Thermodynamic equations |
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Laws of thermodynamics |
Conjugate variables |
Thermodynamic potential
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Material properties |
Maxwell relations |
Bridgman's equations |
Exact differential |
Table of thermodynamic equations |
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A thermodynamic free entropy is an entropic thermodynamic potential analogous to the free energy. Also know as a Massieu, Planck, or Massieu-Planck potentials (or functions), or (rarely) free information. In statistical mechanics, free entropies frequently appear as the logarithm of a partition function. In mathematics, free entropy is the generalization of entropy defined in free probability.
A free entropy is generated by a Legendre transform of the entropy. The different potentials correspond to different constraints to which the system may be subjected. The most common examples are:
Name | Function | Alt. function | Natural variables |
Entropy | |||
Massieu potential \ Helmholtz free entropy | |||
Planck potential \ Gibbs free entropy |
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Note that the use of the terms "Massieu" and "Planck" for explicit Massieu-Planck potentials are somewhat obscure and ambiguous. In particular "Planck potential" has alternative meanings. The most standard notation for an entropic potential is ψ, used by both Planck and Schrödinger. (Note that Gibbs used ψ to denote the free energy.) Free entropies where invented by Massieu in 1869, and actually predate Gibb's free energy (1875).
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By the definition of a total differential,
From the equations of state,
The differentials in the above equation are all of extensive variables, so they may be integrated to yield
Starting over at the definition of Φ and taking the total differential, we have via a Legendre transform (and the chain rule)
The above differentials are not all of extensive variables, so the equation may not be directly integrated. From dΦ we see that
If reciprocal variables are not desired,:222
Starting over at the definition of Ξ and taking the total differential, we have via a Legendre transform (and the chain rule)
The above differentials are not all of extensive variables, so the equation may not be directly integrated. From dΞ we see that
If reciprocal variables are not desired,:222