A manufacturer claims a refrigerator X which

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · The laws of thermodynamics · The laws of thermodynamics

Thermal Physics, Ch Heat engines, refrigerators and the Carnot cycle

Problem

A manufacturer claims a refrigerator X which, working in a cycle between a cold store at

250\,\mathrm{K}

and a room at

300\,\mathrm{K}

, has a coefficient of performance for cooling of

6.50

. A reversible machine R is available between the same two reservoirs; it can be run either way round. Both reservoirs hold their temperatures, and all heats and works are magnitudes. ((a)) Find the coefficient of performance for cooling of R. ((b)) Take

100\,\mathrm{J}

of work delivered to X per cycle of the combination. Find the heat X removes from the store and the heat it delivers to the room. ((c)) R is now run forwards as an engine, arranged to deliver exactly the

100\,\mathrm{J}

of work that X consumes. Find the heat R takes from the room and the heat it rejects to the store. ((d)) Add up what each reservoir gains over one cycle of the pair, and use the Clausius statement to conclude a bound on the coefficient of performance of any refrigerator between these two temperatures. ((e)) Explain briefly, by running two reversible refrigerators each way round against one another, why every reversible machine between these two reservoirs has the same coefficient of performance, so that the ratio

Q_h/Q_c

depends on the two temperatures alone.

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