Free entropy

Short description: Thermodynamic potential of entropy, analogous to the free energy

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 | c= | | T| ∂S

|

N| ∂T
Compressibility | β=− | | 1| ∂V

|

V| ∂p
Thermal expansion | α= | | 1| ∂V

|

V| ∂T

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
U(S,V)
* Enthalpy
H(S,p)=U+pV
* Helmholtz free energy
A(T,V)=U−TS
* Gibbs free energy
G(T,p)=H−TS

* 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
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* Clapeyron
* Clausius
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* Gibbs
* von Helmholtz
* Joule
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* Onsager
* Rankine
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* Stahl
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* Thomson
* van der Waals
* Waterston

* Book
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* 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 | S=1TU+PTV−∑i=1sμiTNi |  | U,V,{Ni}
Massieu potential \ Helmholtz free entropy  | Φ=S−1TU | =−AT | 1T,V,{Ni}
Planck potential \ Gibbs free entropy  | Ξ=Φ−PTV | =−GT | 1T,PT,{Ni}

where

S is entropy
Φ is the Massieu potential[1][2]
Ξ is the Planck potential[1]
U is internal energy

T is temperature
P is pressure
V is volume
A is Helmholtz free energy

G is Gibbs free energy
Ni is number of particles (or number of moles) composing the i-th chemical component
μi is the chemical potential of the i-th chemical component
s is the total number of components
i is the ith 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 ψ, used by both Planck and Schrödinger. (Note that Gibbs used ψ 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

S=S(U,V,{Ni})

By the definition of a total differential,

dS=∂S∂UdU+∂S∂VdV+∑i=1s∂S∂NidNi.

From the equations of state,

dS=1TdU+PTdV+∑i=1s(−μiT)dNi.

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

S=UT+PVT+∑i=1s(−μiNT).

### Massieu potential / Helmholtz free entropy

Φ=S−UT
Φ=UT+PVT+∑i=1s(−μiNT)−UT
Φ=PVT+∑i=1s(−μiNT)

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

dΦ=dS−1TdU−Ud1T,
dΦ=1TdU+PTdV+∑i=1s(−μiT)dNi−1TdU−Ud1T,
dΦ=−Ud1T+PTdV+∑i=1s(−μiT)dNi.

The above differentials are not all of extensive variables, so the equation may not be directly integrated. From dΦ we see that

Φ=Φ(1T,V,{Ni}).

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

dΦ=dS−TdU−UdTT2,
dΦ=dS−1TdU+UT2dT,
dΦ=1TdU+PTdV+∑i=1s(−μiT)dNi−1TdU+UT2dT,
dΦ=UT2dT+PTdV+∑i=1s(−μiT)dNi,
Φ=Φ(T,V,{Ni}).

### Planck potential / Gibbs free entropy

Ξ=Φ−PVT
Ξ=PVT+∑i=1s(−μiNT)−PVT
Ξ=∑i=1s(−μiNT)

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

dΞ=dΦ−PTdV−VdPT
dΞ=−Ud2T+PTdV+∑i=1s(−μiT)dNi−PTdV−VdPT
dΞ=−Ud1T−VdPT+∑i=1s(−μiT)dNi.

The above differentials are not all of extensive variables, so the equation may not be directly integrated. From dΞ we see that

Ξ=Ξ(1T,PT,{Ni}).

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

dΞ=dΦ−T(PdV+VdP)−PVdTT2,
dΞ=dΦ−PTdV−VTdP+PVT2dT,
dΞ=UT2dT+PTdV+∑i=1s(−μiT)dNi−PTdV−VTdP+PVT2dT,
dΞ=U+PVT2dT−VTdP+∑i=1s(−μiT)dNi,
Ξ=Ξ(T,P,{Ni}).

## 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.

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