The Second Law of Thermodynamics — Question 4

Question 4

The cyclic condition distinguishes a forbidden engine from a work store

A proposed cyclic machine absorbs 500 J from a single 300 K reservoir and converts all of it to work, with no other effect. Compare this with an ideal descending weight delivering 500 J of mechanical work while its gravitational energy decreases by 500 J.

1. Calculate the entropy change of the machine–reservoir composite for the cyclic claim.

2. Write energy and entropy balances for the ideal weight. Explain why its work delivery does not contradict the prohibition on a single-reservoir cyclic heat engine.

Question 4 - Solution

The cyclic condition distinguishes a forbidden engine from a work store

(a) Method and calculation

$$\Delta S_{\mathrm{machine}}=0,\quad\Delta S_r=-500/300=-5/3\,\mathrm{J/K}.$$$$\Delta S_{\mathrm{composite}}<0\ \Rightarrow\text{the cyclic claim is impossible}.$$

(b) Interpretation and further calculation

$$Q_{\mathrm{weight}}=0,\quad W_{\mathrm{weight}}=500\,\mathrm J,\quad\Delta E_{\mathrm{grav}}=-500\,\mathrm J.$$$$\Delta S_{\mathrm{ideal\ work\ store}}=0.$$The weight does not return to its starting height. It consumes stored mechanical energy rather than cyclically converting heat from one reservoir into work.

Physical check

The phrase “with no other effect is essential to the single-reservoir statement.