Free entropy

Thermodynamics

The classical Carnot heat engine

Branches

* Classical
* Statistical
* Chemical
* Quantum thermodynamics

* Equilibrium / Non-equilibrium

Laws

* Zeroth
* First
* Second
* Third

Systems

|  State

* Equation of state
* Ideal gas
* Real gas
* State of matter
* Equilibrium
* Control volume
* Instruments

Processes

* Isobaric
* Isochoric
* Isothermal
* Adiabatic
* Isentropic
* Isenthalpic
* Quasistatic
* Polytropic
* Free expansion
* Reversibility
* Irreversibility
* Endoreversibility

Cycles

* Heat engines
* Heat pumps
* Thermal efficiency

System properties

Note: Conjugate variables in italics

* Property diagrams
* Intensive and extensive properties

Process functions

* Work
* Heat

Functions of state

* Temperature / Entropy (introduction)
* Pressure / Volume
* Chemical potential / Particle number
* Vapor quality
* Reduced properties

Material properties

* Property databases

| Specific heat capacity | [math]\displaystyle{ c= }[/math] | | [math]\displaystyle{ T }[/math]| [math]\displaystyle{ \partial S }[/math]

|

[math]\displaystyle{ N }[/math]| [math]\displaystyle{ \partial T }[/math]
Compressibility | [math]\displaystyle{ \beta=- }[/math] | | [math]\displaystyle{ 1 }[/math]| [math]\displaystyle{ \partial V }[/math]

|

[math]\displaystyle{ V }[/math]| [math]\displaystyle{ \partial p }[/math]
Thermal expansion | [math]\displaystyle{ \alpha= }[/math] | | [math]\displaystyle{ 1 }[/math]| [math]\displaystyle{ \partial V }[/math]

|

[math]\displaystyle{ V }[/math]| [math]\displaystyle{ \partial T }[/math]

Equations

* Carnot's theorem
* Clausius theorem
* Fundamental relation
* Ideal gas law

* Maxwell relations
* Onsager reciprocal relations
* Bridgman's equations
* Table of thermodynamic equations

Potentials

* Free energy
* Free entropy

* Internal energy
[math]\displaystyle{ U(S,V) }[/math]
* Enthalpy
[math]\displaystyle{ H(S,p)=U+pV }[/math]
* Helmholtz free energy
[math]\displaystyle{ A(T,V)=U-TS }[/math]
* Gibbs free energy
[math]\displaystyle{ G(T,p)=H-TS }[/math]

* History
* Culture

|  History

* General
* Entropy
* Gas laws

* "Perpetual motion" machines

Philosophy

* Entropy and time
* Entropy and life
* Brownian ratchet
* Maxwell's demon
* Heat death paradox
* Loschmidt's paradox
* Synergetics

Theories

* Caloric theory
* Theory of heat

* Vis viva ("living force")
* Mechanical equivalent of heat
* Motive power

Key publications

* "An Experimental Enquiry
Concerning ... Heat"

* "On the Equilibrium of
Heterogeneous Substances"

* "Reflections on the
Motive Power of Fire"

Timelines

* Thermodynamics
* Heat engines

* Art
* Education

* Maxwell's thermodynamic surface
* Entropy as energy dispersal

Scientists

* Bernoulli
* Boltzmann
* Carnot
* Clapeyron
* Clausius
* Carathéodory
* Duhem
* Gibbs
* von Helmholtz
* Joule
* Maxwell
* von Mayer
* Onsager
* Rankine
* Smeaton
* Stahl
* Thompson
* Thomson
* van der Waals
* Waterston

* Book
* Category

* v
* t
* e

A thermodynamic free entropy is an entropic thermodynamic potential analogous to the free energy. Also known 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. The Onsager reciprocal relations in particular, are developed in terms of entropic potentials. In mathematics, free entropy means something quite different: it is a generalization of entropy defined in the subject of free probability.

A free entropy is generated by a Legendre transformation of the entropy. The different potentials correspond to different constraints to which the system may be subjected.

