A buffer that contains \(0.110 M \mathrm{HY}\) and \(0.220 \mathrm{M} \mathrm{Y}^{-}\) has a \(\mathrm{pH}\) of 8.77 . What is the \(\mathrm{pH}\) after \(0.0015 \mathrm{~mol}\) of \(\mathrm{Ba}(\mathrm{OH})_{2}\) is added to \(0.350 \mathrm{~L}\) of this solution?

The Henderson-Hasselbalch equation is crucial in understanding buffer solutions and their pH. It relates the pH of a buffer to the pKa (acid dissociation constant) and the ratio of the concentrations of the conjugate base to the acid in the buffer. The equation is written as:
\[ \text{pH} = \text{p}K_a + \text{log} \frac{[\text{conjugate base}]}{[\text{acid}]} \]
This equation allows us to predict the effect of adding acids or bases to buffer solutions and is particularly useful for calculating the pH during experiments. By knowing the initial concentrations of the buffer components and using the Henderson-Hasselbalch equation, we can precisely adjust and maintain the pH within a desired range.

Acid-base equilibrium represents the balance between acidic and basic species in a solution. For a buffer system, this equilibrium involves the weak acid (HY) and its conjugate base (Y⁻). In a buffer solution, the acid partially dissociates in water:
\[ \text{HY} \rightleftharpoons \text{H}^+ + \text{Y}^- \]
The conjugate base, Y⁻, can react with any added H⁺ ions to form HY again, minimizing the change in pH. Likewise, the addition of OH⁻ ions will react with H⁺ ions from the dissociation, forming water and shifting the equilibrium to produce more H⁺ ions, thus again stabilizing the pH. This equilibrium behavior is the key to the buffer's ability to resist pH changes.

The pKa value of an acid is a measure of its strength, represented by the negative log of the acid dissociation constant (Ka). It provides insight into how easily the acid donates protons (H⁺) in a solution. To determine the pKa using the initial conditions of our buffer solution, we rearrange the Henderson-Hasselbalch equation:
\[ \text{pKa} = \text{pH} - \text{log} \frac{[\text{conjugate base}]}{[\text{acid}]} \]
Given that the initial pH is 8.77 and the concentrations of HY and Y⁻ are 0.110 M and 0.220 M respectively, we can solve:
\[ 8.77 = \text{p}K_a + \text{log}(2) \] \[ \text{p}K_a = 8.77 - 0.30 = 8.47 \]
This value helps us understand the buffer’s properties and predict its behavior under different conditions.

Buffer capacity refers to the ability of a buffer solution to maintain its pH when small amounts of acid or base are added. It depends on the concentrations of the weak acid and its conjugate base; higher concentrations provide greater buffer capacity.
When Ba(OH)₂ is added to our buffer solution, it dissociates to give Ba²⁺ and 2OH⁻ ions. The OH⁻ ions react with HY, transforming it into Y⁻ and water:
\[ \text{HY} + \text{OH}^- \rightarrow \text{Y}^- + \text{H}_2\text{O} \]
This reaction alters the concentrations of HY and Y⁻, but the buffer's ability to neutralize the added base without a significant pH change demonstrates its capacity. After calculations, the new concentrations are found, and by applying them into the Henderson-Hasselbalch equation again, we find the new pH. The slight change in pH (from 8.77 to around 9.12) reflects the buffer’s efficient capacity to counteract changes and maintain equilibrium.