CHAPTER
3.1 M n Methods
3.1.6 Vapor Phase Osmometry
(Fig. 3.2a), however, since the second virial coefficient reflects polymersolvent interactions and can be related, for example, to the FloryHuggins interaction parameterχ(Chapter 5) by
A25 1 2 2χ
LV10υ22 (3-30)
Here υ2is the specific volume of the polymer, V10 is the molar volume of the solute, and χ is an interaction energy per mole of solvent divided by RT. When χ50.5, A250 and the solvent is a theta solvent for the particular polymer.
Better solvents have lowerχvalues and higher second virial coefficients.
It may be expected also that different molecular weight samples of the same poly- mer should yield the same slopes and different intercepts when the osmotic pressures of their solutions are measured in a common solvent. This situation, which is shown inFig. 3.2b, is not realized exactly in practice because the second virial coefficient is a weakly decreasing function of increasing polymer molecular weight.
can be considered in this connection. The expanded virial form of Eq. (2-72) for this property is
ΔP
P01 5 μ12G01
RT 5 2V10c2
1
M 1Bc21Cc221?
(3-31) (RecallEqs. 3-16 and 3-26.) Direct measurement ofΔPis difficult because of the small magnitude of the effect. (At 10 g/liter concentration in benzene, a poly- mer with Mn equal to 20,000 produces a vapor pressure lowering of about 231023mm Hg at room temperature. The limits of accuracy of pressure mea- surements are about half this value.) It is more accurate and convenient to convert this vapor pressure difference into a temperature difference. This is accomplished in the method called vapor phase osmometry. The procedure is also known as vapor pressure osmometry or more accurately as thermoelectric differential vapor pressure lowering.
In the vapor phase osmometer, two matched thermistors are located in a thermostatted chamber which is saturated with solvent vapor. A drop of solvent is placed on one thermistor and a drop of polymer solution of equal size on the other thermistor. The solution has a lower vapor pressure at the test temperature (Eq. 3-20), and so the solvent condenses on the solution thermistor until the latent heat of vaporization released by this process raises the temperature of the solution sufficiently to compensate for the lower solvent activity. At equilibrium, the sol- vent has the same vapor pressure on the two temperature sensors but is at differ- ent temperatures.
Ideally the vapor pressure difference ΔP inEq. (3-31) corresponds to a tem- perature difference ΔT, which can be deduced from the ClausiusClapeyron equation
ΔT5ΔPRT2=ΔHvP01 (3-32) whereΔHvis the latent heat of vaporization of the solvent at temperatureT. With the previous equation
ΔT c2
5 RT2 ΔHv
V10 1
M 1Bc21Cc221?
(3-33) Thus, the molecular weight of the solute can be determined in theory by mea- suring ΔT/c2and extrapolating this ratio to zero c2. (SinceΔTis small in prac- tice,T may be taken without serious error as the average temperature of the two thermistors or as the temperature of the vapor in the apparatus.)
In fact, thermal equilibrium is not attained in the vapor phase osmometer, and the foregoing equations do not apply as written since they are predicated on the existence of thermodynamic equilibrium. Perturbations are experienced from heat conduction from the drops to the vapor and along the electrical connections.
Diffusion controlled processes may also occur within the drops, and the magnitude
of these effects may depend on drop sizes, solute diffusivity, and the presence of volatile impurities in the solvent or solute. The vapor phase osmometer is not a closed system and equilibrium cannot therefore be reached. The system can be operated in the steady state, however, and under those circumstances an analog of expression (3-33) is
ΔT c2 5ks
1 Mn
1Bc21Cc221?
(3-34) where ks is an instrument constant. Attempts to calculate this constant a priori have not been notably successful and the apparatus is calibrated in use for a given solvent, temperature, and thermistor pair, by using solutes of known molecular weight. The operating equation is
Mn5 k ðΔΩ=cÞc50
(3-35) where k is the measured calibration constant and ΔΩ is the imbalance in the bridge (usually a resistance) that contains the two thermistors.
There is some question as to whether the calibration is independent of the molec- ular weight of the calibration standards in some VPO instruments. It is convenient to use low-molecular-weight compounds, like benzil and hydrazobenzene, as standards since these materials can be obtained in high purity and their molecular weights are accurately known. However, molecular weights of polymeric species which are based on the calibrations of some vapor phase instruments may be erroneously low.
The safest procedure involves the use of calibration standards that are in the same molecular weight range, more or less, as the unknown materials to be determined.
Fortunately, the low-molecular-weight anionic polystyrenes that are usually used as gel permeation chromatography standards (Section 3.4.3) are also suitable for vapor phase osmometry standards. Since these products have relatively narrow molecular weight distributions all measured average molecular weights should be equal to each other to within experimental uncertainty. TheMvaverage (Section 3.3.1) of the poly- styrene should be considered as the standard value if there is any uncertainty as to which average is most suited for calibration in vapor phase osmometry.
Vapor phase osmometers differ in design details. The most reliable instru- ments appear to be those incorporating platinum gauzes on the thermistors in order to ensure reproducible solvent and solution drop sizes. In any case, the highest purity solvents should be used with this technique to ensure a reasonably fast approach to steady-state conditions.
The upper limit of molecular weights to which the vapor phase osmometer can be applied is usually considered to be 20,000 g mol21. Newer, more sensitive machines have extended this limit to 50,000 g mol21or higher. The measurements are convenient and relatively rapid and this is an attractive method to use, with the proper precautions.
102 CHAPTER 3 Practical Aspects of Molecular Weight Measurements