The question: A perfect gas undergoes the following cyclic processes: State 1 to 2 cooling at constant pressure. State 2 to 3 heating at constant volume. State 3 to 1 adiabatic expansion. Deduce an expression for the thermal efficiency of the cycle in terms of r the volume compression ratio (r=V1/V2) and γ (where γ = ratio of specific heats Cp/Cv) η = 1 - γ(r-1)/(rγ-1) My attempt at the solution: First I tried sketching the cycle Bare with me as I present you the silly symbol art. P 3. ^'. |..| |...\ |.....'-. |.........'-._ 2<---------':.1 v I'd like to work in specific terms As it is a perfect gas P1v1= RT1 P1v2= RT2 P2v2= RT3 Heats from 1->2, 2->3, 3->1 Q1->2=Cp(T2-T1) Q2->3=Cv(T3-T2) Q3->1 = 0 ; adiabatic. Also, polytropic relations v2/v1 = (P1/P2)1/n as r = v1/v2⇔ r = (P2/P1)1/n ∴ rn = P2/P1 and 1/rn = P1/P2 Substituting Ideal Gas expressions in terms of Tn Q1->2=Cp((P1v2-P1v1)/R) Q2->3=Cv((P2v2-P1v2)/R) Thermal efficiency This is what I am unsure off. I begin assuming quite a few things. First I assume that heat in the cooling process is the equivalent of heat escaping to a cold reservoir; coincidentally, heat from the pressurization is the heat INPUT from the hot reservoir. As such: η = [Q2->3 - Q1->2]/Q2->3 η = 1 - Q1->2/Q2->3 ∴ η = 1 - Cp(P1v2-P1v1)/Cv(P2v2-P1v2) η = 1 - γ((P1v2-P1v1)/(P2v2-P1v2)) From here: P1v2/P2v2-P1v2 = P1/(P2-P1) = 1/rn-1 And P1v1/P2v2-P1v2 = r/(rn-1) My Result ∴ η = γ(1-r)/(rn-1) =/= η = 1 - γ(r-1)/(rγ-1) Can I assume n=γ in this situation? only 1 process is adiabatic.