A monoatomic ideal gas, initially at temperature  ${\text{T}}_{\text{1}}$  is enclosed in a cylinder fitted with a frictionless piston. The gas is allowed to expand adiabatically to a temperature  ${\text{T}}_{\text{2}}$ by releasing the piston suddenly. If   ${\text{L}}_{\text{1}}$,${\text{L}}_{\text{2}}$  are the lengths of the gas column before and after expansion respectively, then  ${\text{T}}_{\text{1}}{\text{/T}}_{\text{2}}$ 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