\(12.85\) Malonic acid, \(\mathrm{HOOCCH}{ }_{2} \mathrm{COOH}\), a diprotic acid with \(\mathrm{p} K_{\mathrm{a} 1}=2.8\) and \(\mathrm{p} K_{\mathrm{a} 2}=5.7\), is titrated with \(\mathrm{KOH}(\mathrm{aq})\). (a) What is the \(\mathrm{pH}\) when \(\left[\mathrm{HOOCCH}{ }_{2} \mathrm{COOH}\right]=\left[\mathrm{HOOCCH}_{2} \mathrm{CO}_{2}{ }^{-}\right]\)? (b) What is the pH when \(\left[\mathrm{HOOCCH}_{2} \mathrm{CO}_{2}{ }^{-}\right]=\left[{ }^{-} \mathrm{O}_{2} \mathrm{CCH}_{2} \mathrm{CO}_{2}{ }^{-}\right]\) (c) Which is the predominant species at \(\mathrm{pH}=4.2\) ?

Acid-base titration is a laboratory process used to determine the concentration of an unknown acid or base solution by performing a controlled reaction with a solution of known concentration, usually referred to as the titrant. In the case of titrating a diprotic acid like malonic acid, we have to consider that titration involves two steps as each hydrogen ion is released sequentially.

In practice, a color indicator or a pH meter can be used to detect the endpoint of the titration, which occurs when equivalent amounts of acid and base have reacted. The titration curve for a diprotic acid features two distinct buffering regions and two equivalence points, one for each of the protons being titrated. These equivalence points relate closely to the pKa values of each acidic proton, which helps us in determining the pH during the titration process at specific points.

The calculation of pH during a titration is vital to understand the acid-base behavior of the solution. pH is calculated using the concentration of hydrogen ions, \( H^+ \), in the solution. In the state when a diprotic acid is half neutralized, meaning the acid form is in equilibrium with its first dissociated form, the Henderson-Hasselbalch equation is often employed to find pH when concentrations are known:

\[ \text{pH} = \text{p}K_{\text{a}} + \log\left(\frac{\text{[A-]}}{\text{[HA]}}\right) \]
However, the exercise shows an even simpler method by exploiting the characteristic of the titration curve at the halfway point to the first and second equivalence points. Here, the concentration of the acid equals that of its conjugate base, simplifying the log term to 1, which means pH equals pKa at these particular points. The Henderson-Hasselbalch equation is not directly used in this scenario, but it is an underlying concept aiding the understanding of pH changes during titration.

The acid dissociation constant, denoted as Ka or pKa, is a quantitative measure of the strength of an acid in solution. A low pKa value indicates a strong acid that dissociates completely in solution, while a higher pKa value corresponds to a weaker acid that does not dissociate as much. Diprotic acids have two pKa values, one for each hydrogen ion they can donate.

In the context of a titration, these pKa values are critical, as they determine the pH at which the acid is half dissociated. When the pH of the solution equals the pKa value of the acid, as seen in the exercise, the concentrations of the protonated and deprotonated forms are equal. This concept helps predict the predominant form of the acid at different pH levels. For instance, malonic acid at pH 4.2, which is between its two pKa values, predominantly exists as its first ionized form \( \mathrm{HOOCCH}_{2} \mathrm{CO}_{2}^{-} \) because that pH is closer to the first pKa. This guides us in understanding acid-base behavior during a titration and formulating buffers.