Ebulliometry and Cryoscopy

The boiling points of a solution and of the pure solvent are different because of the difference in activities. At equilibrium, equation (9-1) changes from the form d G = AV dp — AS dt into (since dAG = 0) 

For a reversible isothermal isobaric process, on the other hand, the second law of thermodynamics applies in the form AS = AH)t,p/T. If this equation is inserted into equation (9-24), one obtains, after rearrangement 

At the boiling point the volume of vapor is much greater than the volume of liquid AV = K ap — Vu On introducing this expression and the ideal gas law pV p = KTbp, where Tbp is the boiling point, into equation (9-25), one obtains, with subscript bp for thermodynamic parameters at the boiling point, 

According to the derivation, equation (9-28) only applies to solutions at infinite dilution. For finite concentrations, one can, in analogy to the procedure adopted for membrane osmometry measurements, develop a series with virial coefficients. In polymeric solutes, the number-average molecular weight is measured in ebulliometry. (The proof is analogous to that given for osmotic-pressure measurements.) 

9 Determination oj Molecular Weight, Molecular-Weight Distribution 

The Staverman coefficient s has not, to date, been theoretically calculated. Experimentally, it has been found with cellophane 600 membranes and poly(oxyethylenes) of different molar masses that the Staverman coefficients decreases more or less linearly from a value of 1 for molar masses greater than 6000 g/mol to s = 0 for Af = 62 g/mol. 

in order to determine the molar mass of a partially permeating solute, the solution is first dialyzed with the same membrane as is to be used for osmometry. The nondialyzing part is then studied by membrane osmometry, and the dialyzing part is then studied by, for example, vapor phase osmometry. The molar mass of the original sample is then calculated from the mass fractions and molar masses of both the dialyzing and the nondialyzing parts. 

Measurements of the elevation of solvent boiling point ebulliometry) and of the depression of solvent freezing point (cryoscopy) caused by the presence of a polymeric solute enable M to be evaluated in a similar way to VPO. J4owadays, however, such measurements are rarely used to evaluate M for polymers and so will be given only brief consideration here. 

Theoretical treatments lead to equations that are identical to Equation (3.108) derived for VPO except that  

Non-ideality is eliminated in the usual way by extrapolating ATJc data to c = 0 to give an intercept from which M can be calculated. 

Accurate measurements of ATg in ebulliometry and cryoscopy of polymer solutions are made difficult by the small magnitudes of ATg since superheating and supercooling can introduce serious errors. Other practical problems often encountered include separate measurements of Tf and T in cryoscopy, and foaming of the boiling polymer solution in ebulliometry. With curreifi commercial instruments the upper limit for accurate measurement of M is about 5 x 10 gmol .  

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The other methods for determining M, by using the colligative nature of the polymer solutions are ebulliometry and cryoscopy, by both of which the maximum molecular weight obtainable is about 2 x 10. The detailed principles and practical techniques for determination of by these methods are described in refs 12 and 13. Note that these two methods are available only at specific (boiling or melting) temperatures for given solutions and are limited in applications. 

Thus, like ebulliometry or cryoscopy, the method would have a strong thermodynamic basis if heat transfer other than that due to vapor condensation could be prevented. Vapor and drop are, however, in contact with one another, and the temperature thus tends to equilibrate in time by convection, radiation, and conduction. This again causes renewed condensation of solvent vapor, which proceeds until a final steady state with a temperature difference A r is reached. Equation (9-25) becomes, with AT = A Ttn ... 

Osmotic pressure, equilibrium centrifugation, end group analysis, freezing point depression (cryoscopy), boiling point elevation (ebulliometry) and vapor pressure lowering measure the colligative... 

Average of cryoscopy, ebulliometry, 11,000 15,300 18,500 and vapour-phase osmometry Rapid membrane osmometer, first 19,900 22,100 27,900 observable values (5-7 min after sample introduction)... 

Among the techniques employed to estimate the average molecular weight distribution of polymers are end-group analysis, dilute solution viscosity, reduction in vapor pressure, ebulliometry, cryoscopy, vapor pressure osmometry, fractionation, hplc, phase distribution chromatography, field flow fractionation, and gel-permeation chromatography (gpc). For routine analysis of SBR polymers, gpc is widely accepted. Table 1 lists a number of physical properties of SBR (random) compared to natural mbber, solution polybutadiene, and SB block copolymer. 

Equations applicable to membrane osmometry, as also in the case of ebulliometry, cryoscopy, and vapor phase osmometry, can be rigorously derived from the second law of thermodynamics in the form... 

Figure 1.67 shows a schematic of a simple, but effective set-up for cryoscopy, the method for the measurement of the freezing point lowering. Cryoscopy is perhaps the easiest of the molar mass determinations. The main prerequisites are a good temperature control and uniformity, corrections for the common supercooling observed on crystallization, and the usual extrapolation to infinite dilution. The thermodynamic equations are derived in Sect. 2.2.5, together with the equations needed for the ebulliometry. 

In Table 5.1, membrane osmometry, cryoscopy, ebulliometry, end-group analysis, light-scattering and ultracentrifugation belong to the category of so-called absolute methods. Vapour pressure osmometry, viscometry and GPC can be classified as relative methods, which require samples of known molecular weight for calibration. GPC is discussed in chapter 6, and will not be dealt with here. 

Isothermal thermometry is useful for the establishment of the parameters for melting or boiling temperatures. These techniques are called cryoscopy and ebulliometry. 

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