Page 136 - Thermodynamics of Biochemical Reactions

7.5 Dissociation of a Diprotic Acid 133

The plot of NH versus pH is the titration curve for H,A. Note that In P can be
calculated by integrating (mH/[Hf])d[Hf]. Applying equation 7.5-4 to equation
7.5-3 yields

(7.5-5)

At very high pH, the binding of Hi approaches zero, and at very low pH it
approaches 2.
There is another way to look at this binding, and that is to assume that the
two groups are independent. In this case the dissociation reactions are written
HA- = H+ + A2- Kl (7.5-6)

HAH = Hf + HA- K2 (7.5-7)

In other words, K~ is the dissociation constant for the left hydrogen atom and K,
is the dissociation constant for the right hydrogen atom. In this case the binding
is simply the sum of the bindings at the two sites:

(7.5-8)

This equation can be rearranged to

Thus
(7.5- 10)

K, = rcl + K, (7.5-1 1)
If IC~ << K,, K, is essentially equal to K~ and K, is essentially equal to K,. When
the groups are independent, the binding polynomial is given by

K, and K, can be evaluated by curve fitting the plot of P (equation 7.5-3)
versus [H'], or better log P versus log[H+]. If these two dissociation constants
have nearly the same magnitude, a quadratic has to be solved to evaluate K, and
Kz:
K, 5 JK: - 4K,K2
K2 = (7.5- 13)
2
Note that K, cannot be equal to K,. If the two groups are identical with
dissociation constants K (in equations 7.5-6 and 7.5-7), K, = 42 and K2 = 2~.
If K1 = K2,

P=(l+F)2 (7.5- 14)

(7.5- 15)

Equation 7.5-4 yields

(7.5- 1 6)