Why is maximum buffer capacity for some diprotic buffers not when pH = pKa?

Question

According to Rajkovic et al. [1]:

$$β = \frac{[\ce{H+}]}{K_\mathrm{w}} + 2.303\cdot\frac{[\ce{H+}]\cdot K_\mathrm{a}\cdot c}{[\ce{H+}] + K_\mathrm{a}}$$

where $K_\mathrm{a}$ is the dissociation constant of the acid; $c$ is the concentration of the acid in all forms; and $K_\mathrm{w}$ is the ionization constant of water $(1·10^{–14})$. It can be seen that the buffer capacity of the acid is greatest when $[\ce{H+}]$ equals $K_\mathrm{a}$ (or when the $\mathrm{pH}$ is equal to the $\mathrm{p}K_\mathrm{a}$ of the acid). This relationship can be used for mixtures of monoprotic acids and many diprotic acids (by considering them to be made up of two monoprotic acids). Unfortunately, this is not true for most of the diprotic acids found in wines because the second dissociation is not completely independent of the first.

Is it true that for some diprotic acids maximum buffer capacity is not reached when $\mathrm{pH} = \mathrm{p}K_\mathrm{a}$? If so, why is it true? Also, then when is buffer capacity maximum? Can it only be determined experimentally?

Or did I just misinterpret the article? Is it saying that this equation of buffer capacity is invalid for some diprotic acids instead of the fact that buffer capacity is max when $\mathrm{pH} = \mathrm{p}K_\mathrm{a}$?

References

1. Rajkovic, M.; Novakovic, I.; Petrovic, A. Determination of Titratable Acidity in White Wine. Journal of Agricultural Sciences, Belgrade 2007, 52 (2), 169–184. https://doi.org/10.2298/JAS0702169R.

Answer 1

Any aqueous solution increases its buffer capacity toward $\mathrm{pH} = 0$ or $\mathrm{pH} = 14$, as more and more acid or basis is needed to change $\mathrm{pH}$ by a given interval. It happens even without presence of any buffering substance, aside of strong acids or bases.

If an acid, monoprotic or diprotic, is added, a peak(s) of maximum buffer capacity is added. But if $\mathrm pK_\mathrm a$ is small, the peak of maximum buffer capacity does not lay on a steady baseline of the capacity function. It lays on a sloped baseline. If the maximum absolute peak slope is smaller than the absolute baseline slope, the peak does not manifests itself as a peak(=maximum capacity), but rather as just as a shoulder on the slope, without a maximum.

This phenomena is not specific for chemistry, but rather for mathematics and mathematical analysis. If you have a sinusoid function on a large enough slope, its peaks would disappear at some slope value, becoming just a shoulders.

The similar phenomena occurs in case of multiple $\mathrm{p}K_\mathrm{a}$ in the solution, of the same or of different compounds. On the background of non steady capacity function, a particular peak may not have a shape of a peak.

Additionally, the presence of close peak neighbors affects position of the peak maximum.

If 2 peaks of the similar height are close enough each other, their maxima shifts towards each other, and with enough closeness they merge into a single peak.