Optimizing a separation of acids. Benzoic acid containing 16O can be separated from benzoic acid containing 18O by electrophoresis at a suitable pH because they have slightly different acid dissociation constants. The difference in mobility is caused by the different fraction of each acid in the anionic form, A-. Calling this fraction α, we can write

where K is the equilibrium constant. The greater the fraction of acid in the form A-, the faster it will migrate in the electric field.
It can be shown that, for electrophoresis, the maximum separation will occur when ˆ†Î±/ˆšÎ± is a maximum. In this expression, ˆ†Î± = 16a- 18a, and is the average fraction of dissociation [ = ½ (16a + 18a)].
(a) Let us denote the ratio of acid dissociation constants as R 16K/18K. In general, R will be close to unity. For benzoic acid, R = 1.020. Abbreviate 16K as K and write 18K = K/R. Derive an expression for ˆ†Î±/ˆšÎ± in terms of K, [H+], and R. Because both equilibrium constants are nearly equal (R is close to unity), set equal to 16α in your expression.
(b) Find the maximum value ˆ†Î±/ˆšÎ± of by taking the derivative with respect to [H+] and setting it equal to 0. Show that the maximum difference in mobility of isotopic benzoic acids occurs when.

(c) Show that, for , this expression simplifies to [H+] 2K, or pH pK 0.30. That is, the maximum electrophoretic separation should occur when the column buffer has pH = pK - 0.30, regardless of the exact value of R.63

18K I6K fK' 18 I6 18K 16 18 : 16K + [H+I [H'] = (K/2R)(1 + V1 + 8R).