Calculate the degree of hydrolysis of an N/10 KCN solution at `25^C. K_(a) (HCN)=7.2 xx 10^(-10), K_(w)=1 xx 10^(-14)`.

Answer

To calculate the degree of hydrolysis of an N/10 KCN solution at 25°C, we can follow these steps:

### Step 1: Identify the relevant equilibrium constants
Given:
- \( K_a \) (for HCN) = \( 7.2 \times 10^{-10} \)
- \( K_w \) = \( 1 \times 10^{-14} \)

### Step 2: Calculate the \( K_b \) for CN⁻
Since KCN is a salt of a weak acid (HCN) and a strong base (KOH), we can find the base dissociation constant \( K_b \) for CN⁻ using the relation:
\[ K_b = \frac{K_w}{K_a} \]
Substituting the values:
\[ K_b = \frac{1 \times 10^{-14}}{7.2 \times 10^{-10}} = \frac{1}{7.2} \times 10^{-4} \approx 1.39 \times 10^{-5} \]

### Step 3: Set up the hydrolysis equilibrium expression
For the hydrolysis of CN⁻:
\[ CN^- + H_2O \rightleftharpoons HCN + OH^- \]
Let \( x \) be the degree of hydrolysis. The concentration of CN⁻ at equilibrium will be:
\[ [C] = \frac{N}{10} - x \approx \frac{N}{10} \quad (\text{since } x \text{ is small}) \]
The concentration of HCN and OH⁻ at equilibrium will both be \( x \).

### Step 4: Write the expression for \( K_b \)
Using the equilibrium concentrations, we can write:
\[ K_b = \frac{[HCN][OH^-]}{[CN^-]} = \frac{x \cdot x}{\frac{N}{10}} = \frac{x^2}{\frac{N}{10}} \]
Substituting \( K_b \):
\[ 1.39 \times 10^{-5} = \frac{x^2}{\frac{N}{10}} \]

### Step 5: Solve for \( x \)
Rearranging gives:
\[ x^2 = 1.39 \times 10^{-5} \cdot \frac{N}{10} \]
Taking the square root:
\[ x = \sqrt{1.39 \times 10^{-5} \cdot \frac{N}{10}} = \sqrt{1.39 \times 10^{-6} \cdot N} \]

### Step 6: Calculate the degree of hydrolysis
Since we have \( N = 0.1 \) (for N/10 solution):
\[ x = \sqrt{1.39 \times 10^{-6} \cdot 0.1} = \sqrt{1.39 \times 10^{-7}} \approx 1.18 \times 10^{-4} \]

### Step 7: Convert to percentage
The degree of hydrolysis (in percentage) is given by:
\[ \text{Degree of hydrolysis} = \frac{x}{\frac{N}{10}} \times 100 = \frac{1.18 \times 10^{-4}}{0.1} \times 100 = 0.118\% \]

### Final Answer
The degree of hydrolysis of the N/10 KCN solution at 25°C is approximately **0.118%**.

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