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differs from (11.7) by the omission of the solvent’s activity. The mean molal ionic Section 11.3
Reaction Equilibrium
activity coefficient g is defined by (g ) n n (g ) (g ) [Eq. (10.43)]. For the in Electrolyte Solutions
n
n

2
H O ionization, n 1 n , g g g , and K° becomes
2

w

2
2
K° g m1H O 2m1OH 2>1m°2 dil. aq. soln. (11.12)*

w
3
Experiment (Prob. 13.50) gives 1.00 10 14 for K° at 25°C and 1 atm.
w

Approximating g as 1 in pure water, we get m(H O ) m(OH ) 1.00 10 7

3
mol/kg in pure water at 25°C. This gives an ionic strength I 1.00 10 7 mol/kg.
m
The Davies equation (10.68) then gives g 0.9996 in pure water, which is essen-

tially equal to 1. Hence the H O and OH molalities are accurately equal to 1.00

3
10 7 mol/kg in pure water at 25°C. In an aqueous solution that is not extremely dilute,
g in (11.12) will probably not be close to 1.

Since m° for each species in solution depends on pressure, G° for the reaction
depends on pressure and the equilibrium constant for a reaction in solution depends
on pressure. However, this dependence is weak. Ordinarily, equilibrium constants in
solution are determined for P near 1 bar, and this value of P is assumed throughout
this section.
Next, consider the ionization of the weak acid HX in aqueous solution. The ion-
ization reaction and the molality-scale equilibrium constant (11.7) are

HX H O ∆ H O X (11.13)
3
2

3g1H O 2m1H O 2>m°43g1X 2m1X 2>m°4
3
3
K° (11.14)
a
g1HX2m1HX2>m°
where the subscript a (for acid) is traditional and where the activity of the solvent H O
2
is approximated as 1 in dilute solutions. Figure 11.1 plots K° at 25°C and 1 bar for
a
some acids in water. In most applications, the HX molality is rather low, and it is a
good approximation to take g 1 for the uncharged species HX. However, even

though the X and H O molalities are usually much less than the HX molality, we
3
cannot set g 1 for these ions. g for an ion deviates significantly from 1 even in quite
dilute solutions. Using (10.43) to introduce g , we have

2
g m1H O 2m1X 2
3

K dil. soln. (11.15)
a
m1HX2 Figure 11.1

where g is for the pair of ions H O and X and differs from g in (11.12). In Ionization constants of acids in

3

(11.15) we have omitted dividing each molality by the standard molality m°( 1 water at 25°C and 1 atm. The
mol/kg), so K has the dimensions of molality (mol/kg). Correspondingly, the degree values for strong acids are
a
superscript on K is omitted. approximate. For consistency with
a
Eq. (11.15), K for H O is
a
2

2
g m(H O )m(OH )/m(H O),

3
2
EXAMPLE 11.1 Weak-acid ionization which differs from K . The scale
w
is logarithmic. (Data from J.
K 1.75 10 5 mol/kg for acetic acid (HC H O ) in water at 25°C. Find March, Advanced Organic
2
2
a
3

the H O and OH molalities in a 0.200-mol/kg 25°C aqueous solution of Chemistry, 3d ed., Wiley, 1985,
3
pp. 220–222.)
acetic acid.

To solve (11.15) for m(H O ), we need g . To use the Davies equation
3

(10.68) to estimate g , we need the ionic strength I , which can’t be calculated

m

until m(H O ) is known. The solution to this dilemma is to first estimate
3

m(H O ) and m(X ) by setting g 1 in (11.15) and solving for the ionic mo-

3
lalities. With these approximate molalities, we calculate an approximate I and
m
then use the Davies equation to find an approximate g , which we use in (11.15)

to find a more accurate value for the molalities. If necessary, we can then use
these more accurate molalities to find a more accurate I , and so on.
m