If a salt of strong acid and weak base hydrolyses appreciably `(alpha=0.1),` which of the following formula is to be used to calculate degree of hydrolsis 'alpha'?

Options

A. `alpha=sqrt(K_(w))/(K_(a).a)`
B. `alpha=sqrt(K_(w))/(K_(b).a)`
C. `alpha=sqrt(K_(w))/(K_(a).K_(b)`
D. None of these

Answer

To solve the problem regarding the degree of hydrolysis (α) of a salt formed from a strong acid and a weak base, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Salt**:
- The salt formed from a strong acid and a weak base can be represented as NH4Cl (ammonium chloride), where HCl is the strong acid and NH4OH (ammonium hydroxide) is the weak base.

2. **Understand Hydrolysis**:
- When NH4Cl is dissolved in water, it dissociates into NH4⁺ (ammonium ions) and Cl⁻ (chloride ions). The ammonium ion (NH4⁺) will hydrolyze in water:
\[ NH4^+ + H2O \rightleftharpoons NH4OH + H^+ \]
- This reaction produces hydroxide ions and makes the solution acidic due to the formation of H⁺ ions.

3. **Set Up Initial Concentrations**:
- Let the initial concentration of NH4Cl be C. At equilibrium, if α is the degree of hydrolysis, the concentrations will be:
- [NH4⁺] = Cα
- [H⁺] = Cα
- [NH4OH] = C(1 - α)

4. **Write the Hydrolysis Constant Expression**:
- The hydrolysis constant (K_H) can be expressed as:
\[ K_H = \frac{[NH4OH][H^+]}{[NH4^+]} = \frac{Cα \cdot Cα}{C(1 - α)} = \frac{Cα^2}{1 - α} \]

5. **Assume α is Small**:
- Since α = 0.1 is small compared to 1, we can approximate (1 - α) as 1:
\[ K_H \approx Cα^2 \]

6. **Relate K_H to K_a**:
- In an acidic solution, K_H is related to the dissociation constant of the acid (K_a):
\[ K_H = K_a \]
- We can also express K_a in terms of K_w (the ion product of water) and K_b (the base dissociation constant of NH4OH):
\[ K_a = \frac{K_w}{K_b} \]

7. **Final Formula for Degree of Hydrolysis**:
- Rearranging the equation gives:
\[ α^2 = \frac{K_a}{C} \]
- Taking the square root to find α:
\[ α = \sqrt{\frac{K_a}{C}} \]
- Substituting K_a with K_w/K_b gives:
\[ α = \sqrt{\frac{K_w}{K_b \cdot C}} \]

### Final Answer:
The formula to calculate the degree of hydrolysis (α) for a salt of a strong acid and a weak base is:
\[ α = \sqrt{\frac{K_w}{K_b \cdot C}} \]