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Showing mathematical relations for coefficient of performance of refrigeratror

This page shows efficiency of a carnot engine.

http://www.oberlin.edu/physics/dstyer/P111/Carnot.pdf

The refrigerator uses the same process only it run backwards. By using the same recipe as in the link above I want a mathematical derivation of the coefficient of performance of a refrigerator. A little additional information is in the attachment for my question. I am only interested in a mathematical explanation as that is what would work for me. Thank you! I also need the answer to be written in a computer document.

See the attached file.

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Solution Preview

The solution is attached below in two files. the files are identical in content, only differ in format. The first is in MS Word format, while the other is in Adobe pdf format. Therefore you can choose the format that is most suitable to you.

A reverse Carnot engine works as following:
From point 1 to point 2 an isotherm expansion at temperature
From point 2 to point 3 an adiabatic compression.
From point 3 to point 4 an isotherm compression at temperature
From point 4 to point 1 an adiabatic expansion back to the original cycle.

Process 1 to 2: isotherm.
Since the temperature ...

Solution Summary

The solution shows the mathematical relations for coefficient of performance of a refrigerator.

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