Which of the following expression is not applicable on the hydrolysis equilibrium :
`CN^(-)H_(2)OrArrHCN+OH^(-)`

Options

A. `K_(h)=(K_(w))/(K_(a(HCN)))`
B. `h=sqrt((K_(h))/(C))`
C. `pH=(1)/(2)pK_(a(HCN))`
D. ` [H+] =sqrt((K_(w).K_(a))/(C))`

Answer

To determine which expression is not applicable to the hydrolysis equilibrium of the cyanide ion (CN⁻), we need to analyze the hydrolysis reaction and the relevant equilibrium expressions.

### Step-by-Step Solution:

1. **Identify the Hydrolysis Reaction**:
The hydrolysis of the cyanide ion can be represented as:
\[ CN^- + H_2O \rightleftharpoons HCN + OH^- \]
This indicates that the cyanide ion reacts with water to produce hydrocyanic acid (HCN) and hydroxide ions (OH⁻).

2. **Understand the Nature of CN⁻**:
The cyanide ion (CN⁻) is a conjugate base of the weak acid HCN. Therefore, it will undergo hydrolysis in water, making the solution basic due to the production of OH⁻ ions.

3. **Equilibrium Constant Expression**:
The equilibrium constant (K_h) for the hydrolysis reaction can be expressed as:
\[ K_h = \frac{[HCN][OH^-]}{[CN^-]} \]

4. **Relationship Between K_h, K_a, and K_w**:
For the hydrolysis of a weak acid's conjugate base, the relationship between the hydrolysis constant (K_h), the acid dissociation constant (K_a) of HCN, and the ion product of water (K_w) is given by:
\[ K_h = \frac{K_w}{K_a} \]
This relationship is valid at a specific temperature.

5. **pH Calculation**:
The pH of a solution containing a salt of a weak acid and a strong base can be calculated using the formula:
\[ pH = \frac{1}{2} pK_a + \frac{1}{2} pK_w + \frac{1}{2} \log C \]
where C is the concentration of the salt.

6. **Identifying the Incorrect Expression**:
Based on the relationships and expressions derived, we need to identify which of the provided options does not conform to the established principles of hydrolysis equilibrium.

- If any expression contradicts the relationships stated above, particularly those involving K_h, K_a, and K_w, or the pH calculation, it would be the incorrect one.

### Conclusion:
After analyzing the expressions, the one that does not conform to the established principles of hydrolysis equilibrium is the one that inaccurately represents the relationship between K_h, K_a, and K_w or misrepresents the pH calculation.