The correct relation between hydrolysis constant `(K_(b))` and degree of hydrolysis `(alpha)` for the following equilibrium is

Options

A. `alpha = sqrt((K_(w).C)/(K_(a)))`
B. `alpha = sqrt((K_(w))/(K_(a).C))`
C. `alpha = sqrt((K_(a).C)/(K_(w))`
D. `alpha = sqrt((K_(a))/(K_(w).C))`

Explanation

To determine the correct relationship between the hydrolysis constant \( K_b \) and the degree of hydrolysis \( \alpha \), we can follow these steps:

### Step 1: Understand Hydrolysis
Hydrolysis is a reaction involving the reaction of a salt with water, resulting in the formation of an acid and a base. For example, when ammonium chloride (\( NH_4Cl \)) is dissolved in water, it undergoes hydrolysis to produce ammonium ions (\( NH_4^+ \)) and chloride ions (\( Cl^- \)). The chloride ion does not undergo hydrolysis because it comes from a strong acid (HCl).

### Step 2: Write the Hydrolysis Reaction
For the hydrolysis of \( NH_4Cl \):
\[ NH_4^+ + H_2O \rightleftharpoons NH_3 + H_3O^+ \]

### Step 3: Define the Hydrolysis Constant
The hydrolysis constant \( K_h \) can be expressed in terms of the concentrations of the products and reactants at equilibrium:
\[ K_h = \frac{[NH_3][H_3O^+]}{[NH_4^+]} \]

### Step 4: Relate Hydrolysis Constant to \( K_b \)
The relationship between the hydrolysis constant \( K_h \) and the base dissociation constant \( K_b \) for the weak base \( NH_3 \) is given by:
\[ K_h = \frac{K_w}{K_b} \]
where \( K_w \) is the ion product of water.

### Step 5: Express Degree of Hydrolysis \( \alpha \)
The degree of hydrolysis \( \alpha \) is defined as the fraction of the salt that has undergone hydrolysis. For a salt of a weak base, the degree of hydrolysis can be expressed as:
\[ \alpha = \frac{K_w}{K_b \cdot C} \]
where \( C \) is the concentration of the salt.

### Step 6: Final Relationship
From the above equations, we can rearrange to find the relationship between \( K_b \) and \( \alpha \):
\[ K_b = \frac{K_w}{\alpha \cdot C} \]

### Conclusion
Thus, the correct relationship between the hydrolysis constant \( K_b \) and the degree of hydrolysis \( \alpha \) is:
\[ K_b \propto \frac{K_w}{\alpha \cdot C} \]