An ideal gas (gamma = 1.5) is expanded adiaba | vedclass

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(B) The root mean square velocity is given by $v_{rms} = \sqrt{\frac{3RT}{M}}$,which implies $v_{rms} \propto \sqrt{T}$.To reduce $v_{rms}$ by a facto

Vedclass · Accepted Answer

$16$

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(B) The root mean square velocity is given by $v_{rms} = \sqrt{\frac{3RT}{M}}$,which implies $v_{rms} \propto \sqrt{T}$.To reduce $v_{rms}$ by a factor of $2$,the temperature $T$ must be reduced by a factor of $2^2 = 4$. Thus,$\frac{T'}{T} = \frac{1}{4}$.For an adiabatic process,the relation between temperature and volume is $TV^{\gamma - 1} = T'V'^{\gamma - 1}$.Rearranging for the volume ratio,we get $\frac{V'}{V} = \left(\frac{T}{T'}\right)^{\frac{1}{\gamma - 1}}$.Substituting the values $\gamma = 1.5$ and $\frac{T}{T'} = 4$,we get $\frac{V'}{V} = (4)^{\frac{1}{1.5 - 1}} = (4)^{\frac{1}{0.5}} = (4)^2 = 16$.Therefore,the gas must be expanded $16$ times.

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

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