Under an adiabatic process, the volume of an ideal gas gets doubled. Consequently the mean collision time between the gas molecule changes from \(\tau_{1}\) to \(\tau_{2} .\) If \(\frac{C_{p}}{C_{v}}=\gamma\) for this gas then a good estimate for \(\frac{\tau_{2}}{\tau_{1}}\) is given by :

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 \(t \propto V^{\frac{\gamma+1}{2}}\)