# How to calculate the composition of a borate buffer with a defined pH using the Henderson-Hasselbalch equation?

I am struggling with what appears to be an extremely easy pH problem that uses the Henderson-Hasselbalch equation. The problem and answer provided by the book is given below. I understand where the book is getting the answer but it appears that they are overlooking something.

From what I understand, the Henderson-Hasselbalch equation allows you to calculate the pH of a solution based on the concentrations of acid and conjugate base at equilibrium. In the problem, you start off with a solution of just the conjugate base. Then you add pure acid to the solution to get a desired pH. However, the book seems to assume that the acid you add completely contributes to the concentration of undissociated acid at equilibrium [HA]. How can this be the case? Some of the acid you add will surely dissociate. Anyways, here's what the book did...

Consider a weak acid HA and the salt BA. Its easy to show that the concentration of $H^+$ due to ionisation of HA is $$[H^+]=k_a\frac{[HA]}{[A^-]}$$ It is reasonable to assume that the concentration of $A^-$ from BA is much larger than that from HA due to complete ionisation of BA. Hence $[A^-]=[BA]$=[salt] So, $$[H^+]=k_a\frac{[acid]}{[salt]}$$ Taking logarithm, you get the desired equation.