First slide

Different type of processes

Question

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?

Moderate

A

4 times
B

16 times
C

8 times
D

2 times

Solution

$v_{rms} = \sqrt{\frac{3 RT}{M}} or v_{rms} \propto \sqrt{T}$
$v_{rms}$ , 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,
${TV}^{γ - 1} = T^{'} V^{' γ - 1}$
or $\frac{V^{'}}{V} = {(\frac{T}{T^{'}})}^{\frac{γ}{γ - 1}}$
= ${(4)}^{\frac{1}{1.5 - 1}} = 4^{2} = 16$
$∴ V^{'} = 16 V$

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