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 solve the problem regarding the relationship between the hydrolysis constant (K_H) and the dissociation constant (K_B) for magnesium chloride (MgCl₂), we can follow these steps:

### Step 1: Write the Dissociation Reaction
Magnesium chloride (MgCl₂) dissociates in water as follows:
\[ \text{MgCl}_2 \rightarrow \text{Mg}^{2+} + 2\text{Cl}^- \]

### Step 2: Write the Hydrolysis Reaction
The hydrolysis of the magnesium ion (Mg²⁺) in water can be represented as:
\[ \text{Mg}^{2+} + 2\text{H}_2\text{O} \rightleftharpoons \text{Mg(OH)}_2 + 2\text{H}^+ \]

### Step 3: Establish the Hydrolysis Constant (K_H)
The hydrolysis constant (K_H) can be expressed in terms of the concentrations of the products and reactants:
\[ K_H = \frac{[\text{Mg(OH)}_2][\text{H}^+]^2}{[\text{Mg}^{2+}]} \]

### Step 4: Write the Dissociation Constant for the Weak Base
For magnesium hydroxide (Mg(OH)₂), which is a weak base, the dissociation can be represented as:
\[ \text{Mg(OH)}_2 \rightleftharpoons \text{Mg}^{2+} + 2\text{OH}^- \]
The dissociation constant (K_B) for this reaction is given by:
\[ K_B = \frac{[\text{Mg}^{2+}][\text{OH}^-]^2}{[\text{Mg(OH)}_2]} \]

### Step 5: Relate K_H and K_B
To relate K_H and K_B, we can use the ion product of water (K_W):
\[ K_W = [\text{H}^+][\text{OH}^-] \]
From the hydrolysis and dissociation equations, we can derive:
\[ K_H = \frac{K_W}{K_B} \]

### Conclusion
Thus, the correct relationship for magnesium chloride (MgCl₂) is:
\[ K_H = \frac{K_W}{K_B} \]

### Final Answer
The correct formula for MgCl₂ is:
\[ K_H = \frac{K_W}{K_B} \]

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