Assuming an ideal gas, we can calculate what would happen under two types of compression: isothermal (temperature and therefore internal energy remain constant) and adiabatic (no heat). Most compressions in real life will be between one or the other extremes. We derive two simple equations relating heat and work based on the change of volume of the gas.
Playlist: • An Introduction to Thermal Physics by...
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The textbook I am using is: Schroeder, Daniel V. "An Introduction to Thermal Physics" 1st ed., Addison Wesley Longman, 2005.