Physical Chemistry Third Edition

6.4 The Activities of Nonvolatile Solutes 267

6.4 The Activities of Nonvolatile Solutes

For a nonvolatile substance we must find a way to determine its activity coefficient

that does not depend on measuring its vapor pressure. We will discuss three different

methods. The first is through integration of the Gibbs–Duhem equation. The second is

through a theory due to Debye and Hückel, which can be applied to electrolyte solutes.

The third method for electrolyte solutes is an electrochemical method, which we will

discuss in Chapter 8. Published data are available for common electrolytes, and some

values are included in Table A.11 in Appendix A.

The Gibbs–Duhem Integration

For a two-component solution with a volatile solvent such as water and a nonvolatile

solute, values of the activity of the solvent can be determined for several values of

the solvent mole fraction between unity and the composition of interest. Integration

of the Gibbs–Duhem relation can then give the value of the activity coefficient of the

solute. The activity of the solvent is usually determined using theisopiestic method.

The solution of interest and a solution of a well-studied nonvolatile reference solute

in the same solvent are placed in a closed container at a fixed temperature. A solution

of KCl is usually used as the reference solute for aqueous solutions, since accurate

water activity coefficient data are available for KCl solutions. The solutions are left

undisturbed at constant temperature until enough solvent has evaporated from one

solution and condensed into the other solution to equilibrate the solvent in the two

solutions.

At equilibrium, the activity of solvent (substance 1) in the solution of interest

(solution B) is equal to the activity of solvent in the reference solution (solution A),

so that

γ

(B)

1 

a(B) 1

x(B) 1



a(A) 1

x(B) 1



γ 1 (A)x(A) 1

x(B) 1

(6.4-1)

The solutions are analyzed to determine the mole fractions of the solvent in the two

solutions, and the activity of the solvent in the reference solution is determined from

tabulated values. The activity of the solvent in the unknown solution is equal to this

value. The experiment is repeated several times for a range of compositions of solution

B beginning with pure solvent and extending tox 2 x′ 2 , the composition at which we

want the value ofγ 2.

For constant pressure and temperature, the Gibbs–Duhem relation for two compo-

nents is given by Eq. (4.6-11). When we substitute Eq. (6.3-6) into this equation, we

obtain

x 1 RTdln(a 1 )+x 2 RTdln(a 2 ) 0 (6.4-2)

Using convention II,a 1 γ 1 x 1 anda 2 γ 2 x 2 , where we omit the superscript (II).

Using the fact thatxidln(xi)dxi, we obtain

x 1 RTdln(γ 1 )+RTdx 1 +x 2 RTdln(γ 2 )+RTdx 2  0