Question

Consider the dissociation equilibrium of the following weak acid: \[ \mathrm{HA \rightleftharpoons H^+(aq) + A^-(aq)} \] If the \(pK_a\) of the acid is \(4\), then the pH of a \(10\ \text{mM}\) HA solution is ________ (Nearest integer). (Given: The degree of dissociation can be neglected with respect to unity)

Answer 1

Correct Answer: 3

Step 1: Understanding the Concept:
For a weak monobasic acid, the dissociation constant (\(K_a\)) and initial concentration (\(C\)) determine the concentration of hydrogen ions.
When the degree of dissociation (\(\alpha\)) is very small (\(\alpha<5%\)), we can simplify the expression for \([H^+]\).
Step 2: Key Formula or Approach:
1. \(pK_a = -\log K_a \implies K_a = 10^{-pK_a}\).
2. If \(\alpha \ll 1\), then \([H^+] = \sqrt{K_a \cdot C}\).
3. \(pH = -\log [H^+]\).
Step 3: Detailed Explanation:
Given values:
\(pK_a = 4 \implies K_a = 10^{-4}\).
\(C = 10 \text{ mM} = 10 \times 10^{-3} \text{ M} = 10^{-2} \text{ M}\).
Using the approximate formula for hydrogen ion concentration:
\[ [H^+] = \sqrt{K_a \times C} \]
\[ [H^+] = \sqrt{10^{-4} \times 10^{-2}} \]
\[ [H^+] = \sqrt{10^{-6}} = 10^{-3} \text{ M} \]
Now, calculate the pH:
\[ pH = -\log [H^+] \]
\[ pH = -\log (10^{-3}) = 3 \]
Step 4: Final Answer:
The pH of the solution is 3.