The equilibrium constants for the following reactions
`N_(2) (g) + 3H_(2)(g) hArr 2NH_(3) (g) N_(2) (g) + O_(2) (g) hArr 2NO(g)`
`" and " H_(2)(g) +1//2 O_(2) (g) hArr H_(2)O (Ig) " are " K_(1) ,K_(2) " and " K_(3)` respectively.
The equilibrium constant (K) for the reaction
`2NH_(3) (g) + 2^(1)//2) O_(2) (g) hArr 2NO(g) +3H_(2) O(I) " is "`

Options

A. `K_(2) .K_(3)^(3)//K_(1)`
B. `K_(2)^(2) K_(3)//K_(1)`
C. `K_(1).K_(2)//K_(3)`
D. `K_(2).K_(3)//K_(1)`

Explanation

To find the equilibrium constant (K) for the reaction

\[ 2NH_3(g) + \frac{1}{2} O_2(g) \rightleftharpoons 2NO(g) + 3H_2O(l) \]

we will use the equilibrium constants of the given reactions:

1. \( N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g) \) with equilibrium constant \( K_1 \)
2. \( N_2(g) + O_2(g) \rightleftharpoons 2NO(g) \) with equilibrium constant \( K_2 \)
3. \( H_2(g) + \frac{1}{2} O_2(g) \rightleftharpoons H_2O(l) \) with equilibrium constant \( K_3 \)

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

1. \( N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g) \) → \( K_1 \)
2. \( N_2(g) + O_2(g) \rightleftharpoons 2NO(g) \) → \( K_2 \)
3. \( H_2(g) + \frac{1}{2} O_2(g) \rightleftharpoons H_2O(l) \) → \( K_3 \)

### Step 2: Manipulate the reactions to derive the target reaction

We need to manipulate the above reactions to arrive at the target reaction.

1. Start with the reaction for \( NH_3 \):
\[ 2NH_3(g) \rightleftharpoons N_2(g) + 3H_2(g) \]
This is the reverse of the first reaction, so its equilibrium constant will be:
\[ K' = \frac{1}{K_1} \]

2. Next, we will use the second reaction as it is:
\[ N_2(g) + O_2(g) \rightleftharpoons 2NO(g) \]
This has the equilibrium constant \( K_2 \).

3. Finally, we will reverse the third reaction:
\[ H_2O(l) \rightleftharpoons H_2(g) + \frac{1}{2} O_2(g) \]
The equilibrium constant for this reaction will be:
\[ K'' = \frac{1}{K_3} \]

### Step 3: Combine the manipulated reactions

Now, we can combine these reactions:

1. \( 2NH_3(g) \rightleftharpoons N_2(g) + 3H_2(g) \) → \( K' = \frac{1}{K_1} \)
2. \( N_2(g) + O_2(g) \rightleftharpoons 2NO(g) \) → \( K_2 \)
3. \( H_2O(l) \rightleftharpoons H_2(g) + \frac{1}{2} O_2(g) \) → \( K'' = \frac{1}{K_3} \)

Adding these reactions gives:

\[ 2NH_3(g) + O_2(g) \rightleftharpoons 2NO(g) + 3H_2(g) + H_2O(l) \]

### Step 4: Write the overall equilibrium constant

The overall equilibrium constant \( K \) for the target reaction is given by the product of the equilibrium constants of the individual reactions:

\[ K = K' \cdot K_2 \cdot K'' \]

Substituting the manipulated constants:

\[ K = \left(\frac{1}{K_1}\right) \cdot K_2 \cdot \left(\frac{1}{K_3}\right) \]

Thus, the equilibrium constant for the reaction

\[ 2NH_3(g) + \frac{1}{2} O_2(g) \rightleftharpoons 2NO(g) + 3H_2O(l) \]

is

\[ K = \frac{K_2}{K_1 \cdot K_3} \]