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Derivation of Kp and Kc

For the react...

Question

For the reaction, $N_{2} (g) + O_{2} (g) ⇌ 2 N O (g)$ , the equilibrium constant is $K_{1}$ . The equilibrium constant is $K_{2}$ for the reaction, $2 N O (g) + O_{2} (g) ⇌ 2 N O_{2} (g)$ .

What is $K$ for the reaction, $N O_{2} (g) ⇌ \frac{1}{2} N_{2} (g) + O_{2} (g)$ ?

1 / [4 K_{1} K_{2}]

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{[1 / K_{1} K_{2}]}^{1 / 2}

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1 / [K_{1} K_{2}]

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1 / [2 K_{1} K_{2}]

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Solution

The correct option is B ${[1 / K_{1} K_{2}]}^{1 / 2}$
$N_{2} (g) + O_{2} (g) ⇌ 2 N O (g) K_{1}$ .....(1)
$2 N O (g) + O_{2} (g) ⇌ 2 N O_{2} (g) K_{2}$ .....(2)
Add reactions (1) and (2):
$N_{2} (g) + 2 O_{2} (g) ⇌ 2 N O_{2} (g) K = K_{1} K_{2}$
Reverse the resulting reaction:
$2 N O_{2} (g) ⇌ N_{2} (g) + 2 O_{2} (g) K = \frac{1}{K_{1} K_{2}}$
Divide the reaction with 2:
$N O_{2} (g) ⇌ \frac{1}{2} N_{2} (g) + O_{2} (g) K = {[1 / K_{1} K_{2}]}^{1 / 2}$

Hence, option B is correct.

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