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My professor in thermodynamics showed us the proof of the Carnot theory using integrals and a temp vs. entropy plot for a heat engine cycle. We haven't actually learned about entropy yet, so can someone help me understand how this translates into the coefficient of performance β for a heat pump? We were given the rule that β≤β(Carnot) for heat pumps and refrigerators, but I can't prove this is true on my own. Any explanation is appreciated :)

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# Prove:β (heat pump) is always less or equal to β(Carnot HP)

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