Table of thermodynamic equations

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Thermodynamics

The classical Carnot heat engine

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Note: Conjugate variables in italics

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| 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
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* 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

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Common thermodynamic equations and quantities in thermodynamics, using mathematical notation, are as follows:

## Contents

* 1 Definitions
* 1.1 General basic quantities
* 1.2 General derived quantities
* 1.3 Thermal properties of matter
* 1.4 Thermal transfer
* 2 Equations
* 2.1 Thermodynamic processes
* 2.2 Kinetic theory
* 2.2.1 Ideal gas
* 2.3 Entropy
* 2.4 Statistical physics
* 2.5 Quasi-static and reversible processes
* 2.6 Thermodynamic potentials
* 2.7 Maxwell's relations
* 2.8 Quantum properties
* 3 Thermal properties of matter
* 3.1 Thermal transfer
* 3.2 Thermal efficiencies
* 4 See also
* 5 References
* 6 External links

## Definitions

Many of the definitions below are also used in the thermodynamics of chemical reactions.

### General basic quantities

Quantity (common name/s)  | (Common) symbol/s  | SI unit  | Dimension

|
|
|

Number of molecules  | N | 1  | 1
Amount of substance  | n | mol  | N
Temperature | T | K  | Θ
Heat Energy | Q, q | J  | ML2T−2
Latent heat | QL | J  | ML2T−2

### General derived quantities

Quantity (common name/s)  | (Common) symbol/s  | Defining equation  | SI unit  | Dimension

|
|
|
|

Thermodynamic beta, inverse temperature  | β | β=1/kBT | J−1 | T2M−1L−2
Thermodynamic temperature | τ | τ=kBT

τ=kB(∂U/∂S)N 1/τ=1/kB(∂S/∂U)N

| J  | ML2T−2
Entropy | S | S=−kB∑ipiln⁡pi

S=−(∂F/∂T)V,N , S=−(∂G/∂T)P,N

| J⋅K−1 | ML2T−2Θ−1
Pressure | P | P=−(∂F/∂V)T,N

P=−(∂U/∂V)S,N

| Pa  | ML−1T−2
Internal Energy  | U | U=∑ipiEi | J  | ML2T−2
Enthalpy | H | H=U+pV | J  | ML2T−2
Partition Function | Z |  | 1  | 1
Gibbs free energy | G | G=H−TS | J  | ML2T−2
Chemical potential (of component i in a mixture)  | μi | μi=(∂U/∂Ni)Nj≠i,S,V

μi=(∂F/∂Ni)T,V, where F is not proportional to N because μi depends on pressure. μi=(∂G/∂Ni)T,P, where G is proportional to N (as long as the molar ratio composition of the system remains the same) because μi depends only on temperature and pressure and composition. μi/τ=−1/kB(∂S/∂Ni)U,V

| J  | ML2T−2
Helmholtz free energy | A, F | F=U−TS | J  | ML2T−2
Landau potential, Landau free energy, Grand potential | Ω, ΦG | Ω=U−TS−μN | J  | ML2T−2
Massieu potential, Helmholtz free entropy | Φ | Φ=S−U/T | J⋅K−1 | ML2T−2Θ−1
Planck potential, Gibbs free entropy | Ξ | Ξ=Φ−pV/T | J⋅K−1 | ML2T−2Θ−1

### Thermal properties of matter

Main articles: Physics:Heat capacity and Thermal expansion

Quantity (common name/s)  | (Common) symbol/s  | Defining equation  | SI unit  | Dimension

|
|
|
|

General heat/thermal capacity  | C | C=∂Q/∂T | J⋅K−1 | ML2T−2Θ−1
Heat capacity (isobaric)  | Cp | Cp=∂H/∂T | J⋅K−1 | ML2T−2Θ−1
Specific heat capacity (isobaric)  | Cmp | Cmp=∂2Q/∂m∂T | J⋅kg−1⋅K−1 | L2T−2Θ−1
Molar specific heat capacity (isobaric)  | Cnp | Cnp=∂2Q/∂n∂T | J⋅K−1⋅mol−1 | ML2T−2Θ−1N−1
Heat capacity (isochoric/volumetric)  | CV | CV=∂U/∂T | J⋅K−1 | ML2T−2Θ−1
Specific heat capacity (isochoric)  | CmV | CmV=∂2Q/∂m∂T | J⋅kg−1⋅K−1 | L2T−2Θ−1
Molar specific heat capacity (isochoric)  | CnV | CnV=∂2Q/∂n∂T | J⋅K⋅−1 mol−1 | ML2T−2Θ−1N−1
Specific latent heat | L | L=∂Q/∂m | J⋅kg−1 | L2T−2
Ratio of isobaric to isochoric heat capacity, heat capacity ratio, adiabatic index, Laplace coefficient  | γ | γ=Cp/CV=cp/cV=Cmp/CmV | 1  | 1

