Under an adiabatic process, the volume of an ideal gas gets doubled. Consequently the mean collision time between the gas molecule changes from ${\tau }_1$ to ${\tau }_2$ . If $\frac{C_P}{C_V}=\gamma$ for this gas then a good estimate for $\frac{{\tau }_1}{{\tau }_2}$  is given by:

A gas is enclosed in a cylinder with a movable frictionless piston. Its initial thermodynamic state at pressure $P_1={10}^{5}Pa$ and volume $V_i={10}^{-3}{m}^3$  changes to final state at $P_f=\frac{1}{32}\times {10}^{5}pa\&V_f=8\times {10}^{-3}{m}^3$in an adiabatic quasi-static process, such that $P^3{V}^5=$ constant. Consider another thermodynamic process that brings the system form the same initial state to the same final state in two steps: an isobaric expansion at $P_i$ followed by an isochoric (is volumetric) process at volume$V_f$. The amount of heat supplied to the system in the two-step process is approximately