Physical Chemistry Third Edition

274 6 The Thermodynamics of Solutions

the Debye–Hückel result appears as the leading term of a series containing powers

and logarithms of the ionic strength. The work of Mayer gives some credibility to the

Brønsted equation and to an equation of Guggenheim:^7

ln(γ±)−z+|z−|

αI^1 /^4

1 +(I/m◦)^1 /^2

+

2 v+v−

v

bm (6.4-30)

wherebis a parameter that is evaluated experimentally for each electrolyte solute,

not necessarily equal to the constant in Brønsted’s theory. TheDavies equation^8 is

a generalized version of Guggenheim’s equation. For water as the solvent and for a

temperature of 298.15 K, the Davies equation is

log 10 (γ±)− 0. 510 z+|z−|

[

(I/m◦)^1 /^2

1 +(I/m◦)^1 /^2

− 0. 30

I

m◦

]

(6.4-31)

This equation can be used when no experimental information is available. In some cases

it can give usable results for activity coefficients up to ionic strengths of 0.5 mol kg−^1

or beyond, but it is ordinarily in error by several percent in this region. Table A.11

in Appendix A gives experimental values of the mean ionic activity coefficients of

several aqueous electrolytes at various concentrations. It also gives the predictions

of the Debye–Hückel formula withβataken equal to 1.00 kg^1 /^2 mol−^1 /^2 , and of the

Davies equation.

Joseph E. Mayer, 1904–1983, was a
prominent American physical chemist
who was well known for a textbook in
statistical mechanics that he coauthored
with his wife, Maria Goeppert Mayer,
1906–1972, who was one of the 1963
Nobel Prize winners in physics for her
work on the shell theory of nuclei.

EXAMPLE6.15

Calculate the activity coefficient for the solution of Example 6.14 using the Davies equation.

Find the percent difference between the result of the Davies equation and that of the Debye–

Hückel limiting law.

log 10 (γ±)− 0. 510

(

(0.0100)^1 /^2

1 +(0.0100)^1 /^2

−(0.30)(0.0100)

)

 4. 48 × 10 −^2

γ± 0. 902

% difference

0. 902 − 0. 899

0. 899

×100% 0 .3%

Exercise 6.21

Calculate the activity coefficient for an aqueous solution of NaCl with molality 0.250 mol kg−^1

at 298.15 K using the Davies equation. Find the percent difference between the result of the

Davies equation and that of the Debye–Hückel limiting law.

(^7) E. A. Guggenheim,Phil. Mag., 19 , 588 (1935); E. A. Guggenheim and J. C. Turgeon,Trans. Faraday
Soc., 51 , 747 (1955).
(^8) C. W. Davies,Ion Association, Butterworth, London, 1962, pp. 35–52.