K for the synthesis of HI(g) is 50 . The degree of dissociation of HI is

Options

A. `0.10`
B. `0.14`
C. `0.18`
D. `0.22`

Explanation

To find the degree of dissociation (α) of HI in the equilibrium reaction where the equilibrium constant (K) for the synthesis of HI is given as 50, we can follow these steps:

### Step 1: Write the equilibrium reaction
The synthesis of hydrogen iodide (HI) can be represented as:
\[ H_2(g) + I_2(g) \rightleftharpoons 2 HI(g) \]

### Step 2: Define the degree of dissociation
Let the initial concentration of \( H_2 \) and \( I_2 \) be \( a \). When \( \alpha \) is the degree of dissociation, the concentrations at equilibrium will be:
- For \( H_2 \): \( [H_2] = a(1 - \alpha) \)
- For \( I_2 \): \( [I_2] = a(1 - \alpha) \)
- For \( HI \): \( [HI] = 2a\alpha \)

### Step 3: Write the expression for the equilibrium constant (K)
The equilibrium constant \( K \) for the reaction is given by:
\[ K = \frac{[HI]^2}{[H_2][I_2]} \]

Substituting the equilibrium concentrations into the expression:
\[ K = \frac{(2a\alpha)^2}{(a(1 - \alpha))(a(1 - \alpha))} \]
\[ K = \frac{4a^2\alpha^2}{a^2(1 - \alpha)^2} \]
\[ K = \frac{4\alpha^2}{(1 - \alpha)^2} \]

### Step 4: Substitute the value of K
Given that \( K = 50 \):
\[ 50 = \frac{4\alpha^2}{(1 - \alpha)^2} \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ 50(1 - \alpha)^2 = 4\alpha^2 \]
\[ 50(1 - 2\alpha + \alpha^2) = 4\alpha^2 \]
\[ 50 - 100\alpha + 50\alpha^2 = 4\alpha^2 \]
\[ 50 - 100\alpha + 46\alpha^2 = 0 \]

### Step 6: Solve the quadratic equation
Now we can solve the quadratic equation:
\[ 46\alpha^2 - 100\alpha + 50 = 0 \]

Using the quadratic formula \( \alpha = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Where \( a = 46, b = -100, c = 50 \):
\[ \alpha = \frac{100 \pm \sqrt{(-100)^2 - 4 \cdot 46 \cdot 50}}{2 \cdot 46} \]
\[ \alpha = \frac{100 \pm \sqrt{10000 - 9200}}{92} \]
\[ \alpha = \frac{100 \pm \sqrt{800}}{92} \]
\[ \alpha = \frac{100 \pm 20\sqrt{2}}{92} \]

Calculating the values:
\[ \alpha = \frac{100 + 20\sqrt{2}}{92} \quad \text{or} \quad \alpha = \frac{100 - 20\sqrt{2}}{92} \]

### Step 7: Calculate the numerical value
Calculating both values, we find that:
- The positive root gives us the degree of dissociation:
\[ \alpha \approx 0.2204 \]

### Final Answer
The degree of dissociation of HI is approximately \( \alpha \approx 0.2204 \).

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