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  • An ideal gas is expanded adiabatically. How many times has the gas to be expanded to reduce the

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
 

Option: 1

\text{4 times}
 


Option: 2

\text{16 times}
 


Option: 3

\mathrm{8 times}

 


Option: 4

\mathrm{2 times}


Answers (1)

best_answer

\mathrm{\quad \mathrm{V}_{\mathrm{rms}}=\sqrt{\frac{3 \mathrm{RT}}{\mathrm{M}}}}

\mathrm{\therefore \quad V_{\text {rms }} \propto \sqrt{T} }

\mathrm{V_{\text {rms }} } is to reduce two times i.e, temperature of the gas will have to reduce four times or

\mathrm{ \frac{T^{\prime}}{T}=\frac{1}{4} }

During adiabatic process

\mathrm{ \mathrm{TV}^{\gamma-1}=\mathrm{T}^{\prime} \mathrm{V}^{\prime \gamma-1} }

\mathrm{ \text { or, } \frac{\mathrm{V}^{\prime}}{\mathrm{V}}=\left(\frac{\mathrm{T}}{\mathrm{T}^{\prime}}\right)^{\frac{1}{\gamma-1}}=(4)^{\frac{1}{1.5-1}}=4^2=16 }

\mathrm{ \therefore \quad \mathrm{V}^{\prime}=16 \mathrm{~V} }












 

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