An ideal gas is expanded adiabatically. How many times has the gas to be expanded to reduce the r.m.s. speed of molecules 3 times ? ( = 1.5 )

Step-by-step solution

The root mean square speed of gas molecules is given by v_ rms = 3RT M . This implies v_ rms T . Given that the r.m.s. speed is reduced by a factor of 3 , we have v_ rms ' = v_ rms 3 . Therefore, the final temperature is T' = T 9 . For an adiabatic process, the relation between temperature and volume is T V^ - 1 = constant . Substituting the given values T₁ = T , T₂ = T 9 , and = 1.5 : T V^ 1.5 - 1 = T 9 V'^ 1.5 - 1 V^ 0.5 = 1 9 V'^ 0.5 ( V' V )^ 0.5 = 9 Squaring both sides, we get: V' V = 81 The gas has to be expa