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Is there a mathematical proof/explanation of why $\mathrm{p}K_\mathrm{a}$ corresponds to the $\mathrm{pH}$ at $\text{Volume of titrant}/2$ at the equivalence point?

A concise proof will suffice.

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You've got a weak acid, since you're contemplating a positive pKa, which means when you're halfway to the end point you're in the buffer region and you can use the Henderson-Hasselbalch equation:

pH = pKa + log [A-]/[HA]

You've titrated half your initial HA, so half of it is still around and half got turned into A-, which means [A-] = [HA]. For example, if you started with 0.1 mol/L of HA, you now have 0.05 mol/L HA and 0.05 mol/L of A-.

That means [A-]/[HA] = 1, and the log of 1 is zero. The HH equation then tells you pH = pKa.

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