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The following equilibrium constants are given:
$N_2 + 3H_2 \rightleftharpoons 2NH_3$; $K_1$
$N_2 + O_2 \rightleftharpoons 2NO$; $K_2$
$H_2 + \dfrac{1}{2}O_2 \rightleftharpoons H_2O$; $K_3$
The equilibrium constant for the oxidation of $NH_3$ by oxygen to give NO is
$N_2 + 3H_2 \rightleftharpoons 2NH_3$; $K_1$
$N_2 + O_2 \rightleftharpoons 2NO$; $K_2$
$H_2 + \dfrac{1}{2}O_2 \rightleftharpoons H_2O$; $K_3$
The equilibrium constant for the oxidation of $NH_3$ by oxygen to give NO is
A
$\dfrac{K_2 K_3^2}{K_1}$
B
$\dfrac{K_2^2 K_3}{K_1}$
C
$\dfrac{K_1 K_2}{K_3}$
D
$\dfrac{K_2 K_3^3}{K_1}$
Detailed Solution
Given: $N_2 + 3H_2 \rightleftharpoons 2NH_3$; $K_1$ ...(i)
$N_2 + O_2 \rightleftharpoons 2NO$; $K_2$ ...(ii)
$H_2 + \dfrac{1}{2}O_2 \rightleftharpoons H_2O$; $K_3$ ...(iii)
We have to calculate $K$ for $4NH_3 + 5O_2 \rightarrow 4NO + 6H_2O$, or $2NH_3 + \dfrac{5}{2}O_2 \rightarrow 2NO + 3H_2O$
For this equation, $K = \dfrac{[NO]^2[H_2O]^3}{[NH_3]^2[O_2]^{5/2}}$
$K_1 = \dfrac{[NH_3]^2}{[N_2][H_2]^3}$, $K_2 = \dfrac{[NO]^2}{[N_2][O_2]}$ and $K_3 = \dfrac{[H_2O]}{[H_2][O_2]^{1/2}}$, so $K_3^3 = \dfrac{[H_2O]^3}{[H_2]^3[O_2]^{3/2}}$
The target equation = (ii) + 3 $\times$ (iii) $-$ (i). Now operate $\dfrac{K_2 \cdot K_3^3}{K_1}$:
$\dfrac{K_2 K_3^3}{K_1} = \dfrac{[NO]^2}{[N_2][O_2]} \times \dfrac{[H_2O]^3}{[H_2]^3[O_2]^{3/2}} \times \dfrac{[N_2][H_2]^3}{[NH_3]^2}$
$= \dfrac{[NO]^2[H_2O]^3}{[NH_3]^2[O_2]^{5/2}} = K$
$\therefore K = \dfrac{K_2 K_3^3}{K_1}$
$N_2 + O_2 \rightleftharpoons 2NO$; $K_2$ ...(ii)
$H_2 + \dfrac{1}{2}O_2 \rightleftharpoons H_2O$; $K_3$ ...(iii)
We have to calculate $K$ for $4NH_3 + 5O_2 \rightarrow 4NO + 6H_2O$, or $2NH_3 + \dfrac{5}{2}O_2 \rightarrow 2NO + 3H_2O$
For this equation, $K = \dfrac{[NO]^2[H_2O]^3}{[NH_3]^2[O_2]^{5/2}}$
$K_1 = \dfrac{[NH_3]^2}{[N_2][H_2]^3}$, $K_2 = \dfrac{[NO]^2}{[N_2][O_2]}$ and $K_3 = \dfrac{[H_2O]}{[H_2][O_2]^{1/2}}$, so $K_3^3 = \dfrac{[H_2O]^3}{[H_2]^3[O_2]^{3/2}}$
The target equation = (ii) + 3 $\times$ (iii) $-$ (i). Now operate $\dfrac{K_2 \cdot K_3^3}{K_1}$:
$\dfrac{K_2 K_3^3}{K_1} = \dfrac{[NO]^2}{[N_2][O_2]} \times \dfrac{[H_2O]^3}{[H_2]^3[O_2]^{3/2}} \times \dfrac{[N_2][H_2]^3}{[NH_3]^2}$
$= \dfrac{[NO]^2[H_2O]^3}{[NH_3]^2[O_2]^{5/2}} = K$
$\therefore K = \dfrac{K_2 K_3^3}{K_1}$