Consider the reactions `1/2N_2+O_2hArrNO_2``K_1`
`2NO_2hArrN_2O_4``K_2`
Using above equations , write down expression for K of the following reaction `N_2O_4hArrN_2+2O_2 K`

Options

A. `K_1K_2`
B. `(K_2^2)/K_1`
C. `1/(K_1K_2^(2))`
D. `1/(K_1^2K_2)`

Answer

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

\[ N_2O_4 \rightleftharpoons N_2 + 2O_2 \]

we will use the given reactions and their equilibrium constants \( K_1 \) and \( K_2 \).

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

1. The first reaction is:
\[ \frac{1}{2} N_2 + O_2 \rightleftharpoons NO_2 \quad (K_1) \]

2. The second reaction is:
\[ 2 NO_2 \rightleftharpoons N_2O_4 \quad (K_2) \]

### Step 2: Reverse the second reaction

To express \( N_2O_4 \) in terms of \( N_2 \) and \( O_2 \), we need to reverse the second reaction:

\[ N_2O_4 \rightleftharpoons 2 NO_2 \quad \left( \frac{1}{K_2} \right) \]

### Step 3: Use the first reaction

Now, we need to express \( NO_2 \) in terms of \( N_2 \) and \( O_2 \) from the first reaction. We can also reverse the first reaction:

\[ NO_2 \rightleftharpoons \frac{1}{2} N_2 + O_2 \quad \left( \frac{1}{K_1} \right) \]

### Step 4: Combine the reactions

Now, we can combine the reversed reactions:

1. From the reversed second reaction:
\[ N_2O_4 \rightleftharpoons 2 NO_2 \quad \left( \frac{1}{K_2} \right) \]

2. From the reversed first reaction (multiplied by 2):
\[ 2 NO_2 \rightleftharpoons N_2 + 2 O_2 \quad \left( \frac{1}{K_1^2} \right) \]

### Step 5: Add the reactions

When we add these two reactions, we get:

\[ N_2O_4 \rightleftharpoons N_2 + 2 O_2 \]

### Step 6: Calculate the overall equilibrium constant

The equilibrium constant for the overall reaction is the product of the equilibrium constants of the individual reactions:

\[ K = \left( \frac{1}{K_2} \right) \times \left( \frac{1}{K_1^2} \right) = \frac{1}{K_2 K_1^2} \]

Thus, the expression for \( K \) is:

\[ K = \frac{1}{K_2 K_1^2} \]

### Final Answer

\[ K = \frac{1}{K_2 K_1^2} \]