## Contents

* 1 Examples
* 2 Dependence of the potentials on the natural variables
* 2.1 Entropy
* 2.2 Massieu potential / Helmholtz free entropy
* 2.3 Planck potential / Gibbs free entropy
* 3 References
* 4 Bibliography

## Examples

The most common examples are:

Name | Function | Alt. function | Natural variables

|
|
|

Entropy | [math]\displaystyle{ S = \frac {1}{T} U + \frac {P}{T} V - \sum_{i=1}^s \frac {\mu_i}{T} N_i \, }[/math] |  | [math]\displaystyle{ ~~~~~U,V,\\{N_i\\}\, }[/math]
Massieu potential \ Helmholtz free entropy  | [math]\displaystyle{ \Phi =S-\frac{1}{T} U }[/math] | [math]\displaystyle{ = - \frac {A}{T} }[/math] | [math]\displaystyle{ ~~~~~\frac {1}{T},V,\\{N_i\\}\, }[/math]
Planck potential \ Gibbs free entropy  | [math]\displaystyle{ \Xi=\Phi -\frac{P}{T} V }[/math] | [math]\displaystyle{ = - \frac{G}{T} }[/math] | [math]\displaystyle{ ~~~~~\frac{1}{T},\frac{P}{T},\\{N_i\\}\, }[/math]

where

[math]\displaystyle{ S }[/math] is entropy
[math]\displaystyle{ \Phi }[/math] is the Massieu potential[1][2]
[math]\displaystyle{ \Xi }[/math] is the Planck potential[1]
[math]\displaystyle{ U }[/math] is internal energy

[math]\displaystyle{ T }[/math] is temperature
[math]\displaystyle{ P }[/math] is pressure
[math]\displaystyle{ V }[/math] is volume
[math]\displaystyle{ A }[/math] is Helmholtz free energy

[math]\displaystyle{ G }[/math] is Gibbs free energy
[math]\displaystyle{ N_i }[/math] is number of particles (or number of moles) composing the i-th chemical component
[math]\displaystyle{ \mu_i }[/math] is the chemical potential of the i-th chemical component
[math]\displaystyle{ s }[/math] is the total number of components
[math]\displaystyle{ i }[/math] is the [math]\displaystyle{ i }[/math]th components.

|
|

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 [math]\displaystyle{ \psi }[/math], used by both Planck and Schrödinger. (Note that Gibbs used [math]\displaystyle{ \psi }[/math] to denote the free energy.) Free entropies where invented by French engineer François Massieu in 1869, and actually predate Gibbs's free energy (1875).

## Dependence of the potentials on the natural variables

### Entropy

[math]\displaystyle{ S = S(U,V,\\{N_i\\}) }[/math]

By the definition of a total differential,

[math]\displaystyle{ d S = \frac {\partial S} {\partial U} d U + \frac {\partial S} {\partial V} d V + \sum_{i=1}^s \frac {\partial S} {\partial N_i} d N_i. }[/math]

From the equations of state,

[math]\displaystyle{ d S = \frac{1}{T}dU+\frac{P}{T}dV + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i . }[/math]

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

[math]\displaystyle{ S = \frac{U}{T}+\frac{P V}{T} + \sum_{i=1}^s \left(- \frac{\mu_i N}{T}\right). }[/math]

### Massieu potential / Helmholtz free entropy

[math]\displaystyle{ \Phi = S - \frac {U}{T} }[/math]
[math]\displaystyle{ \Phi = \frac{U}{T}+\frac{P V}{T} + \sum_{i=1}^s \left(- \frac{\mu_i N}{T}\right) - \frac {U}{T} }[/math]
[math]\displaystyle{ \Phi = \frac{P V}{T} + \sum_{i=1}^s \left(- \frac{\mu_i N}{T}\right) }[/math]

Starting over at the definition of [math]\displaystyle{ \Phi }[/math] and taking the total differential, we have via a Legendre transform (and the chain rule)

[math]\displaystyle{ d \Phi = d S - \frac {1} {T} dU - U d \frac {1} {T} , }[/math]
[math]\displaystyle{ d \Phi = \frac{1}{T}dU + \frac{P}{T}dV + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i - \frac {1} {T} dU - U d \frac {1} {T}, }[/math]
[math]\displaystyle{ d \Phi = - U d \frac {1} {T}+\frac{P}{T}dV + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i. }[/math]

