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
Famous quotes containing the words mathematical and/or development:
“It is by a mathematical point only that we are wise, as the sailor or the fugitive slave keeps the polestar in his eye; but that is sufficient guidance for all our life. We may not arrive at our port within a calculable period, but we would preserve the true course.”
—Henry David Thoreau (1817–1862)
“I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori. Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction.”
—Gottlob Frege (1848–1925)