Helmholtz Free Energy - Mathematical Development

Mathematical Development

From the first law of thermodynamics we have

,

where is the internal energy, is the energy added by heating and is the work done by the system. From the second law of thermodynamics, for a reversible process we may say that . Also, in case of a reversible change, the work done can be expressed as

Applying the product rule for differentiation to d(TS) = TdS + SdT, we have:

,

and:

The definition of A = U - TS enables to rewrite this as

This relation is also valid for a process that is not reversible because A is a thermodynamic function of state.

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