Thermodynamics closed cycle entropy cycle

[Liquidxlax]

Homework Statement

The figure below shows a plot of temperature T versus entropy S for the closed cycle of a particular heat engine (not necessarily an ideal gas) which consists of 3 processes and which operates between two heater reservoirs, a hot reservoir with temperature T1 and a cold reservoir with temperature T2. Assume that each of the 3 processes is reversible.

A B

C

it's a triangle like that with T in the y direction and entropy in the x direction.

a) show that the heat Q1 absorbed by the engine in the process a to b and the heat Q2 rejected by the engine in process b to c are given by

Q1 = T1(S2 - S1) and Q2 = (S2 - S1)((T1+T2)/2)

b) calculate the work Wby performed by the engine in one complete cycle in terms of T1, T2, S1, and S2.

c) use the def of the efficiency eta = Wby/Q1 to show that

eta= (T1-T2)/2T1

Homework Equations

The Attempt at a Solution

ΔS = ∫dQ/T₁ because from a to b it's constant temp

so pretty simple to get Q1

I'm not sure how Q2 arises especially that it is (T1+T2)/2

ΔS = ∫dQ/T = ∫c_vndT/T but that wouldn't give me the right answer unless there is some algebra involved

any help will be appreciated thanks

 

[Liquidxlax]

Is ther a different equation for a system in contact with both reservoirs?

 

[Liquidxlax]

can anyone give me an idea, I'm still having problems with the Q2 = (S2-S1)((T1+T2)/2)