Thermodynamics and Chemistry

CHAPTER 12 EQUILIBRIUM CONDITIONS IN MULTICOMPONENT SYSTEMS

12.2 SOLVENTCHEMICALPOTENTIALS FROMPHASEEQUILIBRIA 370

To calculatemfromAA, we use Eq.9.6.16on page 267 for a nonelectrolyte
solute, or Eq.10.6.1on page 299 for an electrolyte solute. Both equations are represented
by

mD

AA

RTMAmB

(12.2.1)

wherefor a nonelectrolyte is 1 and for an electrolyte is the number of ions per formula
unit.
The sequence of steps, then, is (1) the determination ofAAover a range of molality
at constantTandp, (2) the conversion of these values tomusing Eq.12.2.1, and (3) the
evaluation of the solute activity coefficient^3 by a suitable integration from infinite dilution
to the molality of interest.
Sections12.2.1and12.2.2will describe freezing-point and osmotic-pressure measure-
ments, two much-used methods for evaluatingAAin a binary solution at a givenTand
p. The isopiestic vapor-pressure method was described in Sec.9.6.4. The freezing-point
and isopiestic vapor-pressure methods are often used for electrolyte solutions, and osmotic
pressure is especially useful for solutions of macromolecules.

12.2.1 Freezing-point measurements

This section explains how we can evaluateAAfor a solution of a given composition
at a givenTandpfrom the freezing point of the solution combined with additional data
obtained from calorimetric measurements.
Consider a binary solution of solvent A and solute B. We assume that when this solution
is cooled at constant pressure and composition, the solid that first appears is pure A. For
example, for a dilute aqueous solution the solid would be ice. The temperature at which
solid A first appears isTf, the freezing point of the solution. This temperature is lower
than the freezing pointTfof the pure solvent, a consequence of the lowering ofAby the
presence of the solute. BothTfandTfcan be measured experimentally.
LetT^0 be a temperature of interest that is equal to or greater thanTf. We wish to
determine the value ofA.l; T^0 /A.sln; T^0 /, whereA.l; T^0 /refers to pure liquid solvent
andA.sln; T^0 /refers to the solution.
Figure12.1on the next page explains the principle of the procedure. The figure shows
A=T for the solvent in the pure solid phase, in the pure liquid phase, and in the fixed-
composition solution, plotted as functions ofT at constantp. SinceAis the same in the
solution and solid phases at temperatureTf, and is the same in the pure liquid and solid
phases at temperatureTf, the curves intersect at these temperatures as shown.
Formulas for the slopes of the three curves, from Eq.12.1.3on page 366 , are included
in the figure. The desired value ofA.l; T^0 /A.sln; T^0 /is the product ofT^0 and the
difference of the values ofA=Tat points e and a. To find this difference, we integrate the

(^3) A measurement of
AAalso gives us thesolventactivity coefficient, based on the pure-solvent reference
state, through the relationADACRTln.
AxA/(Eq.9.5.15).