### Thermal transfer

Main article: Physics:Thermal conductivity

Quantity (common name/s)  | (Common) symbol/s  | Defining equation  | SI unit  | Dimension

|
|
|
|

Temperature gradient | No standard symbol  | ∇T | K⋅m−1 | ΘL−1
Thermal conduction rate, thermal current, thermal/heat flux, thermal power transfer  | P | P=dQ/dt | W  | ML2T−3
Thermal intensity  | I | I=dP/dA | W⋅m−2 | MT−3
Thermal/heat flux density (vector analogue of thermal intensity above)  | q | Q=∬𝐪⋅d𝐒dt | W⋅m−2 | MT−3

## Equations

The equations in this article are classified by subject.

### Thermodynamic processes

Physical situation  | Equations

|

Isentropic process (adiabatic and reversible)  | Q=0,ΔU=−W

For an ideal gas
p1V1γ=p2V2γ
T1V1γ−1=T2V2γ−1
p11−γT1γ=p21−γT2γ

Isothermal process | ΔU=0,W=Q

For an ideal gas
W=kTNln⁡(V2/V1) W=nRTln⁡(V2/V1)

Isobaric process | p1 = p2, p = constant

W=pΔV,Q=ΔU+pδV

Isochoric process | V1 = V2, V = constant

W=0,Q=ΔU

Free expansion | ΔU=0
Work done by an expanding gas  | Process

W=∫V1V2pdV

Net work done in cyclic processes
W=∮cyclepdV=∮cycleΔQ

### Kinetic theory

Ideal gas equations  Physical situation  | Nomenclature  | Equations

|
|

Ideal gas law |

* p = pressure
* V = volume of container
* T = temperature
* n = amount of substance
* R = gas constant
* N = number of molecules
* k = Boltzmann constant

| pV=nRT=kTN

p1V1p2V2=n1T1n2T2=N1T1N2T2

Pressure of an ideal gas  |

* m = mass of one molecule
* Mm = molar mass

| p=Nm⟨v2⟩3V=nMm⟨v2⟩3V=13ρ⟨v2⟩

#### Ideal gas

Quantity  | General Equation  | Isobaric
Δp = 0  | Isochoric
ΔV = 0  | Isothermal
ΔT = 0  | Adiabatic
Q=0

|
|
|
|
|

Work
W | δW=−pdV | −pΔV | 0 | −nRTln⁡V2V1

−nRTln⁡P1P2

| PVγ(Vf1−γ−Vi1−γ)1−γ=CV(T2−T1)
Heat Capacity
C | (as for real gas)  | Cp=52nR
(for monatomic ideal gas)

Cp=72nR
(for diatomic ideal gas)

| CV=32nR
(for monatomic ideal gas)

CV=52nR
(for diatomic ideal gas)

|  |
Internal Energy
ΔU | ΔU=CVΔT | Q+W

Qp−pΔV | Q

CV(T2−T1) | 0
Q=−W | W
CV(T2−T1)
Enthalpy
ΔH | H=U+pV | Cp(T2−T1) | QV+VΔp | 0 | Cp(T2−T1)
Entropy
Δs | ΔS=CVln⁡T2T1+nRln⁡V2V1
ΔS=Cpln⁡T2T1−nRln⁡p2p1[1] | Cpln⁡T2T1 | CVln⁡T2T1 | nRln⁡V2V1
QT | Cpln⁡V2V1+CVln⁡p2p1=0
Constant  |  | VT | pT | pV | pVγ

### Entropy

* S=kBln⁡Ω, where kB is the Boltzmann constant, and Ω denotes the volume of macrostate in the phase space or otherwise called thermodynamic probability.
* dS=δQT, for reversible processes only

### Statistical physics

Below are useful results from the Maxwell–Boltzmann distribution for an ideal gas, and the implications of the Entropy quantity. The distribution is valid for atoms or molecules constituting ideal gases.

