Question

The buffer capacity \((\beta)\) for a weak acid (A) \(-\) conjugate base (B) buffer is defined as the number of moles of strong acid or base needed to change the \(\mathrm{pH}\) of \(1 \mathrm{~L}\) of solution by \(1 \mathrm{pH}\) unit, where \(\beta=\frac{2.303\left(C_{\mathrm{A}}+C_{\mathrm{B}}\right) K_{\mathrm{a}}\left[\mathrm{H}^{+}\right]}{\left(\left[\mathrm{H}^{+}\right]+K_{\mathrm{a}}\right)^{2}} .\) Under what condition will a buffer best resist a change in \(\mathrm{pH}\) ? (a) \(\mathrm{pH}=3 \mathrm{p} \mathrm{Ka}\) (b) \(2 \mathrm{pH}=\mathrm{p} \mathrm{Ka}\) (c) \(\mathrm{pH}=\mathrm{p} \mathrm{Ka}\) (d) \(\mathrm{pH}=2 \mathrm{p} \mathrm{Ka}\)

Short answer

The buffer will best resist a change in pH under the condition (c) \(\mathrm{pH} = \mathrm{pKa}\).

Step-by-step solution

Identify the Condition for Maximal Buffer Capacity

To find the condition under which the buffer best resists changes in pH, one must identify when the buffer capacity \(\beta\) is maximized. The buffer capacity \(\beta\) depends on the concentrations of the weak acid \(C_{\mathrm{A}}\) and its conjugate base \(C_{\mathrm{B}}\), the acid dissociation constant \(K_{\mathrm{a}}\), and the hydrogen ion concentration \([\mathrm{H}^{+}]\).

Express the pH Condition in Terms of [H+]

Each of the given conditions can be transformed into an expression involving the hydrogen ion concentration \([\mathrm{H}^{+}]\) and the acid dissociation constant \(K_{\mathrm{a}}\). Remember that \(\mathrm{pH} = -\log[\mathrm{H}^{+}]\) and \(\mathrm{pKa} = -\log K_{\mathrm{a}}\). From this relationship, we can deduce that at \(\mathrm{pH} = \mathrm{pKa}\), it means that \([\mathrm{H}^{+}] = K_{\mathrm{a}}\).

Find the Optimum Condition

Comparing the derivative expression with the given options, we can see that the optimum condition occurs when the derivative is zero. This happens when \(\left([\mathrm{H}^{+}] + K_{\mathrm{a}}\right)^2\) is largest, which is when \([\mathrm{H}^{+}] = K_{\mathrm{a}}\) since both terms are equal and add constructively. Hence, the buffer capacity is maximized when \(\mathrm{pH} = \mathrm{pKa}\), which corresponds to option (c).

Key concepts

Weak Acid and Conjugate Base Buffer

Understanding how a weak acid and its conjugate base work together to create a buffer system is essential for grasping how buffers maintain a stable pH. A buffer typically consists of a weak acid (HA) and its conjugate base (A^-). When a strong acid (like HCl) is added to this system, the conjugate base can react with the added H⁺ ions to produce more HA, reducing the impact of the additional acid. Likewise, if a strong base (such as NaOH) is added, the weak acid can donate an H⁺ ion to neutralize the OH^- ions, forming water and the conjugate base, A^-.

This buffering action is most effective when the concentrations of the weak acid and its conjugate base are similar because they can neutralize added acids and bases with equal ability. This equilibrium is crucial for the buffer capacity, which indicates the buffer's ability to withstand changes in pH.

• If too much weak acid is present, the buffer is less capable of neutralizing added bases.
• If too much conjugate base is there, then added acids will cause a larger pH change.

Acid Dissociation Constant (Ka)

The acid dissociation constant, abbreviated as Ka, is a quantitative measure of the strength of an acid in solution. It specifically represents the equilibrium constant for the dissociation of a weak acid into its ions. For a generic weak acid HA, this dissociation can be represented by the equation:\[ HA \rightleftharpoons H^{+} + A^{-} \]The Ka expression for this equilibrium is given by:\[ K_{a} = \frac{[H^{+}][A^{-}]}{[HA]} \]A larger Ka value indicates a stronger acid that dissociates more in solution, which implies a higher concentration of H⁺ ions at equilibrium. Weak acids have smaller Ka values because they only partially dissociate in solution. Understanding Ka is crucial because it helps us predict how the acidity of the solution will change when amounts of acid or conjugate base are altered, impacting the buffer's effectiveness.

pH and pKa Relationship

The pH is a logarithmic scale used to specify the acidity or basicity of an aqueous solution. It is related to the concentration of hydrogen ions in the solution by the formula:\[ \text{pH} = -\text{log}[H^{+}] \]Similarly, pKa is the negative logarithm of the acid dissociation constant (Ka):\[ \text{pKa} = -\text{log} K_{a} \]There is a pivotal relationship between pH and pKa in the context of buffer solutions. When the pH equals the pKa, the concentrations of the weak acid and its conjugate base are equal. This is the optimal point for a buffer because at this point, the buffer has equal capacities to neutralize added acids and bases, thus resisting changes in pH most effectively. An intuitive way to remember this is that when the pH is one unit below the pKa, the solution is mainly acidic (more HA present), and when it's one unit above, it's mainly basic (more A^- present).