The average time between the collisions of the gas molecules is nothing but the mean free path divided by the root mean square speed of the gas molecules.

$\mathrm{So}$

$\Longrightarrow\left[\text { Time }=t=\frac{\lambda}{v_{R M S}}\right]$

Now we also know that :-

$v_{R M S} \propto \sqrt{P V}$

Using the above we get :-

$\Longrightarrow t \propto V \times \sqrt{\frac{1}{P V}}$

$\Longrightarrow t \propto \sqrt{\frac{V^{2}}{P V}}$

$\Longrightarrow t \propto \sqrt{\frac{V}{P}}$

adiabatic process then $PV^{\gamma}=constant$

then $t \propto V^{\frac{\gamma+1}{2}}$