An ideal gas expands adiabatically, ( =1 5) . To reduce the r.m.s. velocity of the molecules 4 times, the gas has to be expanded

Step-by-step solution

The root mean square velocity is given by v_ rms = 3RT M , hence v_ rms T . A fourfold reduction in velocity requires v'_ rms v_ rms = 1 4 = T' T , yielding T' = T 16 after squaring both sides. 1 4 = T' T T' = T 16 For adiabatic expansion with = 1.5 , TV^ -1 remains constant. Using T₂ = T₁ 16 : T₁ V₁^ 0.5 = T₁ 16 V₂^ 0.5 V₂^ 0.5 V₁^ 0.5 = 16 V₂ V₁ = 256 The gas must be expanded 256 times, corresponding to option A .