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Derivation of Kp and Kc

The equilibri...

Question

The equilibrium constant for the reaction
$N_{2} + 2 O_{2} ⇌ 2 {N O}_{2}$
at a particular temperature is $100$ . Write down the equilibrium law equations of the following reactions and determine the values of equilibrium constants:
$2 {N O}_{2} ⇌ N_{2} + 2 O_{2} . . . . (i)$
${N O}_{2} ⇌ 1 / 2 N_{2} + O_{2} . . . (i i)$

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Solution

The equilibrium constant for the reaction
$N_{2} + 2 O_{2} ⇌ 2 {N O}_{2}$
$K_{c} = \frac{{[{N O}_{2}]}^{2}}{{[N_{2}]} {[O_{2}]}^{2}} = 100.... (a)$
The equilibrium constant equation for reaction $(i)$
$K_{1} = \frac{{[N_{2}]} {[O_{2}]}^{2}}{{[{N O}_{2}]}^{2}} . . . . (b)$
This equation is reciprocal of eq. $(a)$
So, $K_{1} = \frac{1}{K_{c}} = \frac{1}{100} = 1 \times 10^{- 2}$
The equilibrium constant equation for reaction $(i i)$
$K_{2} = \frac{{[N_{2}]}^{1 / 2} {[O_{2}]}}{{[{N O}_{2}]}} . . . . (c)$
Comparing eqs. $(b)$ and $(c)$
$K_{2} = \sqrt{K_{1}} = \sqrt{10^{- 2}} = 10^{- 1} = 0.1$

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