If a salt of a strong acid and a weak base hydrolyses appreciably, which of the following formulae is to be used to calculate degree of hydrolysis 'X?

Options

A. `X=sqrt((K_(w))/(K_(a).a))`
B. `X=sqrt((K_(w))/(K_(b).a))`
C. `X=sqrt((K_(w))/(K_(a).K_b))`
D. none of these

Answer

To calculate the degree of hydrolysis (X) for a salt of a strong acid and a weak base that hydrolyzes appreciably, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Components**:
- A salt formed from a strong acid (e.g., HCl) and a weak base (e.g., NH₄OH).
- The salt will dissociate in water to produce the corresponding ions.

2. **Write the Hydrolysis Reaction**:
- The hydrolysis of the salt can be represented as:
\[ \text{NH}_4^+ + \text{H}_2\text{O} \rightleftharpoons \text{NH}_4\text{OH} + \text{H}^+ \]
- Here, NH₄⁺ is the cation from the weak base.

3. **Set Up the Initial Concentrations**:
- Let the initial concentration of the salt be C (mol/L).
- At equilibrium, if α is the degree of hydrolysis, the concentrations will be:
- \([NH_4^+] = C(1 - \alpha)\)
- \([H^+] = C\alpha\)
- \([NH_4OH] = C\alpha\)

4. **Write the Expression for Hydrolysis Constant (K_H)**:
- The hydrolysis constant can be expressed as:
\[ K_H = \frac{[H^+][NH_4OH]}{[NH_4^+]} = \frac{C\alpha \cdot C\alpha}{C(1 - \alpha)} = \frac{C\alpha^2}{1 - \alpha} \]

5. **Assume Small Degree of Hydrolysis**:
- If α is small, we can approximate \(1 - \alpha \approx 1\).
- Thus, the equation simplifies to:
\[ K_H \approx C\alpha^2 \]

6. **Solve for Degree of Hydrolysis (α)**:
- Rearranging gives:
\[ \alpha = \sqrt{\frac{K_H}{C}} \]

7. **Relate K_H to K_A and K_B**:
- In acidic medium, the relationship between the hydrolysis constant (K_H), the dissociation constant of the weak base (K_B), and the dissociation constant of water (K_W) is given by:
\[ K_H = \frac{K_W}{K_B} \cdot C \]
- Substituting this into the equation for α gives:
\[ \alpha = \sqrt{\frac{K_W}{K_B} \cdot C} \]

8. **Conclusion**:
- The formula to calculate the degree of hydrolysis (X) for a salt of a strong acid and a weak base is:
\[ \alpha = \sqrt{\frac{K_W}{K_B} \cdot C} \]