# Carnot Engine for Finite Reservoirs

Two solid, finite thermal reservoirs have temperatures of $T_1$ and $T_2$ respectively and an engine operates between the two. Assume $T_1 > T_2$ and that each reservoir has constant heat capacity $C$. What is the maximum work that can be done by an engine between these reservoirs?

I solved this problem eventually using entropy & the second law, but I was wondering if it is possible to do without using entropy? That's what I tried to when I first saw the problem, writing the work in terms of the final temperature by integrating $dW=\text{(Carnot efficiency)}\ C \ dT$ from $T_1$ to $T_f$, then solve for $T_f$ by using the first law. Then plug that $T_f$ back in to find the work done. Although it is a little messier, it feels more intuitive than the solution using entropy.

I ran into trouble because I couldn't express the efficiency in terms of a varying $T$ for one of the reservoirs. Is it possible/reasonable to find the final temperature this way or is the only nice solution with entropy?

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Suggestions: (1) make your question title into an actual question (if possible) whilst being specific and (2) don't have your question title fully in upper-case. – user12345 Sep 16 '12 at 12:06
Using entropy and the second law is the correct way to do it. Why would you want to solve it another way? – Nathaniel Sep 16 '12 at 17:49