Page 328 - Physical Chemistry

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With the definitions (10.43) to (10.45) of g , m°, and n, Eq. (10.42) for the elec- Section 10.5

i
trolyte’s chemical potential m becomes Solutions of Electrolytes
i
n
m m° RT ln 31g 2 1m >m°2 1m >m°2 4 (10.46)
n
n

i

i
q
g 1 (10.47)

where the infinite-dilution behavior of g follows from (10.43) and (10.41).

What is the relation between the ionic molalities m and m in (10.46) and the

electrolyte’s molality? The stoichiometric molality m of electrolyte i is defined as
i
m n >w A (10.48)
i
i
where the solution is prepared by dissolving n moles of electrolyte in a mass w of
i A
solvent. To express m in (10.46) as a function of m , we shall relate m and m to m .
i i i
The formula of the strong electrolyte M X contains n cations and n anions, so the
n n
ionic molalities are m n m and m n m , where m is the electrolyte’s stoi-

i
i
i
chiometric molality (10.48). The molality factor in (10.46) is then
1m 2 1m 2 n 1n m 2 1n m 2 n 1n 2 1n 2 m n i (10.49)
n
n
n
n

i

i
where n n n [Eq. (10.45)]. We define n [analogous to g in (10.43)] as

n
1n 2 1n 2 1n 2 n (10.50)
n

2 1/5
3
For example, for Mg (PO ) , n (3
2 ) 108 1/5 2.551. If n n , then
3 4 2
n n n (Prob. 10.28). With the definition (10.50), Eq. (10.49) becomes
n
(m ) (m ) n (n m ) . The quantity in brackets in (10.46) and the expression
n

i
(10.46) for m become
i
n
31g 2 1m >m°2 1m >m°2 4 1n g m >m°2 n
n
n

i

m m° nRT ln 1n g m >m°2 strong electrolyte (10.51)
i
i

i
y
where ln x y ln x was used. Equation (10.51) expresses the electrolyte’s chemical
potential m in terms of its stoichiometric molality m .
i i
Setting m in (10.51) equal to m°,we see that the standard state of the electrolyte i as
i i
a whole has n g m /m° 1. The standard state of i as a whole is taken as the fictitious

i
state with g 1 and n m /m° 1. This standard state has m (1/n ) mol/kg.

i
i
The activity a of electrolyte i as a whole is defined so that m m° RT ln a
i i i i
[Eq. (10.4)] holds. Therefore (10.51) gives for an electrolyte
a 1n g m >m°2 n (10.52)
i
i

Equation (10.51) is the desired expression for the electrolyte’s chemical potential
in terms of experimentally measurable quantities. The expression (10.51) for m of an
i
electrolyte differs from the expression m m° RT ln (g m /m°) [Eq. (10.28)] for a
i i i i
nonelectrolyte by the presence of n, n , and the expression for g . Even in the infinite-

dilution limit where g 1, the electrolyte and nonelectrolyte forms of m differ.

i
Gibbs Energy of an Electrolyte Solution
Equations (10.37) and (10.38) give
dG S dT V dP m dn m dn i (10.53)
A
A
i
which has the same form as (4.73). Hence, the same reasoning that gave (9.23) and
(10.18) gives for an electrolyte solution
G n m n m i (10.54)
i
A
A
n dm n dm 0 const. T, P (10.55)
A A i i
Equation (10.55) is the Gibbs–Duhem equation for an electrolyte solution.