Physical Chemistry Third Edition

268 6 The Thermodynamics of Solutions

Sincex 1 +x 2 1,dx 1 +dx 2 0, and two terms cancel. We divide byx 2 RT and

obtain the equation

dln(γ 2 )−

x 1

x 2

dln(γ 1 )−

x 1

1 −x 1

dln(γ 1 ) (6.4-3)

We integrate Eq. (6.4-3) fromx 1 x′′ 1 tox 1 x′ 1 (the composition we are interested

in). The lower limitx′′ 1 cannot be taken equal to unity, because the denominator 1−x 1

goes to zero in this case. We choose a value ofx′′ 1 close enough to unity that Raoult’s

law is obeyed by the solvent. At this composition the solute obeys Henry’s law andγ 2

is equal to unity. Integration of Eq. (6.4-3) gives

lnγ 2 (x′ 1 )−

∫x 1 x′

1

x 1 x′′ 1

x 1

1 −x 1

dln(γ 1 ) (6.4-4)

where we consider bothγ 2 andγ 1 to be functions ofx 1. Unless the data are fit to some

formula this integral is approximated numerically.

In the case of an electrolyte solute the Gibbs–Duhem integration is carried out in

a slightly different way. Consider an electrolyte solute represented by the formula

Mν+Xν−, whereν+andν−represent the numbers of cations and anions in its formula.

In the case of CaCl 2 ,ν+1 andν−2. The total number of ions in the formula is

denoted byν, equal toν++ν−. The substance must be electrically neutral:

ν+z++ν−z− 0 (6.4-5)

wherez+is the valence of the cation andz−is thevalenceof the anion. The valence

is the number of proton charges on an ion. It is positive for a cation and negative for

an anion.

It is impossible to measure the chemical potential of an ion since ions of one charge

cannot be added without adding ions of the opposite charge at the same time. The

(unmeasurable) chemical potential of the cation is given by

μ+μ◦++RTln(γ+m+/m◦)

and that of the anion is given by

μ−μ◦−+RTln(γ−m−/m◦)

Since these chemical potentials cannot be measured, we must define the chemical

potential of the neutral electrolyte solute. This is discussed in Chapter 7, but we antic-

ipate the result, which is that the chemical potential of the neutral electrolyte solute

(substance number 2) is given by

μ 2 ν+μ++ν−μ−

ν+μ◦++RTln(γ+m+/m◦)+ν−μ◦−+RTln(γ−m−/m◦)

ν+μ◦++ν−μ◦−+RTln

(

γ+ν+γ−ν−mν++mν−−/m◦ν

)

This is a measurable quantity since the neutral electrolyte substance can be added while

keeping the amounts of all other substances fixed.

In order to write equations that apply to all electrolyte solutes, we define themean

ionic activity coefficient

γ±

(

γ+ν+γ−ν−

) 1 /ν

(6.4-6)