Assertion (A) : For `N_(2)(g)+3H_(2)(g) hArr 2NH_(3)(g)`, the equilibrium constant is `K`. The for `1/2 N_(2)(g)+3/2H_(2)(g) hArr NH_(3)(g)`, the equilibrium constant will be `sqrt(K)`.
Reason (R) : If concentrations are changed to half, the equilibrium constants will be halved.

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.

Answer

To solve the problem, we need to analyze the two statements provided: the Assertion (A) and the Reason (R).

### Step-by-Step Solution:

1. **Understanding the Equilibrium Constant (K)**:
The equilibrium constant \( K \) for a reaction is defined based on the concentrations of the products and reactants at equilibrium. For the reaction:
\[ N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g) \]
The equilibrium constant \( K \) can be expressed as:
\[ K = \frac{[NH_3]^2}{[N_2][H_2]^3} \]

2. **Analyzing the First Reaction**:
The first reaction is:
\[ N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g) \]
The equilibrium constant for this reaction is \( K \).

3. **Analyzing the Second Reaction**:
The second reaction given is:
\[ \frac{1}{2}N_2(g) + \frac{3}{2}H_2(g) \rightleftharpoons NH_3(g) \]
To relate this to the first reaction, we can see that this reaction is essentially the first reaction multiplied by \( \frac{1}{2} \).

4. **Calculating the New Equilibrium Constant**:
When a reaction is multiplied by a factor \( n \), the new equilibrium constant \( K' \) is given by:
\[ K' = K^n \]
Here, since we multiplied the reaction by \( \frac{1}{2} \), we have:
\[ K' = K^{\frac{1}{2}} = \sqrt{K} \]
Thus, the assertion (A) is correct.

5. **Analyzing the Reason (R)**:
The reason states that if the concentrations are halved, the equilibrium constants will also be halved. This statement is incorrect. The equilibrium constant \( K \) is not dependent on the concentrations of the reactants and products at equilibrium; it remains constant at a given temperature regardless of the concentrations of the reactants or products.

6. **Conclusion**:
- Assertion (A) is correct: The equilibrium constant for the second reaction is \( \sqrt{K} \).
- Reason (R) is incorrect: Halving the concentrations does not halve the equilibrium constant.

### Final Answer:
- Assertion (A) is correct, and Reason (R) is incorrect.