Thermodynamics, Statistical Physics, and Quantum Mechanics

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The heat extracted, is then equal to the difference in initialand final

internalenergies ofthe bodies:

c) Here we may calculatethe maximum useful work by using the Carnot
efficiency of a reversibleheat engineoperatingbetween two reservoirs, one
starting at ahightemperature (100°C) and theother fixed (thelake) at
10°C. Theefficiency may bewritten for asmall heattransfer as

where theheat transferredfrom the hotreservoir equals its change in
internal energy –MCdT. We may then find the total work by integrating
dW as follows:

We may also use the method of part (b) and thefact that theentropy is
conserved.Denote themass of the hotwaterMand the lake Equating
the initial and final entropiesgives

Writing thefinaltemperatureTas where is small (it’s a big lake),
and expanding thelogarithm, we obtain

Substituting(S.4.29.10) backinto(S.4.29.9)gives

THERMODYNAMICS AND STATISTICAL PHYSICS