A buffer that contains \(1.05 M \mathrm{~B}\) and \(0.750 M \mathrm{BH}^{+}\) has a pH of \(9.50 .\) What is the \(\mathrm{pH}\) after \(0.0050 \mathrm{~mol}\) of \(\mathrm{HCl}\) is added to \(0.500 \mathrm{~L}\) of this solution?

To calculate the pH of a buffer solution, we use an important tool known as the Henderson-Hasselbalch equation. This equation connects the pH of the solution with the concentrations of the acid and its conjugate base present in the solution. The Henderson-Hasselbalch equation is expressed as:\[ \text{pH} = \text{p}K_a + \text{log} \frac{[\text{B}]}{[\text{BH}^+]} \] Here, \(\text{p}K_a\) is the negative logarithm of the acid dissociation constant (\(K_a\)), which measures the strength of the acid.\( [\text{B}]\) represents the concentration of the base, and \([\text{BH}^+]\) represents the concentration of the conjugate acid.
This equation allows us to understand how the pH of a buffer solution changes when small amounts of acid or base are added. By maintaining specific molar concentrations of \( [\text{B}]\) and \([\text{BH}^+]\), we can keep the pH relatively stable.

An acid-base reaction involves the transfer of hydrogen ions (\(\text{H}^+\)) between the reactants. When a strong acid like hydrochloric acid (HCl) is added to a buffer solution, it dissociates completely to release \(\text{H}^+\) ions in the solution. These \(\text{H}^+\) ions will react with the base present in the buffer. In our example, the base is represented as \(\text{B}\). The reaction can be depicted as: \[ \text{B} + \text{H}^+ \rightarrow \text{BH}^+ \] This reaction illustrates how the base (\(\text{B}\)) neutralizes the added acid by converting into the conjugate acid (\(\text{BH}^+\)). Consequently, the concentrations of \(\text{B}\) and \(\text{BH}^+\) change, affecting the pH calculation. Understanding these reactions is crucial for grasping how buffer solutions work to resist drastic changes in pH when small amounts of acids or bases are introduced.

Molar concentration, or molarity, is a measure of the concentration of a solute in a solution. It is defined as the number of moles of solute per liter of solution and is expressed in units of moles per liter (M). In the given exercise, we start with molar concentrations of \( 1.05 \text{ M} \) for \(\text{B}\) (the base) and \( 0.750 \text{ M} \) for \(\text{BH}^+\) (the conjugate acid).
Adding a known amount of hydrochloric acid (HCl) to a buffer system changes these concentrations. The calculation must account for the volume of the solution and the amount of acid added to find the new concentrations. After reacting with \(\text{B}\), the concentrations are recalculated using: \[ \text{new } [\text{B}] = \frac{\text{new moles of B}}{\text{volume of solution}} \] \[ \text{new } [\text{BH}^+] = \frac{\text{new moles of BH}^+}{\text{volume of solution}} \] Accurate concentration calculations are essential for correctly using the Henderson-Hasselbalch equation to determine the resulting pH of the buffer solution.