Numerical on adiabatic expansion

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An ideal gas with a specific heat ratio (γ) of 1.5 undergoes adiabatic expansion, requiring a reduction in the root mean square (rms) velocity of its molecules by half. To achieve this, the absolute temperature must be reduced to one-fourth of its original value. In an adiabatic process, the relationship T.V^(γ-1) = constant applies, indicating that the volume must increase significantly. Specifically, a 16-fold increase in volume is necessary to achieve the desired temperature reduction. The discussion concludes with appreciation for the collaborative problem-solving effort.
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An ideal gas with g =1.5 is expanded adiabatically. How many times has the gas to be expanded to reduce the rms velocity of molecules 2 times?
 
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rms speed = √(3kT/m), where k is the Boltzmann constant and m the mass of one molecule.
So to halve the speed you need 1/4 of the absolute temperature.
For adiabatic process, T.V^{γ-1} = constant
With γ = 1.5 you'd need 16 times the volume.
 
Thanks for your help. It was great solving the problem.
 

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