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Chapter 10 To relate m and m to m , we use the Gibbs dG equation (4.73), which for the

i
Nonideal Solutions electrolyte solution is
dG S dT V dP m dn m dn m dn (10.36)

A
A

The numbers of moles of cations and anions from M X is given by Eq. (10.33) as
n n
n n n and n n n . Therefore

i
i
dG S dT V dP m dn 1n m n m 2 dn i (10.37)

A

A
Setting dT 0, dP 0, and dn 0 in (10.37) and using m 10G>0n 2
A i i T,P,n A
[Eq. (10.35)], we get
m n m n m (10.38)*

i

which relates the chemical potential m of the electrolyte as a whole to the chemical
i
potentials m and m of the cation and anion. For example, the chemical potential of

2
CaCl in aqueous solution is m(CaCl , aq) m(Ca , aq) 2m(Cl , aq).
2 2
We now consider the explicit expressions for m and m . The chemical potential
A i
m of the solvent can be expressed on the mole-fraction scale [Eq. (10.30)]:
A
q
m m*1T, P2 RT ln g x , 1g x,A 2 1 (10.39)
A
x,A A
A
where g is the mole-fraction activity coefficient and the q superscript denotes
x,A
infinite dilution.
The electrolyte chemical potentials m , m , and m are usually expressed on the
i
molality scale. Let m and m be the molalities of the ions M and X , and let g
z
z

and g be the molality-scale activity coefficients of these ions. The m subscript is

omitted from the g’s since in this section we shall use only the molality scale for solute
species. Equations (10.28) and (10.29) give as the chemical potentials of the ions:
m m° RT ln 1g m >m°2, m m° RT ln 1g m >m°2 (10.40)

q
q
m° 1 mol>kg, g g 1 (10.41)

where m° and m° are the ions’ molality-scale standard-state chemical potentials.

Substitution of (10.40) for m and m into (10.38) gives m as

i
m n m° n m° n RT ln 1g m >m°2 n RT ln 1g m >m°2

i
m n m° n m° RT ln 31g 2 1g 2 1m >m°2 1m >m°2 4 (10.42)
n
n
n
n
i

Because m and m can’t be determined experimentally, the single-ion activity

coefficients g and g in (10.40) can’t be measured. The combination (g ) (g ) that
n
n

occurs in the experimentally measurable quantity m in (10.42) is measurable.
i
Therefore, to get a measurable activity coefficient, we define the molality-scale mean
ionic activity coefficient g of the electrolyte M X as

n
n
1g 2 n n 1g 2 1g 2 n (10.43)*
n

3
2
1/3
2/3
For example, for BaCl , (g ) (g )(g ) and g (g ) (g ) . The definition
2
(10.43) applies also to solutions of several electrolytes. For a solution of NaCl and

KCl, there is a g for the ions K and Cl and a different g for Na and Cl .

To simplify the appearance of (10.42) for m , we define m°(T, P) (the electrolyte’s
i i
standard-state chemical potential) and n (the total number of ions in the electrolyte’s
formula) as
m° n m° n m° (10.44)

i

n n n (10.45)*