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

Question

Dr. Abubacker · Accepted Answer

(b) $v_{rms} = \sqrt{\frac{3RT}{M}}\ \ \ \ \ \ \ \ \ \ \ \ or\ \ \ \ \ \ \ \ \ \ \ \ \ \ v_{rms} \propto \sqrt{T}$
$v_{rms}$ is to reduce two times, $ie$, the temperature of the gas will have to reduce force times or
$\frac{T'}{T} = \frac{1}{4}$
During adiabatic process,
$TV^{\gamma - 1} = T^{'}V^{'\gamma - 1}$
or $\frac{V^{'}}{V} = \left( \frac{T}{T^{'}} ight)^{\frac{1}{\gamma - 1}}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  = (4)^{\frac{1}{1.5 - 1}} = 4^{2} = 16$
$	herefore$ $V^{'} = 16\ V$

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

Explanation

Reviewed by Dr. Abubacker

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