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Show that the coefficients of performance of refrigerators and heat pumps are related by $C O P_{\text { ref }}=C O P_{\text { hp }}-1$Start with the definitions of the $C O P$ s and the conservation of energy relationship between $Q_{\mathrm{h}}, Q_{\mathrm{c}},$ and $W$

$\mathrm{COP}_{\mathrm{ref}}=\mathrm{COP}_{\mathrm{hp}}-1$

Physics 101 Mechanics

Chapter 15

Thermodynamics

Thermal Properties of Matter

The First Law of Thermodynamics

The Second Law of Thermodynamics

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This is problem 46 Chapter 15 on. We're supposed to show that the coefficients of performance of refrigerators and he pumps are related by, uh, the coefficients of rough and coefficients. They're equals the coefficients of HP minus one. And so now we can go ahead and use this equation here and manipulated to actually show that. So the first thing we can actually do is go ahead and divide the entire thing by W we're just gonna be qh over w minus. Q c over w equals one. Um and then from here, we know that c o P. H p is equal to Q H over W. And so because of that, um, and also we know that the c o. P. Uh, refrigerator is equal to que si over w um we can go ahead and substitute these two things in, uh and so this is gonna be C o p H. P minus C o p are yes, is equal to one. And then we can go ahead and add, uh, Seop, are you on this side? Oh, yes. And then subtract one at the same time on both sides. Uh, and so this is gonna isolate c o P R a f is equal to C o P H B minus one on this is what we were trying to confirm.

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