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1) Interesting part: The words talk about energy, which is correct ...
so make the symbols agree with the words. Just get rid of every mention
of "Q" and "d'", and then the rest is correct. Specifically:
"you heat the water, d'Q = CvdT"becomes "you heat the water, dE = Cv dT".
It's just that simple. Thermodynamics is not nearly so complicated and
ugly as the thermo books make it seem.