I read a problem which was as follows:

Equal volumes of $\pu{1M} \ \ce{H3PO4}$ and $\pu{1M} \ \ce{Na3PO4}$ are mixed. The $\mathrm{pH}$ of the resultant mixture is (along with this $K_\mathrm{a1}, K_\mathrm{a2}, K_\mathrm{a3}$ of the acid were mentioned)

The way I approached this problem was as follows: $K_\mathrm{a1} = \frac{[\ce{H2PO4^-}][\ce{H^+}]}{[\ce{H3PO4}]}$, similarly $K_\mathrm{a2} = \frac{[\ce{HPO4^{2-}}][\ce{H^+}]}{[\ce{H2PO4^-}]}$ and so on...

Which gives me $K_\mathrm{a1}\cdot K_\mathrm{a2}\cdot K_\mathrm{a3} = \frac{[\ce{PO4^{3-}}][\ce{H^+}]^3}{[\ce{H3PO4}]} = [\ce{H^+}]^3$ (since $\ce{[H3PO4] = [Na3PO4]}$). Using this, I calculate the $[\ce{H^+}]$ and so the ph which comes out to be $-\log{\sqrt[3]{K_\mathrm{a1}\cdot K_\mathrm{a2}\cdot K_\mathrm{a3}}}$ which is $\frac{\mathrm{p}K_\mathrm{a1} + \mathrm{p}K_\mathrm{a2} + \mathrm{p}K_\mathrm{a3}}{3}$.

However, it seems from the answer that the system is forming a buffer and $\mathrm{pH}$ is $\mathrm{p}K_\mathrm{a2}$

What am I doing wrong here and how does the $\mathrm{pH}$ come out to be $\mathrm{p}K_\mathrm{a2}$. Is there any such general result or theory for polyprotic acids?

I read a problem which was as follows:

Equal volumes of $\pu{1M} \ \ce{H3PO4}$ and $\pu{1M} \ \ce{Na3PO4}$ are mixed. The $\mathrm{pH}$ of the resultant mixture is (along with this $K_\mathrm{a1}, K_\mathrm{a2}, K_\mathrm{a3}$ of the acid were mentioned)

The way I approached this problem was as follows: $K_\mathrm{a1} = \frac{[\ce{H2PO4^-}][\ce{H^+}]}{[\ce{H3PO4}]}$, similarly $K_\mathrm{a2} = \frac{[\ce{HPO4^{2-}}][\ce{H^+}]}{[\ce{H2PO4^-}]}$ and so on...

Which gives me $K_\mathrm{a1}\cdot K_\mathrm{a2}\cdot K_\mathrm{a3} = \frac{[\ce{PO4^{3-}}][\ce{H^+}]^3}{[\ce{H3PO4}]} = [\ce{H^+}]^3$ (since $\ce{[H3PO4] = [Na3PO4]}$). Using this, I calculate the $[\ce{H^+}]$ and so the ph which comes out to be $-\log{\sqrt[3]{K_\mathrm{a1}\cdot K_\mathrm{a2}\cdot K_\mathrm{a3}}}$ which is $\frac{\mathrm{p}K_\mathrm{a1} + \mathrm{p}K_\mathrm{a2} + \mathrm{p}K_\mathrm{a3}}{3}$.

However, it seems from the answer that the system is forming a buffer and $\mathrm{pH}$ is $\mathrm{p}K_\mathrm{a2}$

What am I doing wrong here and how does the $\mathrm{pH}$ come out to be $\mathrm{p}K_\mathrm{a2}$. Is there any such general result or theory for polyprotic acids?

EDIT: I understand now why my solution is wrong. Any help as to how to proceed further will be appreciated.

I read a problem which was as follows

Equal volumes of 1M H3PO4 and 1M Na3PO4 are mixed. The pH of the resultant mixture is (along with this ka1,ka2,ka3 of the acid were mentioned)

The way I approached this problem was as follows: $ka1 = \frac{[\ce{H2PO4^-}][\ce{H^+}]}{[\ce{H3PO4}]}$, similarly $ka2 = \frac{[\ce{HPO4^{-2}}][\ce{H^+}]}{[\ce{H2PO4^-}]}$ and so on...

Which gives me $ka1.ka2.ka3 = \frac{[\ce{PO4^{-3}}][\ce{H^+}]^3}{[\ce{H3PO4}]} = [H^+]^3$ (since $\ce{[H3PO4] = [Na3PO4]}$). Using this I calculate the $[H^+]$ and so the ph which comes out to be $-\log{\sqrt[3]{ka1.ka2.ka3}}$ which is $\frac{pka1 + pka2 + pka3}{3}$

However, it seems from the answer that the system is forming a buffer and pH is $pka2$

What am I doing wrong here and how does the ph come out to be $pka2$. Is there any such general result or theory for polyprotic acids?

I read a problem which was as follows:

Equal volumes of $\pu{1M} \ \ce{H3PO4}$ and $\pu{1M} \ \ce{Na3PO4}$ are mixed. The $\mathrm{pH}$ of the resultant mixture is (along with this $K_\mathrm{a1}, K_\mathrm{a2}, K_\mathrm{a3}$ of the acid were mentioned)

The way I approached this problem was as follows: $K_\mathrm{a1} = \frac{[\ce{H2PO4^-}][\ce{H^+}]}{[\ce{H3PO4}]}$, similarly $K_\mathrm{a2} = \frac{[\ce{HPO4^{2-}}][\ce{H^+}]}{[\ce{H2PO4^-}]}$ and so on...

Which gives me $K_\mathrm{a1}\cdot K_\mathrm{a2}\cdot K_\mathrm{a3} = \frac{[\ce{PO4^{3-}}][\ce{H^+}]^3}{[\ce{H3PO4}]} = [\ce{H^+}]^3$ (since $\ce{[H3PO4] = [Na3PO4]}$). Using this, I calculate the $[\ce{H^+}]$ and so the ph which comes out to be $-\log{\sqrt[3]{K_\mathrm{a1}\cdot K_\mathrm{a2}\cdot K_\mathrm{a3}}}$ which is $\frac{\mathrm{p}K_\mathrm{a1} + \mathrm{p}K_\mathrm{a2} + \mathrm{p}K_\mathrm{a3}}{3}$.

However, it seems from the answer that the system is forming a buffer and $\mathrm{pH}$ is $\mathrm{p}K_\mathrm{a2}$

What am I doing wrong here and how does the $\mathrm{pH}$ come out to be $\mathrm{p}K_\mathrm{a2}$. Is there any such general result or theory for polyprotic acids?