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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...

Henderson-Hasselbalch Problem

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Boric acid and sodium borate forms an acidic buffer (A mixture of weak acid and salt of weak acid and strong base). The acid is weak hence the concentration of basic ion from acid is much less compared to the concentration of basic ion obtained from the salt which can ionize completely. Yes. Some of the acid dissociates to form hydronium ion and a basic ion. That's how a buffer works.

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.

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Never mind, I found the answer! I was correct to assume that the initial concentration will not be equal to the equilibrium concentration. However, they will be extremely close because we are working with weak acids. In fact, I found an article published in a chemistry education journal that discusses this so-called "Henderson-Hasselbalch Approximation" and how it is often overlooked in chemistry textbooks. Here is the link: Henderson-Hasselbalch Approximation. I believe textbooks should make sure students are aware that the equation is only an approximate method for determining the pH of a buffer.

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