PHYSICAL CHEMISTRY IN BRIEF

CHAP. 11: ELECTROCHEMISTRY [CONTENTS] 371

whereeis the absolute value of the electron charge,NAis the Avogadro constant, 0 is the
permittivity of a vacuum,ris the relative permittivity of the solvent, andρ 1 is the solvent
density. It is obvious from the relation that the parameterAdepends solely on the properties of
the solvent and not on the properties of the ions. Equation (11.43) is called theDebye-H ̈uckel
limiting relationfor the activity coefficient of an ion. It may be also applied for ions in a
mixture of electrolytes.

Example

The density of water at the standard pressure and a temperature of 298.15 K is 997.07 kg m−^3 ,

its relative permittivity is 78.303, the permittivity of a vacuum is 8. 8542 × 10 −^12 C^2 N−^1 m−^2 , and

the electron charge is 1.602× 10 −^19 C. Calculate the value of the parameterAfor water at the

given temperature and pressure.

Solution

Substituting into (11.44) yields

A =

(1. 602 × 10 −^19 )^3 ×(6. 022 × 1023 )^2 ×

√

2 / 997. 07

8 × 3. 1416 ×(8. 8542 × 10 −^12 × 78. 303 × 8. 314 × 298 .15)^3 /^2

= 1. 1762 (kg mol)^1 /^2.

By combining equations (11.2), (11.40) and (11.43) we obtain the Debye-H ̈uckel relation for
the mean activity coefficient
lnγ±=−A z+z−

√

I. (11.45)

Relations (11.43) and (11.45) express well the experimental values of activity coefficients in
dilute solutions, whereI < 0 .01 mol kg−^1 , but they fail at higher ionic strengths of the solution.

Example

Calculate the activity coefficients of the ions and the mean activity coefficients in a solution

containing 0.004 mol HCl and 0.002 mol CaCl 2 in 1 kg of water at a temperature of 25◦C, given

thatA= 1.1762 (kg mol−^1 )^1 /^2.