show matthematically why an acetate buffer cannot maintain an absolutely constant pH upon the addition of H+?

Answer 1

This is true for any acid-base buffer in general. Suppose the buffer is given by the mixture of HX and X-, then we have the equilibrium

HX <----> X- + H+

and acid constant

Ka = [X-][H+] / [HX]

Suppose we add an infinitesimal amount dh of H+. Part of it (dx) leads to an increase of [H+]; the remainder (dh - dx) reacts with [X-], leading to an increase in [HX] and a decrease in [X-]. The equilibrium fraction changes with amount

dKa = Ka * { (dx - dh)/[X-] + (dh - dx)/[HX] - dx / [H+] }

Because Ka is a constant, the factor in { .. } must be zero, so

dx * {1/[X-] - 1/[HX] - 1/[H+]} = dh * {1/[HX] - 1/[X-]}

It follows that

dx = {1/[HX] - 1/[X-]} / {1/[X-] - 1/[HX] - 1/[H+]} * dh

which is non-zero except when [HX] = [X-]. That is the inflection point of a titration curve, but it is no absolute constant.