`N_2O_2 hArr NO,K_1,1/2 N_2 +1/2O_2hArr NO,K_2 2NO hArr N_2+O_2k_3 , NO hArr 1/2 N_2 +1/2O_2 K_4`

Options

A. `K_1 xx K_3 =1`
B. `sqrt(K_1)xxK_4=1`
C. `sqrt(K_3)xxK_2=1`
D. none of the above

Answer

To solve the problem involving the equilibrium constants for the given reactions, we will analyze each reaction step by step and derive the expressions for the equilibrium constants.

### Step 1: Write the Equilibrium Constants for Each Reaction

1. **For the reaction**:
\[ N_2O_2 \rightleftharpoons 2NO \quad (K_1) \]
The equilibrium constant \( K_1 \) is given by:
\[ K_1 = \frac{[NO]^2}{[N_2O_2]} \]

2. **For the reaction**:
\[ \frac{1}{2} N_2 + \frac{1}{2} O_2 \rightleftharpoons NO \quad (K_2) \]
The equilibrium constant \( K_2 \) is given by:
\[ K_2 = \frac{[NO]}{[N_2]^{1/2} [O_2]^{1/2}} = \frac{[NO]}{\sqrt{[N_2][O_2]}} \]

3. **For the reaction**:
\[ 2NO \rightleftharpoons N_2 + O_2 \quad (K_3) \]
The equilibrium constant \( K_3 \) is given by:
\[ K_3 = \frac{[N_2][O_2]}{[NO]^2} \]

4. **For the reaction**:
\[ NO \rightleftharpoons \frac{1}{2} N_2 + \frac{1}{2} O_2 \quad (K_4) \]
The equilibrium constant \( K_4 \) is given by:
\[ K_4 = \frac{[N_2]^{1/2}[O_2]^{1/2}}{[NO]} = \frac{\sqrt{[N_2][O_2]}}{[NO]} \]

### Step 2: Analyze the Options Given

Now, we will analyze the options provided in the question:

1. **Option A**: \( K_1 \times K_4 \)
\[ K_1 \times K_4 = \left(\frac{[NO]^2}{[N_2O_2]}\right) \times \left(\frac{\sqrt{[N_2][O_2]}}{[NO]}\right) \]
Simplifying this gives:
\[ = \frac{[NO]}{[N_2O_2]} \times \sqrt{[N_2][O_2]} \]
This does not yield a valid expression related to the other constants.

2. **Option B**: \( \sqrt{K_1 \times K_4} \)
\[ \sqrt{K_1 \times K_4} = \sqrt{\left(\frac{[NO]^2}{[N_2O_2]}\right) \times \left(\frac{\sqrt{[N_2][O_2]}}{[NO]}\right)} \]
This simplifies to:
\[ = \sqrt{\frac{[NO] \cdot \sqrt{[N_2][O_2]}}{[N_2O_2]}} \]
This also does not yield a valid expression.

3. **Option C**: \( \sqrt{K_3 \times K_2} \)
\[ \sqrt{K_3 \times K_2} = \sqrt{\left(\frac{[N_2][O_2]}{[NO]^2}\right) \times \left(\frac{[NO]}{\sqrt{[N_2][O_2]}}\right)} \]
This simplifies to:
\[ = \sqrt{\frac{[N_2][O_2]}{[NO]}} = 1 \]
This is a valid expression.

### Conclusion

Thus, the correct answer is **Option C**: \( \sqrt{K_3 \times K_2} = 1 \).