIDE VIEW FRONT VIEW
Capillaries
Outer
flange
Outer
flange
Membrance
Exploded View
Inner
flange Solution cell
Figure 5.3: A block type, static osmometer.
an accurate equilibrium. The time depends on many factors such as the membrane area and the speed of transport through the membrane. To quicken osmotic pressure experiments, dynamic osmometers are sometimes used. Recall that osmotic pressure develops for the purpose of raising the activity of the solvent in the solution to 1. By applying a pressure it is possible to do the same thing. You will know when you have applied the correct pressure by monitoring flow across the membrane. When you apply enough pressure to stop the flow you have artificially reached equilibrium. The pressure required can be used to get the equilibrium osmotic pressure. This quick method, unfortunately, is less accurate. Finally, we make a few comments about what makes a good semipermeable membrane. The membrane must be permeable to solvent and impermeable to polymer. This requirement limits the low-end applicability of osmometry to MN of 20,000 g/mol or more. Note that we really require all polymers to be above 20,000 g/mol otherwise the low molecular weight tail will pass through membrane and the measured MN will be too high (do you see why it would be too high?). Therefore, polydisperse polymers probably require MN greater than about 50,000 g/mol; for monodisperse polymers it might be possible to go down to 20,000 g/mol. There are also some material concerns for the membrane. An obvious concern is that the mem- brane not be soluble in the solvent. Perhaps the most common membrane material is gel cellulose.
