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Maximum amount of work that can be extracted from two objects at different temperatures

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Consider a system consisting of two objects A and B, when A and B have initial temperature T_A and T_B, and heat capacities C_VA and C_VB independent of temperature. This system is put to work so that it can provide work W, until both reaches a final temperature T_F, where T_A>T_F>T_B.

(a) Find the maximum mechanical work that can be extracted from this system.

(b) Obtain the equilibrium temperature reached after all the work in (a) was extracted.

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We can find the maximum amount of work that can be performed by considering the entropy of the system. If a system is not in thermal equilibrium, then allowing it to get into thermal equilibrium will lead to an entropy increase. If we extract work from that process, then the more work we extract from the system, the less the entropy will increase. This is because after we've extracted work, we are free to put back that work into the system in the form of heat. We would end up at the same final state where we would have arrived at, had we not extracted any work from the system. Obviously, the more work we put back into the system in the form of heat, the more the entropy will increase. Since we must end up at that same state which has some fixed entropy, this means that the state of the system after the work has been extracted will have a an entropy that will be lower if more work is extracted. The theoretical limit for the maximum amount of work that can be extracted ...

Solution Summary

We give a detailed explanation of the relevant thermodynamics in this problem. A detailed solution is provided based on that explanation.