Which of the following is correct for the solution of the salt of weak acid & weak base ?

Options

A. `pH=1/2[pK_w+pK_a-pK_b]`
B. `pH=1/2[pK_w-pK_a-pK_b]`
C. `pH=1/2[pK_w+pK_a+pK_b]`
D. `pH=1/2[pK_wxxpK_axxpK_b]`

Answer

To solve the question regarding the pH of the solution of a salt formed from a weak acid and a weak base, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Hydrolysis of Salt**:
- A salt formed from a weak acid (HA) and a weak base (BOH) will undergo hydrolysis in water. This means that when the salt dissolves, it will dissociate into its ions (A⁻ and B⁺) and these ions will react with water to form the weak acid and weak base.
- The hydrolysis reaction can be represented as:
\[ BA \rightleftharpoons B^+ + A^- \]
where \( B^+ \) is the cation from the weak base and \( A^- \) is the anion from the weak acid.

2. **Setting Up the Equilibrium Expression**:
- At equilibrium, the concentration of the ions will change, and we can express the equilibrium constant for the hydrolysis reaction (K_h) in terms of the concentrations of the products and reactants.
- The equilibrium expression is given by:
\[ K_h = \frac{[H^+][A^-]}{[BA]} \]

3. **Relating K_h to K_a and K_b**:
- The relationship between the hydrolysis constant \( K_h \), the dissociation constant of the weak acid \( K_a \), and the dissociation constant of the weak base \( K_b \) is given by:
\[ K_h = \frac{K_w}{K_a \cdot K_b} \]
- Here, \( K_w \) is the ion product of water.

4. **Finding the Concentration of H⁺**:
- The concentration of hydrogen ions \( [H^+] \) can be derived from the hydrolysis constant:
\[ [H^+] = \sqrt{K_h} = \sqrt{\frac{K_w}{K_a \cdot K_b}} \]

5. **Calculating pH**:
- The pH can be calculated from the concentration of hydrogen ions:
\[ pH = -\log[H^+] \]
- Substituting the expression for \( [H^+] \):
\[ pH = -\log\left(\sqrt{\frac{K_w}{K_a \cdot K_b}}\right) = \frac{1}{2}(-\log(K_w)) - \frac{1}{2}(-\log(K_a \cdot K_b)) \]
- This simplifies to:
\[ pH = \frac{1}{2}pK_w + \frac{1}{2}(pK_a - pK_b) \]

6. **Final Expression**:
- Therefore, the final expression for the pH of the solution of a salt formed from a weak acid and weak base is:
\[ pH = \frac{1}{2}pK_w + pK_a - pK_b \]

7. **Identifying the Correct Option**:
- Among the given options, the correct one is:
\[ pH = \frac{1}{2}pK_w + pK_a - pK_b \]
- This corresponds to the first option.

### Conclusion:
The correct answer is:
**pH = \frac{1}{2}pK_w + pK_a - pK_b**