Physical situation  | Nomenclature  | Equations

|
|

Maxwell–Boltzmann distribution |

* v = velocity of atom/molecule,
* m = mass of each molecule (all molecules are identical in kinetic theory),
* γ(p) = Lorentz factor as function of momentum (see below)
* Ratio of thermal to rest mass-energy of each molecule: θ=kBT/mc2

K2 is the modified Bessel function of the second kind.

| Non-relativistic speeds

P(v)=4π(m2πkBT)3/2v2e−mv2/2kBT

Relativistic speeds (Maxwell–Jüttner distribution)
f(p)=14πm3c3θK2(1/θ)e−γ(p)/θ

Entropy Logarithm of the density of states |

* Pi = probability of system in microstate i
* Ω = total number of microstates

| S=−kB∑iPiln⁡Pi=kBln⁡Ω

where:
Pi=1/Ω

Entropy change  |  | ΔS=∫Q1Q2dQT

ΔS=kBNln⁡V2V1+NCVln⁡T2T1

Entropic force  |  | 𝐅S=−T∇S
Equipartition theorem | df = degree of freedom  | Average kinetic energy per degree of freedom

⟨Ek⟩=12kT

Internal energy U=df⟨Ek⟩=df2kT

Corollaries of the non-relativistic Maxwell–Boltzmann distribution are below.

Physical situation  | Nomenclature  | Equations

|
|

Mean speed  |  | ⟨v⟩=8kBTπm
Root mean square speed  |  | vrms=⟨v2⟩=3kBTm
Modal speed  |  | vmode=2kBTm
Mean free path |

* σ = effective cross-section
* n = volume density of number of target particles
* ℓ = mean free path

| ℓ=1/2nσ

### Quasi-static and reversible processes

For quasi-static and reversible processes, the first law of thermodynamics is:

dU=δQ−δW

where δQ is the heat supplied to the system and δW is the work done by the system.

### Thermodynamic potentials

See also: Physics:Maxwell relations

The following energies are called the thermodynamic potentials,

Name  | Symbol  | Formula  | Natural variables

|
|
|

Internal energy | U | ∫(TdS−pdV+∑iμidNi) | S,V,{Ni}
Helmholtz free energy | F | U−TS | T,V,{Ni}
Enthalpy | H | U+pV | S,p,{Ni}
Gibbs free energy | G | U+pV−TS | T,p,{Ni}
Landau Potential (Grand potential) | Ω, ΦG | U−TS−∑iμiNi | T,V,{μi}

and the corresponding fundamental thermodynamic relations or "master equations"[2] are:

Potential  | Differential

|

Internal energy  | dU(S,V,Ni)=TdS−pdV+∑iμidNi
Enthalpy  | dH(S,p,Ni)=TdS+Vdp+∑iμidNi
Helmholtz free energy  | dF(T,V,Ni)=−SdT−pdV+∑iμidNi
Gibbs free energy  | dG(T,p,Ni)=−SdT+Vdp+∑iμidNi

### Maxwell's relations

The four most common Maxwell's relations are:

Physical situation  | Nomenclature  | Equations

|
|

Thermodynamic potentials as functions of their natural variables  |

* U(S,V) = Internal energy
* H(S,p) = Enthalpy
* F(T,V) = Helmholtz free energy
* G(T,p) = Gibbs free energy

| (∂T∂V)S=−(∂p∂S)V=∂2U∂S∂V

(∂T∂p)S=+(∂V∂S)p=∂2H∂S∂p

+(∂S∂V)T=(∂p∂T)V=−∂2F∂T∂V

−(∂S∂p)T=(∂V∂T)p=∂2G∂T∂p

More relations include the following.

(∂S∂U)V,N=1T | (∂S∂V)N,U=pT | (∂S∂N)V,U=−μT

|
|

(∂T∂S)V=TCV | (∂T∂S)p=TCp |
−(∂p∂V)T=1VKT |  |

Other differential equations are:

Name  | H | U | G

|
|
|

Gibbs–Helmholtz equation | H=−T2(∂(G/T)∂T)p | U=−T2(∂(F/T)∂T)V | G=−V2(∂(F/V)∂V)T
| (∂H∂p)T=V−T(∂V∂T)p | (∂U∂V)T=T(∂p∂T)V−p |

### Quantum properties

* U=NkBT2(∂ln⁡Z∂T)V
* S=UT+NkBln⁡Z−Nkln⁡N+Nk Indistinguishable Particles

where N is number of particles, h is that Planck constant, I is moment of inertia, and Z is the partition function, in various forms:

Degree of freedom  | Partition function

|

Translation  | Zt=(2πmkBT)32Vh3
Vibration  | Zv=11−e−hω2πkBT
Rotation  | Zr=2IkBTσ(h2π)2

* where:
* σ = 1 (heteronuclear molecules)
* σ = 2 (homonuclear)

