The Second Law of Thermodynamics — Question 2

Question 2

A prescribed entropy decrease sets a minimum heat rejection

A closed system changes between equilibrium states with $\Delta S=-2.00$ J/K. Its only entropy exchange is heat rejected through a boundary held at 300 K. Let $Q_{\mathrm{out}}>0$ and use $\Delta S=-Q_{\mathrm{out}}/300+S_{\mathrm{gen}}$. Work transfer may occur.

1. Find the smallest allowed heat rejection.

2. If the actual rejection is 900 J, calculate entropy generation. Can 500 J be rejected for the same endpoint entropy change under these assumptions?

Question 2 - Solution

A prescribed entropy decrease sets a minimum heat rejection

(a) Method and calculation

$$S_{\mathrm{gen}}=\Delta S+Q_{\mathrm{out}}/300=-2+Q_{\mathrm{out}}/300\ge0.$$$$Q_{\mathrm{out}}\ge600\,\mathrm J.$$

(b) Interpretation and further calculation

$$Q_{\mathrm{out}}=900:\quad S_{\mathrm{gen}}=1.00\,\mathrm{J/K}.$$$$Q_{\mathrm{out}}=500:\quad S_{\mathrm{gen}}=-1/3\,\mathrm{J/K}<0.$$Additional work cannot carry away entropy, so it does not rescue the 500 J proposal unless the specified entropy-exchange assumptions change.

Physical check

Irreversibility requires more entropy export to achieve a fixed negative system entropy change.