Question

An ideal gas $\left(\gamma = 1 .5\right)$ is expanded adiabatically. How many times its initial volume should be expanded to reduce the root-mean-square velocity of molecules by a factor of $\text{2}$ ?

• A. $\text{4}$ times
• B. $\text{16}$ times
• C. $\text{8}$ times
• D. $\text{2}$ times

Answer 1

Answer: (B)

$v_{rms}=\sqrt{\frac{3 RT}{M}}orv_{rms} \propto \sqrt{T}$
$v_{rms}$ is to reduce two times, i.e. the temperature of the gas will have to reduce four times or
$\frac{T '}{T}=\frac{1}{4}$
During the adiabatic process,
$T V⁡^{\gamma - 1}=T⁡^{'}V⁡^{' \gamma - 1}$
or $\frac{V ^{'}}{V}=\left(\frac{T}{T^{'}}\right)^{\frac{1}{\gamma - 1}}$
$ \, =\left(4\right)^{\frac{1}{1.5 - 1}}=4^{2}=16$
$\therefore $ $V^{'}=16 \, V$