Physical Chemistry Third Edition

6.4 The Activities of Nonvolatile Solutes 271

Debye and Hückel devised a theory for the activity coefficients of ionic solutes. Their
theory begins with the expression for the force between a pair of ions in Eq. (6.4-19),
assuming a constant value of the permittivity of the solvent. It is also assumed that the
centers of two ions cannot approach each other more closely than some distance of
closest approach, calleda. The solution is assumed to behave like an ordinary dilute
solution except for the effects of the electrostatic forces.
The Debye–Hückel theory was devised
by Peter J. W. Debye, 1884–1966, a
Dutch-American physicist and chemist
who received the Nobel Prize for
chemistry in 1936 for his work on the
dipole moments of molecules, and Erich
Hückel, 1896–1980, a German chemist
who is also known for an approximation
scheme used in molecular quantum
mechanics.

We will not discuss the details of the Debye–Hückel theory.^2 The main idea of

the theory was to pretend that the ions in a solution could have their charges var-

ied reversibly from zero to their actual values. This charging process created anion

atmospherearound a given ion with an excess of ions of the opposite charge. The

reversible net work of creating the ion atmosphere was calculated from electrostatic

theory. According to Eq. (4.1-32) the reversible net work is equal to∆G, which leads

to equations for the electrostatic contribution to the chemical potential and the activity

coefficient for the central ion. The principal result of the Debye–Hückel theory is a

formula for the activity coefficient of ions of typei:

ln(γi)−

z^2 iαI^1 /^2

1 +βaI^1 /^2

(6.4-21)

whereziis the valence of the ion (the number of proton charges on the ion). The

quantitiesαandβare functions of temperature, and of the properties of the solvent:

α(2πNAvρ 1 )^1 /^2

(

e^2

4 πεkBT

)

(6.4-22)

βe

[

2 NAvρ 1

εkBT

]

(6.4-23)

whereeis the charge on a proton,NAvis Avogadro’s constant,ρ 1 is the density of the

solvent,kBis Boltzmann’s constant (equal toR/NAv),Tis the absolute temperature,

andεis the permittivity of the solvent. The quantityIis theionic strength, defined as

a sum over all of thesdifferent charged species in the solution:

I

1

2

∑s

i 1

miz^2 i (definition of ionic strength) (6.4-24)

wheremiis the molality of speciesiandziis its valence. All of the ions in the solution

must be included in the ionic strength. The formula in Eq. (6.4-21) is for the activity

coefficient in the molality description, but for a relatively dilute solution it can be used

for the activity coefficient in the concentration description.

EXAMPLE6.13

Calculate the ionic strength of a solution that is 0.100 mol kg−^1 in NaCl and 0.200 mol kg−^1

in CaCl 2. Assume complete dissociation.

(^2) An account of the development is found in T. L. Hill,Statistical Thermodynamics, Addison-Wesley,
Reading, MA, 1960, pp. 321ff. The original reference is P. Debye and E. Hückel,Physik. Z., 24 , 185 (1923).