\(\mathrm{~A} 30.0-\mathrm{mL}\) sample of \(0.20 \mathrm{M} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COOH}(\mathrm{aq})\) solution is titrated with \(0.30 \mathrm{M} \mathrm{KOH}(\mathrm{aq})\). (a) What is the initial \(\mathrm{pH}\) of the \(0.20 \mathrm{M} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COOH}(\mathrm{aq})\) solution? (b) What is the \(\mathrm{pH}\) after the addition of \(15.0 \mathrm{~mL}\) of \(0.30 \mathrm{M} \mathrm{KOH}(\mathrm{aq})\) ? (c) What volume of \(0.30 \mathrm{M}\) \(\mathrm{KOH}(\mathrm{aq})\) is required to reach halfway to the stoichiometric point? (d) Calculate the \(\mathrm{pH}\) at the halfway point. (e) What volume of \(0.30 \mathrm{M} \mathrm{KOH}(\mathrm{aq})\) is required to reach the stoichiometric point? (f) Calculate the \(\mathrm{pH}\) at the stoichiometric point.

Understanding the equilibrium constant (Ka) is crucial when studying acid-base reactions, especially during titrations. It represents the strength of an acid in solution, dictating how well the acid donates protons to become its conjugate base. In essence, it is a ratio of the concentration of the products (hydrogen ions, \(H+\), and the conjugate base, \(A-\)) to that of the reactants (the undissociated acid, \(HA\)), not including water since its concentration is constant. Mathematically, it is expressed as \(Ka = \frac{[H+][A-]}{[HA]}\). A larger Ka value indicates a stronger acid. The initial pH of a weak acid solution can be computed using Ka and the initial concentration of the acid, applying the approximation that the concentration of hydrogen ions and the conjugate base at equilibrium (\(x\)) is much less than the initial concentration of the acid. This step simplifies calculations, as in our original problem for \(C_6H_5COOH\).

The Henderson-Hasselbalch equation is a derived logarithmic equation that simplifies the process of pH calculation for buffer solutions, which contain a weak acid and its conjugate base. It is especially handy after the addition of a titrant like \(KOH\) in an acid-base titration, as it helps to calculate the pH at various points before reaching the stoichiometric point. The equation is given by \(pH = pKa + \log(\frac{[A-]}{[HA]})\), where \(pKa\) is the negative logarithm of the acid dissociation constant (Ka), \(\log\) indicates the base-10 logarithm, \(A-\) is the concentration of the conjugate base, and \(HA\) is the concentration of the acid. When the concentrations of the acid and the conjugate base are equal, as is the case at the halfway point to the stoichiometric point in a titration, the log term becomes zero, and thus \(pH = pKa\).

The stoichiometric point in an acid-base titration, often referred to as the equivalence point, is the moment where the amount of acid originally in solution has been exactly neutralized by the added base, or vice versa. This means the number of moles of hydrogen ions equals the number of moles of hydroxide ions added. At this juncture, for a weak acid-strong base titration, the solution contains only water and the conjugate base of the weak acid. It's a pivotal moment in the titration process because it tells us the volume of titrant used to neutralize the solution, which we use to determine the mole-to-mole reactions and ultimately the concentration of the unknown solution. Calculating the stoichiometric point helps to determine the concentration of the acid or base in the solution, which is precisely what we did in part (e) of our original problem when titrating with \(KOH\).

pH calculation is a core concept in acid-base chemistry, reflecting the acidity or basicity of a solution. pH is the negative base-10 logarithm of the hydrogen ion concentration in a solution, defined as \(pH = -\log[H+]\). The pH scale generally ranges from 0 to 14, with 7 being neutral (pure water), values less than 7 indicating acidity, and values greater than 7 indicating basicity. In the context of titrations, calculating pH is critical at various stages: initially, after partial neutralization, at the halfway point, at the stoichiometric point, and beyond. For weak acids or bases, the pH might be derived using the equilibrium expression or the Henderson-Hasselbalch equation, as demonstrated in steps 1, 2, and 4 of the provided solution. At the stoichiometric point, especially in a titration involving a weak acid and a strong base, you will need to calculate the pH using the basicity constant (Kb) of the conjugate base as shown in step 6 of our original problem.