Which is the constant equation for hydrolysis constant of `NH_(4)CN`?

Options

A. `K_(a) xx (K_(b))/((K_(w))`
B. `sqrt((K_(w))/(K_(a)))`
C. `(K_(b))/(K_(a) xx K_(w))`
D. `(K_(w))/(K_(a) xx K_(b))`

Answer

To derive the hydrolysis constant equation for ammonium cyanide (NH₄CN), we need to follow a systematic approach. Here’s how we can do it step by step:

### Step-by-Step Solution:

1. **Identify the Components**:
- Ammonium cyanide (NH₄CN) is a salt formed from a weak acid (hydrogen cyanide, HCN) and a weak base (ammonium hydroxide, NH₄OH).

2. **Understand Hydrolysis**:
- When NH₄CN dissolves in water, it undergoes hydrolysis, producing NH₄OH and HCN:
\[ NH₄^+ + CN^- + H_2O \rightleftharpoons NH₄OH + HCN \]

3. **Write the Hydrolysis Constant Expression**:
- The hydrolysis constant (K_h) can be expressed as:
\[ K_h = \frac{[NH₄OH][HCN]}{[NH₄^+][CN^-]} \]
- Here, the concentrations of the ions and the products are considered.

4. **Identify the Ionization Constants**:
- For the weak base (NH₄OH), the ionization constant (K_b) is:
\[ K_b = \frac{[NH₄^+][OH^-]}{[NH₄OH]} \]
- For the weak acid (HCN), the ionization constant (K_a) is:
\[ K_a = \frac{[H^+][CN^-]}{[HCN]} \]

5. **Use the Ionic Product of Water**:
- The ionic product of water (K_w) is defined as:
\[ K_w = [H^+][OH^-] \]

6. **Combine the Equations**:
- By multiplying the expressions for K_h, K_a, and K_b, we can relate them:
\[ K_h = \frac{K_w}{K_a \cdot K_b} \]

7. **Final Expression**:
- Thus, the hydrolysis constant for NH₄CN can be expressed as:
\[ K_h = \frac{K_w}{K_a \cdot K_b} \]

### Final Answer:
The constant equation for the hydrolysis constant of NH₄CN is:
\[ K_h = \frac{K_w}{K_a \cdot K_b} \]