Physical Chemistry Third Edition

324 7 Chemical Equilibrium

We must solve two simultaneous equations. At 298.15 K the equilibrium relation for the acetic

acid is:

Ka 1. 75 × 10 −^5 

γ

(

H+

)

meq

(

H+

)

/m◦γ

(

A−

)

meq

(

A−

)

/m◦

γ(HA)(meq(HA)/m◦)

(7.3-11a)

The equilibrium relation for water is

Kw 1. 0 × 10 −^14 

(

γ(H+)meq(H+)/m◦

)(

γ(OH−)meq(OH−)/m◦

)

(7.3-11b)

We assume thatγ(HA)≈1. Since all of the ions are univalent, we assume that all of their

activity coefficients are equal and denote them byγ. We let

z

meq

(

H+

)

m◦

, y

meq

(

OH−

)

m◦

We express the other molalities in terms ofzandy. For each OH−ion, there is one H+ion

released from water, and for each A−ion there is an H+ion released from the acid, so that

meq

(

A−

)

m◦

z−y,

meq(HA)

m◦



m

m◦

−(z−y)

wheremis the stoichiometric molality. From Eq. (7.3-11b),

y

Kw

γ^2 z

We use this expression to replaceyin the acid equilibrium expression and obtain a single

equation:

Ka

γ^2 z

[

z−Kw/

(

γ^2 z

)]

m/m◦−z+Kw/

(

γ^2 z

) (7.3-12a)

When we multiply this equation out, we obtain a cubic equation:

γ^2 z^3 +Kz^2 −(m/m◦)K+Kwz−KKw/γ^2  0 (7.3-12b)

For our first approximation, we assume thatγ1 and solve forz. A numerical approximation

to the root of this cubic equation givesmeq(H+) 1. 61 × 10 −^7 mol kg−^1 , only 61% higher

than in pure water. Use of the Davies equation for this ionic strength givesγ± 0. 999999

so that no second approximation is necessary. The value of the pH is 6.79. If the hydrogen

ions from water are ignored, the result is thatmeq(H+) 9. 9 × 10 −^8 mol kg−^1 , which is in

error by 38% and is smaller than the value ofm(H+) in pure water.

Another situation that requires the solution of simultaneous equations is the ioniza-

tion of a polyprotic acid, which means that one molecule of the acid ionizes successively

to give more than one hydrogen ion.

Exercise 7.10

Find the pH of a solution made from 1.000 kg of water and 0.100 mol of phosphoric acid, H 3 PO 4 ,

for which the three acid dissociation constants are