For the reaction $2\mathrm{NO}+{\mathrm{Cl}}_2\to 2\mathrm{NOCl},$the following mechanism has been proposed 

$\mathrm{NO}+{\mathrm{Cl}}_2\mathrm{F} {\mathrm{NOCl}}_2\left(\mathrm{fast}\right)$

${\mathrm{NOCl}}_2+ \mathrm{NO}\to 2\mathrm{NOCl} \left(\mathrm{slow}\right)$

The rate law for the reaction is

For the reaction , $2\mathrm{NO} +{\mathrm{Cl}}_2\to 2\mathrm{NOCl}$

$\mathrm{NO}+{\mathrm{Cl}}_2\to {\mathrm{NOCl}}_2; \mathrm{fast}\to \mathrm{eqn}\left(1\right)$

$\mathrm{NO}+{\mathrm{NOCl}}_2\to 2\mathrm{NOCl}; \mathrm{slow} \to \mathrm{eqn}\left(2\right)$

According to rate law the rate has been determined from slowest step from the reaction mechanism.

$\mathrm{Rate},\mathrm{r}=\mathrm{K}\left[\mathrm{NO}\right]\left[{\mathrm{NOCl}}_2\right]; \mathrm{from} \mathrm{slow} \mathrm{step}\to \mathrm{eqn}\left(3\right)$From eqn (1), equilibrium constant is

${\mathrm{K}}_1=\frac{\left[{\mathrm{NOCl}}_2\right]}{\left[\mathrm{NO}\right]\left[{\mathrm{Cl}}_2\right]}$

$\mathrm{So},\left[{\mathrm{NOCl}}_2\right]={\mathrm{K}}_1\left[\mathrm{NO}\right]\left[{\mathrm{Cl}}_2\right]$

Substitute this value in eqn(3) , then we will get

$\mathrm{Rate},\mathrm{r}=\mathrm{K}\left[\mathrm{NO}\right]\left[\mathrm{NO}\right]\left[{\mathrm{Cl}}_2\right]$

$\therefore \mathrm{Rate},\mathrm{r}=\mathrm{K}{\left[\mathrm{NO}\right]}^2\left[{\mathrm{Cl}}_2\right]$