Equilivalent amounts of `H_(2)` and `I_(2)` are heated in a closed vessel till equilibrium is obtained. If `80%` of the hydrogen is converted to `HI`, the `K_(c)` at this temperature is

Options

A. 64
B. 16
C. `0.25`
D. 14

Answer

To solve the problem, we need to determine the equilibrium constant \( K_c \) for the reaction between hydrogen \( H_2 \) and iodine \( I_2 \) to form hydrogen iodide \( HI \). The reaction can be represented as:

\[ H_2(g) + I_2(g) \rightleftharpoons 2 HI(g) \]

### Step-by-Step Solution:

1. **Initial Setup**:
Assume we start with 1 mole of \( H_2 \) and 1 mole of \( I_2 \) in a closed vessel.

2. **Determine the Amount Reacted**:
It is given that 80% of the hydrogen is converted to \( HI \). Since we started with 1 mole of \( H_2 \):
\[ \text{Amount of } H_2 \text{ reacted} = 0.8 \text{ moles} \]

3. **Calculate Remaining \( H_2 \)**:
The amount of \( H_2 \) left after the reaction is:
\[ \text{Remaining } H_2 = 1 - 0.8 = 0.2 \text{ moles} \]

4. **Calculate Remaining \( I_2 \)**:
Since 1 mole of \( H_2 \) reacts with 1 mole of \( I_2 \), the amount of \( I_2 \) that reacts is also 0.8 moles. Thus, the remaining \( I_2 \) is:
\[ \text{Remaining } I_2 = 1 - 0.8 = 0.2 \text{ moles} \]

5. **Calculate Amount of \( HI \) Formed**:
For every mole of \( H_2 \) and \( I_2 \) that reacts, 2 moles of \( HI \) are produced. Therefore, the amount of \( HI \) formed from 0.8 moles of \( H_2 \) is:
\[ \text{Amount of } HI = 2 \times 0.8 = 1.6 \text{ moles} \]

6. **Write the Expression for \( K_c \)**:
The equilibrium constant \( K_c \) is given by the formula:
\[ K_c = \frac{[HI]^2}{[H_2][I_2]} \]

7. **Substituting the Concentrations**:
At equilibrium, we have:
- Concentration of \( HI = 1.6 \) moles
- Concentration of \( H_2 = 0.2 \) moles
- Concentration of \( I_2 = 0.2 \) moles

Substituting these values into the \( K_c \) expression:
\[ K_c = \frac{(1.6)^2}{(0.2)(0.2)} \]

8. **Calculating \( K_c \)**:
\[ K_c = \frac{2.56}{0.04} = 64 \]

Thus, the equilibrium constant \( K_c \) at this temperature is **64**.

### Final Answer:
The correct option is **A - 64**.