Heat Engine Cycles: Understanding Reversible Changes

[nokure]

Homework Statement

Taken from adkins Introduction to thermal physics

A reversible heat engine is operated between two bodies, one
of heat capacity C1 initially at temperature T1 and the other of
heat capacity C2 initially at temperature T2. As the engine
operates, the warmer body gradually cools and the cooler one
is warmed.
(a) By considering the changes that occur in one cycle of the
engine, show that infinitesimal changes of temperature of the
two bodies are related by

0 = C1 dT₁/T1 + C2 dT₂/T2

Eventually, the bodies reach the same temperature Tf and
the heat engine ceases to run. Show that Tf is given by

Tf^{C1 + C2} = T1^C1 T2^C2

Homework Equations

U = Q + W

The Attempt at a Solution

For the first part

I am taking the U to be 0 and W = 0 ... This probably is not be correct.

0 = C1 (T1 + dt) + C2(T2-dt)... this does lead me to the correct solution.

Any hints would be greatly appreciated!

 

[nokure]

Alright I figured out this question ...

Its a simple manipulation of the dQ = T ds equation for each Temperature. Then you can equate them by using S1 = -S2 and finally integrate.

The last part of this question ask what the Tf would be if the process was irreversible. I think this would just be the average:

Tf = (T1 + T2)/2

however I have a feeling I am missing something. Any suggestions would be great!