Question
If the equilibrium constant for $${N_2}\left( g \right) + {O_2}\left( g \right) \rightleftharpoons 2NO\left( g \right)$$ is $$K,$$ the equilibrium constant for $$\frac{1}{2}{N_2}\left( g \right) + \frac{1}{2}{O_2}\left( g \right) \rightleftharpoons NO\left( g \right)$$ will be,
A.
$${K^{\frac{1}{2}}}$$
B.
$$\frac{1}{2}K$$
C.
$$K$$
D.
$${K^2}$$
Answer :
$${K^{\frac{1}{2}}}$$
Solution :
Plan As wecan see the reaction for which we have to find out equilibrium constant is different only in stoichiometric coefficient as compared to the given reaction. Hence, we can find equilibrium constant for the required reaction with the help of mentioned equilibrium constant in the problem.
Given, equilibrium constant for the reaction,
$${N_2}\left( g \right) + {O_2}\left( g \right) \rightleftharpoons 2NO\left( g \right)\,{\text{is}}\,K$$
i.e. $$K = \frac{{{{\left[ {NO} \right]}^2}}}{{\left[ {{N_2}} \right]\left[ {{O_2}} \right]}}\,\,\,...(i)$$
Let equilibrium constant for the reaction,
$$\frac{1}{2}{N_2}\left( g \right) + \frac{1}{2}{O_2}\left( g \right) \rightleftharpoons NO\left( g \right)\,{\text{is}}\,K'$$
i.e. $$K' = \frac{{\left[ {NO} \right]}}{{{{\left[ {{N_2}} \right]}^{\frac{1}{2}}}{{\left[ {{O_2}} \right]}^{\frac{1}{2}}}}}$$
On squaring both sides
$$K{'^2} = \frac{{{{\left[ {NO} \right]}^2}}}{{\left[ {{N_2}} \right]\left[ {{O_2}} \right]}}\,\,\,...(ii)$$
On comparing Eqs. (i) and (ii), we get
$$K = K{'^2}$$
or $$K' = \sqrt K $$