An ideal gas (γ = 1.5) is expanded adiabatically. How many times has the gas to be expanded to reduce the root mean square velocity of molecules 2.0 times

A
4 times
B
16 times
C
8 times
D
2 times

Solution:

$v_{rms} = \sqrt{\frac{3 RT}{M}} \Rightarrow v_{rms} \propto \sqrt{T}$
v_rms is to reduce two times i.e. temperature of the gas will have to reduce four times or $\frac{T^{'}}{T} = \frac{1}{4}$
During adiabatic process
$\begin{array}{l} {TV}^{γ - 1} = T^{'} V^{' γ - 1} \\ \Rightarrow \frac{V^{'}}{V} = {(\frac{T}{T^{'}})}^{\frac{1}{γ - 1}} = (4)^{\frac{1}{1 .5 - 1}} = (4)^{2} = 16 \\ \Rightarrow V^{'} = 16 V \end{array}$

An ideal gas ( $γ$ = 1.5) is expanded adiabatically. How many times has the gas to be expanded to reduce the root mean square velocity of molecules 2.0 times

Solution:

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