A gas is enclosed in a cylinder with a movable frictionless piston. Its initial thermodynamic state at pressure `P_i=10^5` Pa and volume `V_i=10^-3m^3` changes to a final state at `P_f=(1//32)xx10^5Pa and V_f=8xx10^-3m^3` in an adiabatic quasi-static process, such that `P^3V^3=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 (isovolumetric ) process at volume `V_r.` The amount of heat supplied to the system i the two-step process is approximately

Options

A. `112J`
B. `294J`
C. `588J`
D. `813J`

Answer

C

Explanation

In adiabatic process `P^(3)V^(5)` = constant
`implies PV^(5//3)` = constant
`implies gamma=(5)/(3) implies C_(V) = (3)/(2)R "and" C_(P) = (5)/(2)R`
In another process

`DeltaQ = nC_(P)DeltaT + nC_(V)DeltaT = (5)/(2)nR (T_(B) - T_(A)) + (3)/(2)nR(T_(C)- T_(B))`
`DeltaQ = (5)/(2)(P_(B)V_(B) - P_(A)V_(A))+ (3)/(2)(P_(C)V_(C) - P_(B)V_(B))`
Putting value
`DeltaQ = 587.5J approx 588J`