The following equilibrium constant are given: $\begin{aligned} \mathrm{N}_2+3 \mathrm{H}_2 &…

The following equilibrium constant are given: $\begin{aligned} \mathrm{N}_2+3 \mathrm{H}_2 & \rightleftharpoons 2 \mathrm{NH}_3 ; \mathrm{K}_1 \\ \mathrm{~N}_2+\mathrm{O}_2 & \rightleftharpoons 2 \mathrm{NO} ; \mathrm{K}_2 \\ \mathrm{H}_2+1 / 2 \mathrm{O}_2 & \rightleftharpoons \mathrm{H}_2 \mathrm{O} ; \mathrm{K}_3 \end{aligned}$ The equilibrium constant for the oxidation of $\mathrm{NH}_3$ by oxygen to give $\mathrm{NO}$ is:
  1. $\frac{\mathrm{K}_2 \mathrm{~K}_3^2}{\mathrm{~K}_1}$
  2. $\frac{\mathrm{K}_2^2 \mathrm{~K}_3}{\mathrm{~K}_1}$
  3. $\frac{\mathrm{K}_1 \mathrm{~K}_2}{\mathrm{~K}_3}$
  4. $\frac{K_2 K_3^3}{K_1}$

Solution

Given,
$\begin{aligned}
& \mathrm{N}_2+3 \mathrm{H}_2 \rightleftharpoons 2 \mathrm{NH}_3 ; \mathrm{K}_1 \\
& \mathrm{~N}_2+\mathrm{O}_2 \rightleftharpoons 2 \mathrm{NO} ; \mathrm{K}_2 \\
& \mathrm{H}_2+\frac{1}{2} \mathrm{O}_2 \rightleftharpoons \mathrm{H}_2 \mathrm{O} ; \mathrm{K}_3
\end{aligned}$
We have to calculate
$\begin{aligned}
& 4 \mathrm{NH}_3+5 \mathrm{O}_2 \longrightarrow 4 \mathrm{NO}+6 \mathrm{H}_2 \mathrm{O} ; \mathrm{K}=\text { ? } \\
& \text { or } 2 \mathrm{NH}_3+\frac{5}{2} \mathrm{O}_2 \longrightarrow 2 \mathrm{NO}+3 \mathrm{H}_2 \mathrm{O}
\end{aligned}$
For this equation $\mathrm{K}=\frac{[\mathrm{NO}]^2\left[\mathrm{H}_2 \mathrm{O}\right]^3}{\left[\mathrm{NH}_3\right]^2\left[\mathrm{O}_2\right]^{5 / 2}}$ but
$\begin{aligned}
\mathrm{K}_1 & =\frac{\left[\mathrm{NH}_3\right]^2}{\left[\mathrm{~N}_2\right]\left[\mathrm{H}_2\right]^3} \\
\mathrm{~K}_2 & =\frac{[\mathrm{NO}]^2}{\left[\mathrm{~N}_2\right]\left[\mathrm{O}_2\right]} \\
\mathrm{K}_3 & =\frac{\left[\mathrm{H}_2 \mathrm{O}\right]}{\left[\mathrm{H}_2\right]\left[\mathrm{O}_2\right]^{1 / 2}} \\
\mathrm{~K}_3 & =\frac{\left[\mathrm{H}_2 \mathrm{O}\right]^3}{\left[\mathrm{H}_2\right]^3\left[\mathrm{O}_2\right]^{3 / 2}}
\end{aligned}$
Now
$\begin{aligned}
& \frac{\mathrm{K}_2 \cdot \mathrm{K}_3{ }^3}{\mathrm{~K}_1} =\frac{[\mathrm{NO}]^2}{\left[\mathrm{~N}_2\right]\left[\mathrm{O}_2\right]} \times \frac{\left[\mathrm{H}_2 \mathrm{O}\right]^3}{\left[\mathrm{H}_2\right]^3\left[\mathrm{O}_2\right]^{3 / 2}} \times \frac{\left[\mathrm{N}_2\right]\left[\mathrm{H}_2\right]^3}{\left[\mathrm{NH}_3\right]^2} \\
& =\frac{[\mathrm{NO}]^2\left[\mathrm{H}_2 \mathrm{O}\right]^3}{\left[\mathrm{NH}_3\right]^2\left[\mathrm{O}_2\right]^{5 / 2}}=\mathrm{K} \\
& \therefore \mathrm{K}=\frac{\mathrm{K}_2 \cdot \mathrm{K}_3{ }^3}{\mathrm{~K}_1}
\end{aligned}$

Asked in: NEET 2007

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