(i) `N_(2)(g)+O_(2)(g)hArr2NO(g), K_(1)`
(ii) `((1)/(2))N_(2)(g)+((1)/(2))O_(2)(g)hArrNO(g), K_(2)`
(iii) `2NO(g)hArrN_(2)(g)+O_(2)(g), K_(3)`
(iv) `No(g)hArr((1)/(2))N_(2)(g)+((1)/(2))O_(2)(g), K_(4)`
Correct relation between `K_(1),K_(2),K_(3) "and" K_(4)` is//are:

Options

A. `K_(1)xxK_(3)=1`
B. `sqrtK_(1)xxK_(4)=1`
C. `sqrtK_(3)xxK_(2)=1`
D. None

Answer

To find the correct relationships between the equilibrium constants \( K_1, K_2, K_3, \) and \( K_4 \) for the given reactions, we will analyze each reaction and derive the expressions for the equilibrium constants step by step.

### Step 1: Write the Equilibrium Expressions

1. **For Reaction (i)**:
\[ N_2(g) + O_2(g) \rightleftharpoons 2NO(g), \quad K_1 = \frac{[NO]^2}{[N_2][O_2]} \]

2. **For Reaction (ii)**:
\[ \frac{1}{2}N_2(g) + \frac{1}{2}O_2(g) \rightleftharpoons NO(g), \quad K_2 = \frac{[NO]}{[N_2]^{1/2}[O_2]^{1/2}} \]

3. **For Reaction (iii)**:
\[ 2NO(g) \rightleftharpoons N_2(g) + O_2(g), \quad K_3 = \frac{[N_2][O_2]}{[NO]^2} \]

4. **For Reaction (iv)**:
\[ NO(g) \rightleftharpoons \frac{1}{2}N_2(g) + \frac{1}{2}O_2(g), \quad K_4 = \frac{[N_2]^{1/2}[O_2]^{1/2}}{[NO]} \]

### Step 2: Analyze the Relationships

1. **Relationship between \( K_1 \) and \( K_3 \)**:
- From the definitions:
\[ K_1 \cdot K_3 = \left(\frac{[NO]^2}{[N_2][O_2]}\right) \cdot \left(\frac{[N_2][O_2]}{[NO]^2}\right) = 1 \]
Thus, \( K_1 \cdot K_3 = 1 \).

2. **Relationship between \( K_1 \) and \( K_4 \)**:
- From the definitions:
\[ K_1 \cdot K_4 = \left(\frac{[NO]^2}{[N_2][O_2]}\right) \cdot \left(\frac{[N_2]^{1/2}[O_2]^{1/2}}{[NO]}\right) = \frac{[NO]}{[N_2]^{1/2}[O_2]^{1/2}} = 1 \]
Thus, \( \sqrt{K_1} \cdot K_4 = 1 \).

3. **Relationship between \( K_2 \) and \( K_3 \)**:
- From the definitions:
\[ K_2 \cdot K_3 = \left(\frac{[NO]}{[N_2]^{1/2}[O_2]^{1/2}}\right) \cdot \left(\frac{[N_2][O_2]}{[NO]^2}\right) = 1 \]
Thus, \( \sqrt{K_3} \cdot K_2 = 1 \).

### Conclusion

From the above analysis, we have derived the following relationships:
- \( K_1 \cdot K_3 = 1 \)
- \( \sqrt{K_1} \cdot K_4 = 1 \)
- \( \sqrt{K_3} \cdot K_2 = 1 \)

Therefore, all three options provided in the question are correct.