Free entropy

{{see also|List of thermodynamic properties}}

The most common examples are:

class="wikitable"

Name |Function |Alt. function |Natural variables

Entropy |$dS\; =\; \backslash frac\; \{1\}\{T\}\; dU\; +\; \backslash frac\; \{P\}\{T\}\; dV\; -\; \backslash sum\_\{i=1\}^s\; \backslash frac\; \{\backslash mu\_i\}\{T\}\; dN\_i\; \backslash ,$ | |align="center"|$~~~~~U,V,\backslash \{N\_i\backslash \}\backslash ,$

Massieu potential \ Helmholtz free entropy |$\backslash Phi\; =S-\backslash frac\{1\}\{T\}\; U$ |$=\; -\; \backslash frac\; \{A\}\{T\}$ |align="center"|$~~~~~\backslash frac\; \{1\}\{T\},V,\backslash \{N\_i\backslash \}\backslash ,$

Planck potential \ Gibbs free entropy |$\backslash Xi=\backslash Phi\; -\backslash frac\{P\}\{T\}\; V$ |$=\; -\; \backslash frac\{G\}\{T\}$ |align="center"|$~~~~~\backslash frac\{1\}\{T\},\backslash frac\{P\}\{T\},\backslash \{N\_i\backslash \}\backslash ,$

where

{{Col-begin}}

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::$S$ is entropy

::$\backslash Phi$ is the Massieu potential{{cite web |author=Antoni Planes |author2=Eduard Vives |date=2000-10-24 |publisher=Universitat de Barcelona |url=http://www.ecm.ub.es/condensed/eduard/papers/massieu/node2.html |title=Entropic variables and Massieu-Planck functions |access-date=2007-09-18 |work=Entropic Formulation of Statistical Mechanics |archive-date=2008-10-11 |archive-url=https://web.archive.org/web/20081011011717/http://www.ecm.ub.es/condensed/eduard/papers/massieu/node2.html |url-status=dead }}{{cite journal |author=T. Wada |author2=A.M. Scarfone |date=December 2004 |title=Connections between Tsallis' formalisms employing the standard linear average energy and ones employing the normalized q-average energy |journal=Physics Letters A |volume=335 |issue=5–6 |pages=351–362 |doi=10.1016/j.physleta.2004.12.054 |arxiv=cond-mat/0410527|bibcode = 2005PhLA..335..351W |s2cid=17101164 }}

::$\backslash Xi$ is the Planck potential

::$U$ is internal energy

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::$T$ is temperature

::$P$ is pressure

::$V$ is volume

::$A$ is Helmholtz free energy

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::$G$ is Gibbs free energy

::$N\_i$ is number of particles (or number of moles) composing the i-th chemical component

::$\backslash mu\_i$ is the chemical potential of the i-th chemical component

::$s$ is the total number of components

::$i$ is the $i$^th components.

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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 $\backslash psi$, used by both Planck and Schrödinger. (Note that Gibbs used $\backslash psi$ to denote the free energy.) Free entropies were invented by French engineer François Massieu in 1869, and actually predate Gibbs's free energy (1875).

:$S\; =\; S(U,V,\backslash \{N\_i\backslash \})$

By the definition of a total differential,

:$d\; S\; =\; \backslash frac\; \{\backslash partial\; S\}\; \{\backslash partial\; U\}\; d\; U\; +\; \backslash frac\; \{\backslash partial\; S\}\; \{\backslash partial\; V\}\; d\; V\; +\; \backslash sum\_\{i=1\}^s\; \backslash frac\; \{\backslash partial\; S\}\; \{\backslash partial\; N\_i\}\; d\; N\_i\; .$

From the equations of state,

:$d\; S\; =\; \backslash frac\{1\}\{T\}dU+\backslash frac\{P\}\{T\}dV\; +\; \backslash sum\_\{i=1\}^s\; \backslash left(-\; \backslash frac\{\backslash mu\_i\}\{T\}\backslash right)\; d\; N\_i\; .$

The differentials in the above equation are all of extensive variables, so they may be integrated to yield

