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267.


Problem 10.12 (Zemansky)

A body of finite mass is originally at a temperature , which is higher than that of a reservoir at the temperature . Suppose an engine operates in a cycle between the body and the reservoir until it lowers the temperature of the body from to , thus extracting heat Q from the body. If the engine does work W, then it will reject heat Q-W to the reservoir at . Applying the entropy principle, we have to prove that the maximum work obtainable from the engine is

where is the entropy decrease of the body.


Solution:           Click For PDF Version

The entropy principle is that the change of entropy of the universe as a result of any kind of process is greater than or equal to zero. That is

It is given that a body of finite mass is originally at a temperature , which is higher than that of a reservoir at the temperature . An engine operates in a cycle between the body and the reservoir until it lowers the temperature of the body from to , thus extracting heat Q from the body. If the engine does work W, then it will reject heat Q-W to the reservoir at . Let the change in entropy of the body be

.

As we are considering that the body is connected to a reversible cyclic engine, the change in entropy of the engine is zero.

As an amount of heat Q-W is added to the reservoir, which is at temperature , the change in entropy of the reservoir will be

The change in entropy of the universe will be

According to the entropy principle

Or

Therefore,