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

by mek09e
Tags: beta, carnot, heat pumps, proof 2nd law, thermodynamics
 P: 1 Hello All! 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 :)
 PF Patron Sci Advisor P: 765 Don't you have a textbook? Basically, you start from the definition of the coefficient of performance and use the 1st law to write it in terms of ##Q_C## and ##Q_H##, the heat coming from the cold reservoir and that going to the hot reservoir, respectively. Then, you use the 2nd law to translate ##Q_C## and ##Q_H## to ##T_C## and ##T_H##. This gives you the highest ##\beta## possible according to the 2nd law. Then you prove that a Carnot cycle working between ##T_C## and ##T_H## has a value of ##\beta## that is the highest possible. Therefore, ##\beta \le \beta(\mathrm{Carnot})##.

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