Assertion `:-` Adding inert gas at equilibrium to dissociation
of `PCl_(5)` at constant pressure increases the degree of dissociation.
Reason `:-` Since degree of dissociation `x` `alphasqrt(v)`

Options

A. If both Assertion & Reason are True & the Reason is a correct explanation of the Assertion.
B. If both Assertion & Reason are True but Reason is not a correct explanation of the Assertiion.
C. If Assertion is True but the Reason is False.
D. If both Assertion & Reason are False

Answer

To solve the assertion and reason-based question regarding the dissociation of \( PCl_5 \) and the effect of adding an inert gas at equilibrium, we can break down the solution into clear steps:

### Step 1: Understand the Reaction
The dissociation of \( PCl_5 \) can be represented as:
\[ PCl_5 \rightleftharpoons PCl_3 + Cl_2 \]
Let the initial number of moles of \( PCl_5 \) be \( A \). Upon dissociation, if \( x \) moles of \( PCl_5 \) dissociate, the moles at equilibrium will be:
- Moles of \( PCl_5 \) = \( A - x \)
- Moles of \( PCl_3 \) = \( x \)
- Moles of \( Cl_2 \) = \( x \)

### Step 2: Write the Expression for Equilibrium Constant \( K_c \)
The equilibrium constant \( K_c \) for the reaction can be expressed in terms of concentrations:
\[ K_c = \frac{[PCl_3][Cl_2]}{[PCl_5]} = \frac{\left(\frac{x}{V}\right)\left(\frac{x}{V}\right)}{\frac{A-x}{V}} = \frac{x^2}{V(A-x)} \]
Assuming \( x \) is very small compared to \( A \), we can approximate \( A - x \approx A \):
\[ K_c \approx \frac{x^2}{VA} \]

### Step 3: Relate Degree of Dissociation to Volume
From the expression derived, we can rearrange it to find the degree of dissociation \( x \):
\[ x^2 = K_c \cdot VA \implies x \propto \sqrt{V} \]
This indicates that the degree of dissociation \( x \) is directly proportional to the square root of the volume \( V \).

### Step 4: Effect of Adding Inert Gas
When an inert gas is added to the system at constant pressure, the volume of the system must increase. According to the relationship derived, as the volume \( V \) increases, the degree of dissociation \( x \) also increases.

### Step 5: Conclusion
Both the assertion and reason provided in the question are true:
- **Assertion**: Adding inert gas at equilibrium to the dissociation of \( PCl_5 \) at constant pressure increases the degree of dissociation.
- **Reason**: The degree of dissociation \( x \) is directly proportional to the square root of \( V \).

### Final Answer
Both the assertion and reason are true, and the reason is the correct explanation for the assertion.

---