Under an adiabatic process, the volume of an ideal gas gets doubled. Consequently the mean collision time between the gas molecule changes from τ1 to τ2 . If CPCV=γ for this gas then a good estimate for τ1τ2  is given by:

Mean free time =λVrms=RT2πd2NAP3RTMt∝TP∝TRTV∝VTT∝VT∴t∝V1Vγ−1∝V1Vγ-12∝VVγ-12∝V1+γ-12∝V2+γ-12Mean Collision time , t∝VT                         ..........(1)For adiabatic process, TVγ−1=constant       .................(2)⇒t∝Vγ+12 ⇒t2t1=V2V1γ+12⇒t2t1=21γ+12 ⇒t1t2=12γ+12