The pressure and density of a diatomic gas $(\mathrm{\gamma }=7/5)$ change adiabatically from (P, d) to (P', d'). If  $\frac{\mathrm{d}\text{'}}{\mathrm{d}}=32$, then $\frac{\mathrm{P}\text{'}}{\mathrm{P}}$ should be

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