Step 1: Understanding the Question:
We need to determine the mathematical expression for the hydroxide ion concentration, \( [\text{OH}^-] \), for a weak monoacidic base dissolving in water.
Step 2: Key Formula or Approach:
Consider a weak monoacidic base, \( \text{BOH} \), dissociating in water:
\[ \text{BOH} \rightleftharpoons \text{B}^+ + \text{OH}^- \]
Let \( c \) be the initial concentration and \( \alpha \) be the degree of dissociation.
According to Ostwald's dilution law for weak electrolytes, \( [\text{OH}^-] = c \alpha \) and \( \alpha = \sqrt{\frac{\text{K}_\text{b}}{c}} \).
Step 3: Detailed Explanation:
The dissociation constant, \( \text{K}_\text{b} \), is given by:
\[ \text{K}_\text{b} = c \alpha^2 \text{ (for weak bases where } \alpha \text{ is small)} \]
Solving for \( \alpha \):
\[ \alpha = \sqrt{\frac{\text{K}_\text{b}}{c}} \]
We know that the hydroxide ion concentration is \( [\text{OH}^-] = c \alpha \). Substituting \( \alpha \) into this expression:
\[ [\text{OH}^-] = c \times \sqrt{\frac{\text{K}_\text{b}}{c}} \]
\[ [\text{OH}^-] = \sqrt{c^2 \times \frac{\text{K}_\text{b}}{c}} \]
\[ [\text{OH}^-] = \sqrt{\text{K}_\text{b} \cdot c} \]
Step 4: Final Answer:
The correct formula is \( \sqrt{\text{K}_\text{b} \cdot \text{c}} \).