The above differentials are not all of extensive variables, so the equation may not be directly integrated. From [math]\displaystyle{ d \Phi }[/math] we see that

[math]\displaystyle{ \Phi = \Phi(\frac {1}{T},V, \\{N_i\\}) . }[/math]

If reciprocal variables are not desired,[3]:222

[math]\displaystyle{ d \Phi = d S - \frac {T d U - U d T} {T^2} , }[/math]
[math]\displaystyle{ d \Phi = d S - \frac {1} {T} d U + \frac {U} {T^2} d T , }[/math]
[math]\displaystyle{ d \Phi = \frac{1}{T}dU + \frac{P}{T}dV + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i - \frac {1} {T} d U + \frac {U} {T^2} d T, }[/math]
[math]\displaystyle{ d \Phi = \frac {U} {T^2} d T + \frac{P}{T}dV + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i , }[/math]
[math]\displaystyle{ \Phi = \Phi(T,V,\\{N_i\\}) . }[/math]

### Planck potential / Gibbs free entropy

[math]\displaystyle{ \Xi = \Phi -\frac{P V}{T} }[/math]
[math]\displaystyle{ \Xi = \frac{P V}{T} + \sum_{i=1}^s \left(- \frac{\mu_i N}{T}\right) -\frac{P V}{T} }[/math]
[math]\displaystyle{ \Xi = \sum_{i=1}^s \left(- \frac{\mu_i N}{T}\right) }[/math]

Starting over at the definition of [math]\displaystyle{ \Xi }[/math] and taking the total differential, we have via a Legendre transform (and the chain rule)

[math]\displaystyle{ d \Xi = d \Phi - \frac{P}{T} d V - V d \frac{P}{T} }[/math]
[math]\displaystyle{ d \Xi = - U d \frac {2} {T} + \frac{P}{T}dV + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i - \frac{P}{T} d V - V d \frac{P}{T} }[/math]
[math]\displaystyle{ d \Xi = - U d \frac {1} {T} - V d \frac{P}{T} + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i. }[/math]

The above differentials are not all of extensive variables, so the equation may not be directly integrated. From [math]\displaystyle{ d \Xi }[/math] we see that

[math]\displaystyle{ \Xi = \Xi \left(\frac {1}{T}, \frac {P}{T}, \\{N_i\\} \right) . }[/math]

If reciprocal variables are not desired,[3]:222

[math]\displaystyle{ d \Xi = d \Phi - \frac{T (P d V + V d P) - P V d T}{T^2} , }[/math]
[math]\displaystyle{ d \Xi = d \Phi - \frac{P}{T} d V - \frac {V}{T} d P + \frac {P V}{T^2} d T , }[/math]
[math]\displaystyle{ d \Xi = \frac {U} {T^2} d T + \frac{P}{T}dV + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i - \frac{P}{T} d V - \frac {V}{T} d P + \frac {P V}{T^2} d T , }[/math]
[math]\displaystyle{ d \Xi = \frac {U + P V} {T^2} d T - \frac {V}{T} d P + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i , }[/math]
[math]\displaystyle{ \Xi = \Xi(T,P,\\{N_i\\}) . }[/math]

## References

1. ↑ 1.0 1.1 Antoni Planes; Eduard Vives (2000-10-24). "Entropic variables and Massieu-Planck functions". Entropic Formulation of Statistical Mechanics. Universitat de Barcelona. http://www.ecm.ub.es/condensed/eduard/papers/massieu/node2.html.
2. ↑ T. Wada; A.M. Scarfone (December 2004). "Connections between Tsallis' formalisms employing the standard linear average energy and ones employing the normalized q-average energy". Physics Letters A 335 (5–6): 351–362. doi:10.1016/j.physleta.2004.12.054. Bibcode: 2005PhLA..335..351W.
3. ↑ 3.0 3.1 The Collected Papers of Peter J. W. Debye. New York, New York: Interscience Publishers, Inc.. 1954.

## Bibliography

* Massieu, M.F. (1869). Compt. Rend. 69. pp. 1057.

* Thermodynamics and an Introduction to Thermostatistics (2nd ed.). New York: John Wiley & Sons. ISBN 0-471-86256-8.

0.00

(0 votes)

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Free_entropy&oldid=125723"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template