If the equilibrium constant for ${\text{N}}_2(g)+{\text{O}}_2(g)\rightleftharpoons 2\text{NO}(g)$ is $K$, the equilibrium constant for $\frac{1}{2}{\text{N}}_2(g)+\frac{1}{2}{\text{O}}_2(g)\rightleftharpoons \text{NO}(g)$ will be:

Given the reaction:${\text{N}}_2(g)+{\text{O}}_2(g)\rightleftharpoons 2\text{NO}(g)$ with equilibrium constant $K=\frac{[\text{NO}{]}^2}{[{\text{N}}_2][{\text{O}}_2]}$ The new reaction is obtained by multiplying the coefficients of the original reaction by $\frac{1}{2}$:$\frac{1}{2}{\text{N}}_2(g)+\frac{1}{2}{\text{O}}_2(g)\rightleftharpoons \text{NO}(g)$ The new equilibrium constant $K^{\prime }$ is:$K^{\prime }=\frac{[\text{NO}]}{[{\text{N}}_2{]}^{1/2}[{\text{O}}_2{]}^{1/2}}={\left(\frac{[\text{NO}{]}^2}{[{\text{N}}_2][{\text{O}}_2]}\right)}^{1/2}=K^{1/2}$ or $\sqrt{K}$.