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 $\frac{{\tau }_1}{{\tau }_2}$  is given by:

$Mean free time =\frac{\lambda }{V_{rms}}=\frac{{\displaystyle \frac{RT}{\sqrt{2}\pi d^2{N}_{A}P}}}{\sqrt{{\displaystyle \frac{3RT}{M}}}}$

$t\propto \frac{\sqrt{T}}{P}\propto \frac{\sqrt{T}}{{\displaystyle \frac{RT}{V}}}\propto \frac{V\sqrt{T}}{T}\propto \frac{V}{\sqrt{T}}$

$\therefore t\propto \frac{V}{\sqrt{{\displaystyle \frac{1}{V^{\gamma -1}}}}}\propto \frac{V}{{\displaystyle \frac{1}{V^{{\displaystyle \frac{\gamma -1}{2}}}}}}\propto V{V}^{\frac{\gamma -1}{2}}\propto V^{1+\frac{\gamma -1}{2}}\propto V^{\frac{2+\gamma -1}{2}}$

$Mean Collision time , t\propto \frac{V}{\sqrt{T}} ..........\left(1\right)$

$For adiabatic process, T{V}^{\gamma -1}=\text{constant .................(2)}$

$$\begin{gathered} \Rightarrow t\propto V^{\frac{\gamma +1}{2}} \\ \Rightarrow \frac{t_2}{t_1}={\left(\frac{V_2}{V_1}\right)}^{\frac{\gamma +1}{2}} \end{gathered}$$

$$\begin{gathered} \Rightarrow \frac{t_2}{t_1}={\left(\frac{2}{1}\right)}^{\frac{\gamma +1}{2}} \\ \Rightarrow \frac{t_1}{t_2}={\left(\frac{1}{2}\right)}^{\frac{\gamma +1}{2}} \end{gathered}$$