The reversible cooling of coffee

If coffee cannot cool spontaneously without increasing the entropy of the universe, a natural question arises: is there a way to cool it without waste? The answer is yes, and it is rich in practical consequences: just insert, between the coffee and the atmosphere, a cyclic machine that, instead of letting the heat disperse uselessly, extracts useful work from it.

The idea is that an irreversible transformation “throws away” an opportunity. When the coffee cools directly in the air, the total entropy grows by a few J/K and that heat simply warms the atmosphere a little. If instead we make the heat pass through a heat engine, part of it can be converted into mechanical work before reaching the environment. The ideal limit is reached when the machine operates reversibly, that is, by imposing $\Delta S_\text{tot} = 0$: this is the condition that wastes zero entropy and therefore produces the maximum possible work.

The quantitative reasoning (worked out in the exercise maximum work from the cooling coffee) is: the machine completes a cycle, so its entropy does not change ($\Delta S_\text{macchina} = 0$); we impose $\Delta S_\text{tot} = 0$ to find the heat given to the atmosphere; finally the first law gives the work as the difference between the heat that left the coffee and that given to the air. For the cup in the example this gives a maximum work of about $2256\;\text{J}$. Any real machine, having $\Delta S_\text{tot} > 0$, will produce less work than this.

Where this idea leads

In the next chapter we will use exactly the condition $\Delta S_\text{tot} = 0$ to define the reversible transformations of a gas — isothermal, isobaric, isochoric, adiabatic — and to derive their formulas. Only these transformations have a well-defined curve in the $P$-$V$ diagram. By combining more than one of them into cycles we will build heat engines and refrigerators, and we will find again that the maximum efficiency is that of the reversible Carnot cycle.

Collegamenti

Argomenti: Entropy and the second law Concetti: Second law of thermodynamics · Entropy · Efficiency Competenze: Entropy balance Oggetti: Heat engine

Esercizi collegati: Cooled iron: maximum work · Worked exercise — maximum work from the cooling coffee · Carnot engine: ΔS of the universe