Try it on paper first. A hint, the key idea and the worked solution are here when you need them.
of heat flows from a very large body at to a very large body at . Find the change of entropy of each body and the total. What would the total be if the heat went from the cold body to the hot one?
Heat taken in slowly at a steady temperature raises the entropy by . Warming from to : . A gas at a steady temperature: . Entropy depends only on the state, and the total for an isolated system never falls.
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. Hot body. .
Step 2. Cold body. .
Step 3. Total. .
Step 4. The other way. Every sign flips: the total would be , which the second law forbids.
Answer: , and ; backwards it would fall by , so it never happens by itself.
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