Relation between hydrolysis constant and dissociation constant are given. Which is the correct formula for `MgCl_(2)` ?

Options

A. `K_(h)=K_(w)/K_(a)`
B. `K_(h)=K_(w)^(2)`
C. `K_(h)=K_(w)/(K_(a)xxK_(b))`
D. `K_(w)xxK_(h)/K_(b)`

Explanation

To find the correct formula relating the hydrolysis constant (K_h) and the dissociation constant (K_a or K_b) for \( MgCl_2 \), we need to analyze the hydrolysis of the ions produced when \( MgCl_2 \) dissolves in water.

### Step-by-Step Solution:

1. **Identify the Dissociation of \( MgCl_2 \)**:
- When \( MgCl_2 \) dissolves in water, it dissociates into \( Mg^{2+} \) ions and \( Cl^- \) ions:
\[ MgCl_2 \rightarrow Mg^{2+} + 2Cl^- \]

2. **Determine the Hydrolysis Reaction**:
- The \( Cl^- \) ion comes from a strong acid (HCl) and does not undergo hydrolysis. Therefore, we focus on the hydrolysis of the \( Mg^{2+} \) ion:
\[ Mg^{2+} + 2H_2O \rightleftharpoons Mg(OH)_2 + 2H^+ \]

3. **Write the Hydrolysis Constant Expression**:
- The hydrolysis constant \( K_h \) for the above reaction can be expressed as:
\[ K_h = \frac{[Mg(OH)_2][H^+]^2}{[Mg^{2+}]} \]

4. **Consider the Dissociation of \( Mg(OH)_2 \)**:
- The dissociation of \( Mg(OH)_2 \) can be written as:
\[ Mg(OH)_2 \rightleftharpoons Mg^{2+} + 2OH^- \]
- The dissociation constant \( K_b \) for this reaction is:
\[ K_b = \frac{[Mg^{2+}][OH^-]^2}{[Mg(OH)_2]} \]

5. **Relate \( K_h \) and \( K_b \)**:
- From the above expressions, we can relate \( K_h \) and \( K_b \) using the ion product of water \( K_w \):
\[ K_h \cdot K_b = K_w \]
- Rearranging gives:
\[ K_h = \frac{K_w}{K_b} \]

6. **Final Expression**:
- Therefore, the relationship between the hydrolysis constant and the dissociation constant for \( MgCl_2 \) is:
\[ K_h = \frac{K_w}{K_b} \]