For the following gaseous phase eqilibrium `PCl_(5(s)) hArr PCl_(3(g))+Cl_(2(g)) ` Kp is found to be equal to `K_(x)` `K_(x)` is equilibrium constant when concentratiion are taken in interms of mole fraction). This attained when

Options

A. 1 atm
B. 0.5 atm
C. 2 atm
D. 4 atm

Answer

To solve the problem, we need to analyze the equilibrium reaction and the relationship between the equilibrium constants \( K_p \) and \( K_x \).

### Step-by-Step Solution:

1. **Write the Equilibrium Reaction**:
The given reaction is:
\[ PCl_5 (s) \rightleftharpoons PCl_3 (g) + Cl_2 (g) \]

2. **Define the Equilibrium Constants**:
The equilibrium constant \( K_p \) is defined in terms of the partial pressures of the gases involved:
\[ K_p = \frac{P_{PCl_3} \cdot P_{Cl_2}}{P_{PCl_5}} \]
Since \( PCl_5 \) is a solid, its activity is 1, so we can simplify this to:
\[ K_p = P_{PCl_3} \cdot P_{Cl_2} \]

3. **Relate \( K_p \) and \( K_x \)**:
The equilibrium constant \( K_x \) is defined in terms of mole fractions:
\[ K_x = \frac{X_{PCl_3} \cdot X_{Cl_2}}{X_{PCl_5}} \]
where \( X \) represents the mole fractions of the respective species.

4. **Express Partial Pressures in Terms of Mole Fractions**:
The partial pressures can be expressed in terms of mole fractions and total pressure \( P_t \):
\[ P_{PCl_3} = X_{PCl_3} \cdot P_t \]
\[ P_{Cl_2} = X_{Cl_2} \cdot P_t \]

5. **Substitute into the \( K_p \) Expression**:
Substituting these expressions into the equation for \( K_p \):
\[ K_p = (X_{PCl_3} \cdot P_t) \cdot (X_{Cl_2} \cdot P_t) = X_{PCl_3} \cdot X_{Cl_2} \cdot P_t^2 \]

6. **Set \( K_p \) Equal to \( K_x \)**:
From the problem, we know that \( K_p = K_x \). Therefore:
\[ X_{PCl_3} \cdot X_{Cl_2} \cdot P_t^2 = K_x \]

7. **Determine Conditions for \( K_p = K_x \)**:
For \( K_p \) to equal \( K_x \), the total pressure \( P_t \) must be equal to 1 atm. This is because:
\[ K_p = K_x \cdot P_t^2 \]
If \( P_t = 1 \), then \( K_p = K_x \).

### Conclusion:
The equilibrium constant \( K_p \) is equal to \( K_x \) when the total pressure \( P_t \) is 1 atm.

### Final Answer:
The correct answer is when the total pressure is 1 atm.