PHYSICAL CHEMISTRY IN BRIEF

CHAP. 11: ELECTROCHEMISTRY [CONTENTS] 386

wherecis the acid concentration andαis the degree of dissociation in the second stage. We
obtain it by solving equation (11.65)

α=

−(c+K′) +

√

(c+K′)^2 + 4K′c

2 c

, K′=

K cstγHA−

γH+γA 2 −

≈

K cst

γ^4 ±

, (11.89)

whereγ±is given in11.5.6.

Note: IfcK′orcK′, the numerator of (11.89) is a difference of close numbers.

Either use a sufficient number of digits in the evaluation, or a numerically stable equivalent

formα= 2K′/[c+K′+

√

(c+K′)^2 + 4K′c]. The cases of weak acids and bases are similar.

For a strong diacidic base we have

pH =−log

(

Kwcst

c γOH−

)

+ log (1 +α), (11.90)

where relations (11.89) apply for the degree of dissociation of the base in the second stageα.
Relations (11.88) and (11.90) apply on condition that it is not necessary to consider the
dissociation of water. This has to be taken into account only at extremely low concentrations,
forc < 10 −^6 in order. In such cases we may assume that the dissociation in the second stage is
complete, i.e.α→1, and proceed similarly as in the Example in section11.6.4.

11.6.7 pH of a weak monobasic acid

A weak monobasic acid dissociates up to the attainment of a chemical equilibrium between the
ions and undissociated acid molecules, see11.5.3. We have

pH =−

1

2

log

(

K′c γ±^2

(cst)^2

)

−

1

2

log (1−α), (11.91)

wherecis the initial concentration of the acid andαis the degree of dissociation for which it
follows from equation (11.53)

α=

−K′+

√

K′^2 + 4K′c

2 c

, K′=

K cstγHA

γ±^2

, (11.92)