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The mass balance for a generic amphiprotic substance $[\ce{AH^-}]$ this:

$$ [\ce{HA^-}]=C \cdot \dfrac{K_{a1} [\ce{H_3O^+}]}{K_{a1}\cdot [\ce{H_3O^+}] + [\ce{H_3O^+}]^2 + K_{a1} \cdot K_{a2}} \quad Eq.1 $$

where: C = initial concentration of the amphitrofic substance

The Eq.1 derivative is:

$$ \dfrac{d[\ce{HA^-}]}{d[\ce{H_3O^+}]} = -\dfrac{C \cdot [\ce{H_3O^+}] \cdot K_{a1} \cdot (2 [\ce{H_3O^+}] + K_{a1}}{(K_{a1}\cdot [\ce{H_3O^+}] + [H_3O^+]^2 + K_{a1} \cdot K_{a2})^2} + \dfrac{C \cdot \cdot K_{a1}}{K_{a1}\cdot [\ce{H_3O^+}] + [\ce{H_3O^+}]^2 + K_{a1} \cdot K_{a2}} \quad Eq.2 $$

The zero of this derivative is found at $[\ce{H_3O^+}] = \sqrt{K_{a1} \cdot K_{a2}}$, where is the maximum of $[\ce{AH^-}]$.

The buffer power (BP) can be defined $BP = \dfrac{dC_B}{dp \ce{H}} = -\dfrac{dC_A}{dp \ce{H}} = - 2.3 \cdot [\ce{H_3O^+}] \cdot \dfrac{d[\ce{HA^-}]}{d[\ce{H_3O^+}]} \quad Eq.3$

For example:

If $K_{a1} = 6.2 \times 10^{-8}$ and $K_{a2} = 4.8 \times 10^{-13}$ and C=0.1, the maximum oncentration of the specie $\ce{AH^-}$ is found at $[\ce{H_3O^+}] = \sqrt{6.2 \times 10^{-8} \cdot 4.8 \times 10^{-13}} = 1.725 \times 10^{-10} \quad pH=9.76$

The curve that represents the buffer power, plotting the Eq.3, is:

enter image description here

In this way I can say that at maximum concentration of the amphiprotic substance the buffer power is zero. Is this statement true?

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As the curve that represents the buffer capacity does not simply rely on one fraction of the whole system but on all species of the acid-base system it is not enough to only consider the derivative of $c(\ce{HA-})$.

You also have to take care of the other deprotonated species in your system. For the case of a diprotic acid this means that $c(\ce{A^2-})$ is also important albeit not much as you can see in the following figure.

comparison of different simplifications onto the buffer capacity of a diprotic acid Figure 1: Buffer capacities for a diprotic acid with different amounts of assumptions taken into account – denoted by their plot labels. The lower graphs are zoomed versions of the upper graphs.

As you have come to the mass balance for $c(\ce{HA-})$ I don't have to show you how to come to the mass balance for $c(\ce{A^2-})$: $$c(\ce{A^2-}) = \frac{c_0(\ce{H2A})~k_{a1}~k_{a2}}{c^2(\ce{H3O+})+c(\ce{H3O+})~k_{a1}+k_{a1}~k_{a2}}$$

This means that you have to derive the following equation: $$C_B=\frac{c_0(\ce{H2A})~\left(c(\ce{H3O+})~k_{a1}+2~k_{a1}~k_{a2}\right)}{c^2(\ce{H3O+})+c(\ce{H3O+})~k_{a1}+2~k_{a1}~k_{a2}}$$ or if you want to include he self-ionization of water it has to be $$C_B=\frac{k_W}{c(\ce{H3O+})}-c(\ce{H3O+})+\frac{c_0(\ce{H2A})~\left(c(\ce{H3O+})~k_{a1}+2~k_{a1}~k_{a2}\right)}{c^2(\ce{H3O+})+c(\ce{H3O+})~k_{a1}+2~k_{a1}~k_{a2}}$$

Let's derive (with the help of D. D. van Slyke, J. Biol. Chem. 1922, (52), 525-570 and Mathematica (or by hand if you've got time for that)) $$\beta=-\frac{\mathrm{dC_B}}{\mathrm{dpH}}=-\ln(10)~c(\ce{H3O+})~\left(\frac{\mathrm{d C_B}}{\mathrm{d}c(\ce{H3O+})}\right)$$ $$\frac{\mathrm{d C_B}}{\mathrm{d}c(\ce{H3O+})}=-\frac{k_W}{x}-1-\frac{c_0(\ce{H2A})~k_{a1}~c(\ce{H3O+})~(2~c(\ce{H3O+})+k_{a1})}{(c^2(\ce{H3O+})+c(\ce{H3O+})~k_{a1}+k_{a1}~k_{a2})^2}+\frac{c_0(\ce{H2A})~k_{a1}}{c^2(\ce{H3O+})+c(\ce{H3O+})~k_{a1}+k_{a1}~k_{a2}}$$

Using this final equation you should come up with plots like mine and see that the buffer capacity (for $c_0(\ce{H2A})=0.1~\ce{mol/L}$) at $\ce{pH}=\frac{1}{2}\left(\ce{pK_{a1}}+\ce{pK_{a2}}\right)$ is not zero albeit not being very high.

  • complete: $\beta=0.0014~\ce{mol/L}$
  • complete w/o water: $\beta=0.0013~\ce{mol/L}$
  • w/o $c(\ce{A^2-})$: $\beta=0.0001~\ce{mol/L}$
  • w/o $c(\ce{A^2-})$ and water: $\beta=1.1 \cdot 10^{-10}~\ce{mol/L}$

So ... is it correct to say that at maximum concentration of the amphiprotic substance the buffer capacity is zero? No but for your to acidic constants it's nearly zero.

But is it always zero at $c_\text{max}(\ce{HA^-})$?

Let's look at those systems a little bit closer. Imagine a few systems and their corresponding buffer capacity curves with $pK_{a1}=4$ and $pK_{a2}$ values between 4 and 10 like in the following figure.

enter image description here

If you create a plot with the buffer capacity over $\frac{1}{2}\left(pK_{a1}+pK_{a2}\right)$ you would end up with something like this

enter image description here

What you can see now is that the buffer capacity at $c_\text{max}(\ce{HA-})$ depends clearly on the distance between both $pK_a$ values. If they are close together the buffer capacity at $c_\text{max}(\ce{HA-})$ is high and if $\Delta pK_a \approx 3$ the buffer capacity at $c_\text{max}(\ce{HA-})$ is nearly zero but never exactly zero.

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