Consider an ideal gas confined in an isolated closed chamber. As the gas undergoes adiabatic expansion, the average time of collision between molecules increases as Vq, where V is the volume of the gas. The value of q is : (γ=Cp/Cv)

For an adiabatic process TγV1-γ = constant, we know that average time of collision between molecules τ = 1/nπ√2Vrmsd2 where n = no. of molecules per unit volume Vrms = rms velocity of molecules. As n ∝ 1/V and Vrms ∝ √T Thus we can write: n = k1V-1 and Vrms = k2T1/2 where k1 and k2 are constant. For adiabatic process TVγ-1 = constant. Thus we can write τ ∝ VT1/2V(1-γ)/2 or τ ∝ V(3-γ)/2. Average time between collision = means free path/Vrms t = 1/πd2N/V √3RT/M; t = CV√T where C = √M/πd2√3R ⇒ T ∝ Vγ-1 t2 For adiabatic TVγ-1 = k Vγ-1t2 = k, t ∝ V(γ-1)/2 so q = (γ-1)/2 + 1/2 = γ/2