Assertion (A) : For `PCl_(5)(g) hArr PCl_(3)(g)+Cl_(2)(g)`, if more `Cl_(2)` is added the equilibrium will shift in backward direction. Hence, equilibrium constant will decrease.
Reason (R) : Addition of inert gas to the equilibrium mixture at constant volume does not alter the equilibrium.

Options

A. If both (A) and (R) are correct, and (R) is the correct explanation for (A)
B. If both (A) and (R) are correct, but (R) is not the correct explanation of (A)
C. If (A) is correct, but (R) is incorrect
D. If (A) is incorrect, but (R) is correct.

Explanation

To solve the given question, we need to analyze both the assertion (A) and the reason (R) provided.

### Step-by-Step Solution:

1. **Understanding the Reaction**:
The reaction given is:
\[ PCl_5(g) \rightleftharpoons PCl_3(g) + Cl_2(g) \]
This is a dynamic equilibrium where \(PCl_5\) decomposes into \(PCl_3\) and \(Cl_2\).

2. **Analyzing the Assertion (A)**:
The assertion states that if more \(Cl_2\) is added to the system, the equilibrium will shift in the backward direction, and hence the equilibrium constant will decrease.
- According to Le Chatelier's principle, if we add more of a product (\(Cl_2\)), the equilibrium will shift to the left (backward direction) to counteract the change and reduce the concentration of \(Cl_2\).
- When the equilibrium shifts to the left, the concentration of \(PCl_5\) increases, while the concentration of \(PCl_3\) and \(Cl_2\) decreases.

3. **Equilibrium Constant (K)**:
The equilibrium constant \(K_c\) for the reaction is given by:
\[ K_c = \frac{[PCl_3][Cl_2]}{[PCl_5]} \]
- When the equilibrium shifts to the left, the concentration of \(PCl_5\) increases (denominator increases), and the concentrations of \(PCl_3\) and \(Cl_2\) decrease (numerator decreases).
- Therefore, the overall value of \(K_c\) decreases.

4. **Conclusion for Assertion (A)**:
The assertion is **true** because adding \(Cl_2\) causes the equilibrium to shift left, leading to a decrease in the equilibrium constant.

5. **Analyzing the Reason (R)**:
The reason states that the addition of an inert gas to the equilibrium mixture at constant volume does not alter the equilibrium.
- This statement is also **true**. Adding an inert gas at constant volume does not change the partial pressures of the reactants and products, hence does not affect the equilibrium position.

6. **Relationship Between Assertion and Reason**:
Although both the assertion and the reason are true, the reason does not explain the assertion. The assertion is about the effect of adding \(Cl_2\), while the reason discusses the effect of adding an inert gas, which is unrelated to the assertion.

7. **Final Answer**:
The correct conclusion is:
- Both A and R are correct, but R is not the correct explanation of A.