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.
Read more about this topic: Helmholtz Free Energy
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“The most distinct and beautiful statement of any truth must take at last the mathematical form.”
—Henry David Thoreau (1817–1862)
“The work of adult life is not easy. As in childhood, each step presents not only new tasks of development but requires a letting go of the techniques that worked before. With each passage some magic must be given up, some cherished illusion of safety and comfortably familiar sense of self must be cast off, to allow for the greater expansion of our distinctiveness.”
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