:$S\; =\; \backslash frac\{U\}\{T\}+\backslash frac\{P\; V\}\{T\}\; +\; \backslash sum\_\{i=1\}^s\; \backslash left(-\; \backslash frac\{\backslash mu\_i\; N\}\{T\}\backslash right)\; +\; \backslash textrm\{constant\}.$

:$\backslash Phi\; =\; S\; -\; \backslash frac\; \{U\}\{T\}$

:$\backslash Phi\; =\; \backslash frac\{U\}\{T\}+\backslash frac\{P\; V\}\{T\}\; +\; \backslash sum\_\{i=1\}^s\; \backslash left(-\; \backslash frac\{\backslash mu\_i\; N\}\{T\}\backslash right)\; -\; \backslash frac\; \{U\}\{T\}$

:$\backslash Phi\; =\; \backslash frac\{P\; V\}\{T\}\; +\; \backslash sum\_\{i=1\}^s\; \backslash left(-\; \backslash frac\{\backslash mu\_i\; N\}\{T\}\backslash right)$

Starting over at the definition of $\backslash Phi$ and taking the total differential, we have via a Legendre transform (and the chain rule)

:$d\; \backslash Phi\; =\; d\; S\; -\; \backslash frac\; \{1\}\; \{T\}\; dU\; -\; U\; d\; \backslash frac\; \{1\}\; \{T\}\; ,$

:$d\; \backslash Phi\; =\; \backslash frac\{1\}\{T\}dU\; +\; \backslash frac\{P\}\{T\}dV\; +\; \backslash sum\_\{i=1\}^s\; \backslash left(-\; \backslash frac\{\backslash mu\_i\}\{T\}\backslash right)\; d\; N\_i\; -\; \backslash frac\; \{1\}\; \{T\}\; dU\; -\; U\; d\; \backslash frac\; \{1\}\; \{T\},$

:$d\; \backslash Phi\; =\; -\; U\; d\; \backslash frac\; \{1\}\; \{T\}+\backslash frac\{P\}\{T\}dV\; +\; \backslash sum\_\{i=1\}^s\; \backslash left(-\; \backslash frac\{\backslash mu\_i\}\{T\}\backslash right)\; d\; N\_i.$

The above differentials are not all of extensive variables, so the equation may not be directly integrated. From $d\; \backslash Phi$ we see that

:$\backslash Phi\; =\; \backslash Phi(\backslash frac\; \{1\}\{T\},V,\; \backslash \{N\_i\backslash \})\; .$

If reciprocal variables are not desired,

{{cite book

| title=The Collected Papers of Peter J. W. Debye

| publisher=Interscience Publishers, Inc.

| place=New York, New York

| year=1954

}}

{{rp|222}}

:$d\; \backslash Phi\; =\; d\; S\; -\; \backslash frac\; \{T\; d\; U\; -\; U\; d\; T\}\; \{T^2\}\; ,$

:$d\; \backslash Phi\; =\; d\; S\; -\; \backslash frac\; \{1\}\; \{T\}\; d\; U\; +\; \backslash frac\; \{U\}\; \{T^2\}\; d\; T\; ,$

:$d\; \backslash Phi\; =\; \backslash frac\{1\}\{T\}dU\; +\; \backslash frac\{P\}\{T\}dV\; +\; \backslash sum\_\{i=1\}^s\; \backslash left(-\; \backslash frac\{\backslash mu\_i\}\{T\}\backslash right)\; d\; N\_i\; -\; \backslash frac\; \{1\}\; \{T\}\; d\; U\; +\; \backslash frac\; \{U\}\; \{T^2\}\; d\; T,$

:$d\; \backslash Phi\; =\; \backslash frac\; \{U\}\; \{T^2\}\; d\; T\; +\; \backslash frac\{P\}\{T\}dV\; +\; \backslash sum\_\{i=1\}^s\; \backslash left(-\; \backslash frac\{\backslash mu\_i\}\{T\}\backslash right)\; d\; N\_i\; ,$

:$\backslash Phi\; =\; \backslash Phi(T,V,\backslash \{N\_i\backslash \})\; .$

:$\backslash Xi\; =\; \backslash Phi\; -\backslash frac\{P\; V\}\{T\}$

:$\backslash Xi\; =\; \backslash frac\{P\; V\}\{T\}\; +\; \backslash sum\_\{i=1\}^s\; \backslash left(-\; \backslash frac\{\backslash mu\_i\; N\}\{T\}\backslash right)\; -\backslash frac\{P\; V\}\{T\}$