## Thermal properties of matter

Coefficients  | Equation

|

Joule-Thomson coefficient | μJT=(∂T∂p)H
Compressibility (constant temperature)  | KT=−1V(∂V∂p)T,N
Coefficient of thermal expansion (constant pressure)  | αp=1V(∂V∂T)p
Heat capacity (constant pressure)  | Cp=(∂Qrev∂T)p=(∂U∂T)p+p(∂V∂T)p=(∂H∂T)p=T(∂S∂T)p
Heat capacity (constant volume)  | CV=(∂Qrev∂T)V=(∂U∂T)V=T(∂S∂T)V
Derivation of heat capacity (constant pressure)

Since

(∂T∂p)H(∂p∂H)T(∂H∂T)p=−1
(∂T∂p)H=−(∂H∂p)T(∂T∂H)p=−1(∂H∂T)p(∂H∂p)T
Cp=(∂H∂T)p
⇒(∂T∂p)H=−1Cp(∂H∂p)T
Derivation of heat capacity (constant volume)

Since

dU=δQrev−δWrev,

(where δWrev is the work done by the system),

δS=δQrevT,δWrev=pδV
dU=TδS−pδV
(∂U∂T)V=T(∂S∂T)V−p(∂V∂T)V;CV=(∂U∂T)V
⇒CV=T(∂S∂T)V

### Thermal transfer

Physical situation  | Nomenclature  | Equations

|
|

Net intensity emission/absorption  |

* Texternal = external temperature (outside of system)
* Tsystem = internal temperature (inside system)
* ε = emissivity

| I=σϵ(Texternal4−Tsystem4)
Internal energy of a substance  |

* CV = isovolumetric heat capacity of substance
* ΔT = temperature change of substance

| ΔU=NCVΔT
Meyer's equation  |

* Cp = isobaric heat capacity
* CV = isovolumetric heat capacity
* n = amount of substance

| Cp−CV=nR
Effective thermal conductivities  |

* λi = thermal conductivity of substance i
* λnet = equivalent thermal conductivity

| Series

λnet=∑jλj

Parallel 1λnet=∑j(1λj)

### Thermal efficiencies

Physical situation  | Nomenclature  | Equations

|
|

Thermodynamic engines  |

* η = efficiency
* W = work done by engine
* QH = heat energy in higher temperature reservoir
* QL = heat energy in lower temperature reservoir
* TH = temperature of higher temp. reservoir
* TL = temperature of lower temp. reservoir

| Thermodynamic engine:

η=|WQH|

Carnot engine efficiency:
ηc=1−|QLQH|=1−TLTH

Refrigeration  | K = coefficient of refrigeration performance  | Refrigeration performance

K=|QLW|

Carnot refrigeration performance KC=|QL||QH|−|QL|=TLTH−TL

## See also

* List of thermodynamic properties
* Antoine equation
* Bejan number
* Bowen ratio
* Bridgman's equations
* Clausius–Clapeyron relation
* Departure functions
* Duhem–Margules equation
* Ehrenfest equations
* Gibbs–Helmholtz equation
* Phase rule
* Kopp's law
* Noro–Frenkel law of corresponding states
* Onsager reciprocal relations
* Stefan number
* Thermodynamics
* Timeline of thermodynamics
* Triple product rule
* Exact differential

## References

1. ↑ Keenan, Thermodynamics, Wiley, New York, 1947
2. ↑ Physical chemistry, P.W. Atkins, Oxford University Press, 1978, ISBN 0 19 855148 7

* Atkins, Peter and de Paula, Julio Physical Chemistry, 7th edition, W.H. Freeman and Company, 2002 ISBN 0-7167-3539-3.
* Chapters 1–10, Part 1: "Equilibrium".
* Bridgman, P. W. (1 March 1914). "A Complete Collection of Thermodynamic Formulas". Physical Review (American Physical Society (APS)) 3 (4): 273–281. doi:10.1103/physrev.3.273. ISSN 0031-899X. Bibcode: 1914PhRv....3..273B. https://babel.hathitrust.org/cgi/pt?id=uc1.31210014450082&view=1up&seq=289.
* Landsberg, Peter T. Thermodynamics and Statistical Mechanics. New York: Dover Publications, Inc., 1990. (reprinted from Oxford University Press, 1978).
* Lewis, G.N., and Randall, M., "Thermodynamics", 2nd Edition, McGraw-Hill Book Company, New York, 1961.
* Reichl, L.E., A Modern Course in Statistical Physics, 2nd edition, New York: John Wiley & Sons, 1998.
* Schroeder, Daniel V. Thermal Physics. San Francisco: Addison Wesley Longman, 2000 ISBN 0-201-38027-7.
* Silbey, Robert J., et al. Physical Chemistry, 4th ed. New Jersey: Wiley, 2004.
* Callen, Herbert B. (1985). Thermodynamics and an Introduction to Themostatistics, 2nd edition, New York: John Wiley & Sons.

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