Question

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 $\bigg(\gamma =\frac{C_p}{C_v}\bigg)$

• A. $\frac{3\gamma +5 }{6}$
• B. $\frac{\gamma +1}{2}$
• C. $\frac{3 \gamma-5}{6}$
• D. $\frac{\gamma-1}{2}$

Answer 1

Answer: (B)

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

Answer 2

mean free path
$\lambda=\frac{1}{\sqrt{2}\, \pi d ^{2} n}$
$n =\frac{\text { no. of molecules }}{\text { volume }}$
$v _{\text {avg. }} \propto \sqrt{ T }\,\,\,\,\,\,\,T .V ^{\gamma-1}= C$
$t =\frac{\lambda}{ V _{\text {avg. }}} \propto \frac{ V }{\sqrt{ T }} \,\,\,\,\,\,\,v \rightarrow$ is volume
$\frac{ V }{\sqrt{\frac{ C }{ v ^{ r -1}}}} \propto V ^{\frac{\gamma+1}{2}}$
$v^{q} \propto v^{\frac{\gamma+1}{2}}$
$q =\frac{\gamma+1}{2}$