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For a system coupled to a work reservoir and heat bath of temperature T, show that that the work done by the system (on the work reservoirs) is bounded from above by minus the change in Helmholtz free energy. 2
We will use the equation relating Helmholtz free energy with internal energy, temperature and entropy as follows:
F = U - T.S
Where F is the Helmholtz free energy
U is the internal energy
T is the temperature
and S is the entropy
Since the temperature is maintained constant by the bath, it is an isothermal process
Given below is the detailed derivation: