For the reaction `N_(2)(g)+O_(2)(g) hArr 2NO(g)`, the equilibrium constant is `K_(1)`. The equilibrium constant is `K_(2)` for the reaction
`2NO(g)+O_(2) hArr 2NO_(2)(g)`
What is `K` for the reaction
`NO_(2)(g) hArr 1/2 N_(2)(g)+O_(2)(g)`?

Options

A. `1//(4K_1K_2)`
B. `[1//K_1K_2]^(1//2)`
C. `1//(K_1K_2)`
D. `1//(2K_1K_2)`

Explanation

To find the equilibrium constant \( K \) for the reaction

\[ NO_2(g) \rightleftharpoons \frac{1}{2} N_2(g) + O_2(g) \]

we will use the equilibrium constants \( K_1 \) and \( K_2 \) from the given reactions:

1. \( N_2(g) + O_2(g) \rightleftharpoons 2NO(g) \) with equilibrium constant \( K_1 \)
2. \( 2NO(g) + O_2(g) \rightleftharpoons 2NO_2(g) \) with equilibrium constant \( K_2 \)

### Step 1: Write down the reactions and their equilibrium constants

- **Reaction 1:**
\[ N_2(g) + O_2(g) \rightleftharpoons 2NO(g) \quad (K_1) \]

- **Reaction 2:**
\[ 2NO(g) + O_2(g) \rightleftharpoons 2NO_2(g) \quad (K_2) \]

### Step 2: Combine the reactions

To derive the desired reaction, we will first add Reaction 1 and Reaction 2.

From Reaction 1, we have:
\[ N_2 + O_2 \rightarrow 2NO \]

From Reaction 2, we can rewrite it as:
\[ 2NO_2 \rightarrow 2NO + O_2 \]

Adding these two reactions gives:
\[ N_2 + O_2 + 2NO_2 \rightarrow 2NO + 2NO \]

This simplifies to:
\[ N_2 + 2O_2 \rightarrow 2NO_2 \]

### Step 3: Find the equilibrium constant for the combined reaction

The equilibrium constant for the combined reaction is given by:
\[ K' = K_1 \cdot K_2 \]

### Step 4: Reverse and modify the combined reaction

Now we need to reverse the reaction to get \( NO_2 \) on the reactant side:
\[ 2NO_2 \rightarrow N_2 + 2O_2 \]

The equilibrium constant for the reversed reaction is:
\[ K'' = \frac{1}{K'} \]

### Step 5: Halve the reaction

Now, we need to divide the entire equation by 2 to match the desired reaction:
\[ NO_2 \rightarrow \frac{1}{2}N_2 + O_2 \]

When we halve the reaction, the equilibrium constant becomes:
\[ K = \frac{1}{K'}^{1/2} = \frac{1}{(K_1 \cdot K_2)}^{1/2} = \frac{1}{\sqrt{K_1 \cdot K_2}} \]

### Conclusion

Thus, the equilibrium constant \( K \) for the reaction \( NO_2(g) \rightleftharpoons \frac{1}{2}N_2(g) + O_2(g) \) is given by:

\[ K = \frac{1}{\sqrt{K_1 \cdot K_2}} \]