Physical Chemistry Third Edition

7.5 Buffer Solutions 333

We letmeq(OH−)/m◦yandmeq(H+)/m◦x.

Ka

γ^2 x

(

ms/m◦+x−y

)

ma/m◦−x+y

Whenyis replaced byKw/γ^2 x, we obtain the equation

γ^2 x^3 +(γ^2 ms/m◦+K)x^2 −(Kw+Kma/m◦)x−KaKw/γ^2  0 (7.5-6)

The ionic strength of the solution is equal toms+x, so that the activity coefficientγ

can be estimated by settingImsin the Davies equation. The cubic equation is solved

numerically and the result is thatx 2. 13 × 10 −^5 ,γ 0 .740,a(H+) 1. 576 × 10 −^5 ,

and pH 4 .802. The pH value differs from that in part a by 0.130. If the activity coefficient

is included butxandyare neglected compared withmsandmathe resulting pH is 4.800.

The most significant source of error in the Henderson–Hasselbalch equation is the neglect

of the activity coefficient, not the neglect of ionization and hydrolysis.

Exercise 7.13

a.Verify Eq. (7.5-6).

b.Substitute the numerical result of part b of Example 7.16 into the equilibrium expression,

Eq. (7.5-3), to verify its correctness.

c.Carry out the solution of Example 7.16 including the activity coefficient but neglectingxand

ycompared withmaandms.

Biological Buffering

The principal buffering system in mammalian blood is that of carbonic acid (H 2 CO 3 ).

We have the following equilibria:

CO 2 (g)
CO 2 (aq) (7.5-7a)

CO 2 (aq)+H 2 O
H 2 CO 3 (aq) (7.5-7b)

H 2 CO 3 (aq)
H++HCO− 3 (7.5-7c)

HCO− 3 
H++CO^23 − (7.5-7d)

EXAMPLE7.17

The Henry’s law constant for CO 2 in water at 25◦C is equal to 1. 24 × 106 torr. The equi-

librium constant for reaction (7.5-7b) is equal to 2. 58 × 10 −^3 , that for reaction (7.5-7c)

is equal to 1. 70 × 10 −^4 , and that for reaction in Eq. (7.5-7d) is equal to 4. 69 × 10 −^11 .It

is not possible by chemical analysis to distinguish CO 2 (aq) from H 2 CO 3 , but it has been

determined that the molality of CO 2 (aq) is larger than that of H 2 CO 3 (aq). Find the value

of the first ionization constant of carbonic acid, using the combined molalities of CO 2 (aq)

and H 2 CO 3 (aq) in the equilibrium expression instead of the molality of H 2 CO 3 (aq). Explain

why reaction in Eq. (7.5-7c) can act as an effective buffering reaction near the normal pH of

blood, around pH 7.4.