Three identical bodies of constant heat capacity C = 10 J/K are initially at temperatures of 300 K, 300 K, and 100 K. The bodies can interact with each other and with their surroundings only by exchanging heat with one or more heat engines, and no net work or heat is available from outside sources. Go through the following steps to find the maximum amount of work that can be obtained from the system. [HINT: Note that here the temperatures of the bodies are changing. Think about changes in total entropy and total internal energy of the whole system.]
a) Explain how you know that the final temperatures of the three bodies must be equal once you've extracted the maximum amount of work.
b) Explain how you know that the process to extract the most work must be reversible.
c) Given that the process to extract the work must be reversible and that the 3-body system is thermally isolated from its surroundings, what do you know about ΔS for the whole 3-body system?
d) Now write ΔS in terms of T_1i, T_2i, T_3i, T_f, and C. Use this expression and your result from the previous part to find an equation for T_f in terms of the three initial temperatures.
e) Now that you have T_f, you just need to calculate the maximum amount of work. Note that this work will be the same no matter how reversibly you get to the final temperature. Use the first law for the three-body system to get the maximal work extracted, first in terms of T_1i, T_2i, T_3i, T_f, and C, and finally, as a numeric answer (in Joules).