An ideal gas expands adiabatically ( = 1.5 ). To reduce the r.m.s. velocity of the molecules 3 times the gas has to be expanded

Step-by-step solution

The root mean square velocity of gas molecules is given by v_ rms = 3RT M , which implies v_ rms T . Given that the r.m.s. velocity is reduced 3 times, the new temperature T' is related to the initial temperature T by: T' = T 3 T' = T 9 For an adiabatic process, the relation between temperature and volume is TV^ - 1 = constant . T V^ - 1 = T' (V')^ - 1 Substituting = 1.5 and T' = T 9 : T V^ 1.5 - 1 = ( T 9 ) (V')^ 1.5 - 1 V^ 0.5 = 1 9 (V')^ 0.5 ( V' V )^ 0.5 = 9 V' V = 9^2 = 81 The gas has to be expanded 81 times.