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

To determine the pH of a \(10\ \text{mM}\) solution of the weak acid HA, given its \(pK_a = 4\), follow these steps:

1. Write the dissociation equilibrium for the acid: \(\mathrm{HA \rightleftharpoons H^+(aq) + A^-(aq)}\). 
2. Use the expression for the acid dissociation constant \(K_a\): \[K_a = [\mathrm{H^+}][\mathrm{A^-}]/[\mathrm{HA}]\]
3. Given \(pK_a = 4\), calculate \(K_a\):
   • \(K_a = 10^{-pK_a} = 10^{-4}\)
4. Assume the initial concentration of HA is \([HA]_0 = 10\ \text{mM} = 0.01\ \text{M}\).
5. Calculate the concentration of \([\mathrm{H^+}]\) using the assumption that degree of dissociation is negligible:
   • \(K_a = [\mathrm{H^+}]^2/[\mathrm{HA}]_0\)
   • Rearrange to find \([\mathrm{H^+}] = \sqrt{K_a \times [\mathrm{HA}]_0}\)
   • \([\mathrm{H^+}] = \sqrt{10^{-4} \times 0.01} = \sqrt{10^{-6}} = 10^{-3}\ \text{M}\)
6. Determine the pH of the solution:
   • \(\text{pH} = -\log_{10}([\mathrm{H^+}]) = -\log_{10}(10^{-3}) = 3\)
7. Confirm the calculated pH falls within the specified range (3, 3), verifying it as correct.

The pH of the solution is 3.