DETERMINATION OF THE ACTIVITY OF ONE COMPONENT FROM THE ACTIVITY OF THE OTHER

17.5 DETERMINATION OF THE ACTIVITY OF ONE COMPONENT FROM THE ACTIVITY OF THE OTHER

The fundamental relationship between the chemical potentials of the two components of a solution at a fixed temperature and pressure is the Gibbs – Duhem Equation (9.34):

n 1 dm 1 þn 2 dm 2 ¼0

From the relationship of the chemical potential to the activity [Equation (16.1)], we can write Equation (9.34) as

n 1 d ln a 1 þn 2 d ln a 2 ¼0

If Equation (17.26) is divided by (n 1 þn 2 ), the result is

X 1 d ln a 1 þX 2 d ln a 2 ¼0

DETERMINATION OF NONELECTROLYTE ACTIVITIES AND EXCESS GIBBS FUNCTIONS

Calculation of Activity of Solvent from That of Solute If adequate data are available for the activity of the solute, the activity of the solvent

can be obtained by rearranging Equation (17.27) to

d ln a 1 d ln a 2 (17 :28)

and integrating. As a 2 approaches zero as X 2 approaches zero, ln a 2 is an indetermi- nate quantity at one of the limits of integration. Although both a 2 and X 2 approach zero, their ratio a 2 /X 2 ¼g 2 approaches one [Equation (16.3)]. Thus, it is necessary to convert Equation (17.28) into a corresponding equation for the activity coefficients.

X 2 (17 1 :29) The subtraction of Equation (17.29) from Equation (17.28) gives the expression

d ln X 1 d ln X

d ln 1 2 d ln 2 (17 :30)

Integrating Equation (17.31) from the infinitely dilute solution to some finite concen- tration, we obtain, with the assumption of a Raoult’s-law standard state for the solvent and a Henry’s-law standard state for the solute,

1 d ln g 2 (17 X :32)

17.5 DETERMINATION OF THE ACTIVITY OF ONE COMPONENT

399 If X 2 /X 1 is plotted against ln g 2 , the integration of Equation (17.32) can be carried out

graphically, or a numeric integration can be performed directly from tabulated values of X 2 /X 1 and ln g 2. . (See Appendix A.)

Calculation of Activity of Solute from That of Solvent To calculate the activity of the solute from that of the solvent, it is useful to rearrange

Equation (17.31) to the form

ln g

2 d ln g 1 (17 X :33)

which, on integration, gives

ln g ð 1

ln g

2 d X ln g 1 (17 :34)

However, the integral in Equation (17.34) is divergent because X 1 /X 2 approaches infinity in the limit of infinitely dilute solutions. One method of overcoming this difficulty is as follows. Instead of setting the lower limit in the integration of Equation (17.33) at infinite dilution, let us use a temporary lower limit at a finite concentration X 0 2 . Thus, in place of Equation (17.34), we obtain

ln g ð 1

ln

g 0 1 (17 2 :35) X 2

1 d ln g

ln g 1 0

The evaluation of the integral in Equation (17.35) offers no difficulties because X 1 /X 2 is finite at the lower limit. Using Equation (17.35), we can obtain precise values of the ratio g 2 /g 2 0 as a function of X 2 . If g 2 /g 0 2 is plotted against X 2 , for a solution of water in dimethyl sulfoxide [4], as shown in Figure 17.11, where the reference solution with X 0 2 equal to 0.0365 was chosen, the values can be extrapolated to a finite-limiting value.

From Equations (16.3) and (16.7), we know that, if Henry’s law is followed,

a lim 2

2 !0 X x !0 2 2 2

¼ lim g ¼1

Thus,

lim 2 ¼ 1

2 !0 g 0 g 2 0 2

DETERMINATION OF NONELECTROLYTE ACTIVITIES AND EXCESS GIBBS FUNCTIONS

Figure 17.11. Extrapolation of relative activity coefficients to obtain 1 /g 0 2 for the calculation of solute activity coefficients. Data from Ref. 4. Dimethyl sulfoxide is the solvent, and water is the solute.

Once a value for g 2 0 , 0.9911 in this case, is obtained by extrapolation, values for g 2 in each of the other solutions can be calculated from the values of g 2 /g 2 0 using Equation (17.35). The activity of the solvent often can be obtained by an experimental technique known as the “isopiestic method” [5]. With this method we compare solutions of two different nonvolatile solutes; for one of which, the reference solution, the activity of the solvent has been determined previously with high precision. If both solutions are placed in an evacuated container, solvent will evaporate from the solution with higher vapor pressure and condense into the solution with lower vapor pressure until equilibrium is attained. The solute concentration for each solution then is deter- mined by analysis. Once the molality of the reference solution is known, the activity of the solvent in the reference solution can be read from records of previous experi- ments with reference solutions. As the standard state of the solvent is the same for all solutes, the activity of the solvent is the same in both solutions at equilibrium. Once

the activity of the solvent is known as a function of m 2 for the new solution, the activity of the new solute can be calculated by the methods discussed previously in this section.

17.6 MEASUREMENTS OF FREEZING POINTS Perhaps the method of most general applicability for determining activities of none-

lectrolytes in solutions is the one based on measurements of the lowering of the freezing point of a solution. As measurements are made of the properties of the solvent, activities of the solute are calculated by methods described in the preceding section.

401 Elaborate procedures have been developed for obtaining activity coefficients from

EXERCISES

freezing-point and thermochemical data. However, to avoid duplication, the details will not be outlined here, because a completely general discussion, which is appli- cable to solutions of electrolytes as well as to nonelectrolytes, is presented in Chapter 21 of the Third Edition of this book [6].