Entropy in the Real World: Refrigerators — Question 9

Question 9

A reversible engine can power cooling without external electrical work

A reversible engine operates between 600 K and 300 K. All its work drives a reversible refrigerator lifting heat from 250 K to 300 K. The engine receives 1000 J per cycle; the devices are scaled to exchange equal work.

1. Find engine work, refrigerator cooling, and total heat delivered to the 300 K reservoir.

2. Check the composite energy and entropy balances. Explain why cooling can exceed the high-temperature heat input without violating either law.

Question 9 - Solution

A reversible engine can power cooling without external electrical work

(a) Method and calculation

$$\eta_E=1-300/600=.500\ \Rightarrow\ W_E=500\,\mathrm J.$$$$K_R=250/(300-250)=5\ \Rightarrow\ Q_{c,R}=2500\,\mathrm J.$$$$Q_{300}=(1000-500)+(2500+500)=3500\,\mathrm J.$$

(b) Interpretation and further calculation

$$Q_{300}=Q_{600}+Q_{250}=1000+2500,\quad W_{\mathrm{external}}=0.$$$$\Delta S_{\mathrm{reservoirs}}=3500/300-1000/600-2500/250=0.$$The cooling heat comes from the 250 K reservoir, while high-temperature heat supplies the ability to drive that transfer.

Physical check

Internal work cancels only after both complete device balances are included.