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(B) The average time between collisions $	au$ is given by $	au = \frac{\lambda}{v_{rms}}$,where $\lambda$ is the mean free path and $v_{rms}$ is the

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$\frac{\gamma + 1}{2}$

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(B) The average time between collisions $	au$ is given by $	au = \frac{\lambda}{v_{rms}}$,where $\lambda$ is the mean free path and $v_{rms}$ is the root mean square speed.$\lambda = \frac{1}{\sqrt{2} \pi d^2 (N/V)} \propto V$.$v_{rms} = \sqrt{\frac{3RT}{M}} \propto \sqrt{T}$.Thus,$	au \propto \frac{V}{\sqrt{T}}$.For an adiabatic process,$TV^{\gamma-1} = 	ext{constant}$,which implies $T \propto V^{1-\gamma}$.Substituting this into the expression for $	au$:$	au \propto \frac{V}{(V^{1-\gamma})^{1/2}} = \frac{V}{V^{(1-\gamma)/2}} = V^{1 - \frac{1-\gamma}{2}} = V^{\frac{2-1+\gamma}{2}} = V^{\frac{\gamma+1}{2}}$.Comparing this with $V^q$,we get $q = \frac{\gamma+1}{2}$.

Consider an ideal gas confined in an isolated closed chamber. As the gas undergoes an adiabatic expansion,the average time of collision between molecules increases as $V^q$,where $V$ is the volume of the gas. The value of $q$ is $\left( \gamma = \frac{C_P}{C_V} \right)$.

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