For an adiabatic process TVγ−1= constant.
We know that average time of collision between molecules
τ=nπ2vrmsd21
where, n= number of molecules per unit volume Vrms= rms velocity of molecules
As n∝V1 and vrms∝T
τ∝TV
Thus, we can write
n=K1V−1 and Vrms=K2T21.
where, K1 and K2 are constants.
For adiabatic process, TVγ−1= constant. Thus, we can write
τ∝VT−21∝V(V1−γ)2−1
or τ∝V2γ+1