The Second Law of Thermodynamics — Question 1

Question 1

Energy conservation alone cannot validate a proposed cyclic engine

A cyclic device has zero net internal-energy and entropy changes. Take heat magnitudes $Q_h,Q_c>0$ for absorption from a hot reservoir and rejection to a cold one. Then $W=Q_h-Q_c$ and $S_{\mathrm{gen}}=Q_c/T_c-Q_h/T_h\ge0$. A proposed engine absorbs 1000 J from 500 K, rejects heat to 300 K and claims 500 J of work each cycle.

1. Test the claim using both energy and entropy balances.

2. Find the maximum work and minimum rejected heat compatible with these reservoirs and this heat input.

Question 1 - Solution

Energy conservation alone cannot validate a proposed cyclic engine

(a) Method and calculation

$$Q_c=1000-500=500\,\mathrm J.$$$$S_{\mathrm{gen}}=500/300-1000/500=-1/3\,\mathrm{J/K}<0.$$The claim conserves energy but violates the entropy inequality.

(b) Interpretation and further calculation

$$Q_c\ge Q_hT_c/T_h=600\,\mathrm J.$$$$W\le1000-600=400\,\mathrm J.$$Equality requires an overall reversible device. Actual losses increase the required rejection above 600 J.

Physical check

Entropy generation must never be made negative to close an otherwise convenient energy balance.