:$\backslash Xi\; =\; \backslash sum\_\{i=1\}^s\; \backslash left(-\; \backslash frac\{\backslash mu\_i\; N\}\{T\}\backslash right)$

Starting over at the definition of $\backslash Xi$ and taking the total differential, we have via a Legendre transform (and the chain rule)

:$d\; \backslash Xi\; =\; d\; \backslash Phi\; -\; \backslash frac\{P\}\{T\}\; d\; V\; -\; V\; d\; \backslash frac\{P\}\{T\}$

:$d\; \backslash Xi\; =\; -\; U\; d\; \backslash frac\; \{2\}\; \{T\}\; +\; \backslash frac\{P\}\{T\}dV\; +\; \backslash sum\_\{i=1\}^s\; \backslash left(-\; \backslash frac\{\backslash mu\_i\}\{T\}\backslash right)\; d\; N\_i\; -\; \backslash frac\{P\}\{T\}\; d\; V\; -\; V\; d\; \backslash frac\{P\}\{T\}$

:$d\; \backslash Xi\; =\; -\; U\; d\; \backslash frac\; \{1\}\; \{T\}\; -\; V\; d\; \backslash frac\{P\}\{T\}\; +\; \backslash sum\_\{i=1\}^s\; \backslash left(-\; \backslash frac\{\backslash mu\_i\}\{T\}\backslash right)\; d\; N\_i.$

The above differentials are not all of extensive variables, so the equation may not be directly integrated. From $d\; \backslash Xi$ we see that

:$\backslash Xi\; =\; \backslash Xi\; \backslash left(\backslash frac\; \{1\}\{T\},\; \backslash frac\; \{P\}\{T\},\; \backslash \{N\_i\backslash \}\; \backslash right)\; .$

If reciprocal variables are not desired,{{rp|222}}

:$d\; \backslash Xi\; =\; d\; \backslash Phi\; -\; \backslash frac\{T\; (P\; d\; V\; +\; V\; d\; P)\; -\; P\; V\; d\; T\}\{T^2\}\; ,$

:$d\; \backslash Xi\; =\; d\; \backslash Phi\; -\; \backslash frac\{P\}\{T\}\; d\; V\; -\; \backslash frac\; \{V\}\{T\}\; d\; P\; +\; \backslash frac\; \{P\; V\}\{T^2\}\; d\; T\; ,$

:$d\; \backslash Xi\; =\; \backslash frac\; \{U\}\; \{T^2\}\; d\; T\; +\; \backslash frac\{P\}\{T\}dV\; +\; \backslash sum\_\{i=1\}^s\; \backslash left(-\; \backslash frac\{\backslash mu\_i\}\{T\}\backslash right)\; d\; N\_i\; -\; \backslash frac\{P\}\{T\}\; d\; V\; -\; \backslash frac\; \{V\}\{T\}\; d\; P\; +\; \backslash frac\; \{P\; V\}\{T^2\}\; d\; T\; ,$

:$d\; \backslash Xi\; =\; \backslash frac\; \{U\; +\; P\; V\}\; \{T^2\}\; d\; T\; -\; \backslash frac\; \{V\}\{T\}\; d\; P\; +\; \backslash sum\_\{i=1\}^s\; \backslash left(-\; \backslash frac\{\backslash mu\_i\}\{T\}\backslash right)\; d\; N\_i\; ,$

:$\backslash Xi\; =\; \backslash Xi(T,P,\backslash \{N\_i\backslash \})\; .$

• {{cite journal

| first =M.F. |last = Massieu|year=1869 |title= Compt. Rend

| volume=69

| issue= 858

| pages= 1057 }}

• {{cite book

| first = Herbert B. | last = Callen | author-link = Herbert Callen | year = 1985

| title = Thermodynamics and an Introduction to Thermostatistics | edition = 2nd

| publisher = John Wiley & Sons | location = New York | isbn = 0-471-86256-8 }}

Category:Thermodynamic entropy