# Table of thermodynamic equations

This article is a summary of common equations and quantities in thermodynamics (see thermodynamic equations for more elaboration). SI units are used for absolute temperature, not Celsius or Fahrenheit.

## 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 Units Dimension
Number of molecules N dimensionless dimensionless
Number of moles n mol [N]
Temperature T K [Θ]
Heat Energy Q, q J [M][L]2[T]−2
Latent Heat QL J [M][L]2[T]−2

### General derived quantities

Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
Thermodynamic beta, Inverse temperature β $\beta =1/k_{B}T\,\!$  J−1 [T]2[M]−1[L]−2
Thermodynamic temperature τ $\tau =k_{B}T\,\!$

$\tau =k_{B}\left(\partial U/\partial S\right)_{N}\,\!$  $1/\tau =1/k_{B}\left(\partial S/\partial U\right)_{N}\,\!$

J [M] [L]2 [T]−2
Entropy S $S=-k_{B}\sum _{i}p_{i}\ln p_{i}$

$S=-\left(\partial F/\partial T\right)_{V}\,\!$  , $S=-\left(\partial G/\partial T\right)_{N,P}\,\!$

J K−1 [M][L]2[T]−2 [Θ]−1
Pressure P $P=-\left(\partial F/\partial V\right)_{T,N}\,\!$

$P=-\left(\partial U/\partial V\right)_{S,N}\,\!$

Pa M L−1T−2
Internal Energy U $U=\sum _{i}E_{i}\!$  J [M][L]2[T]−2
Enthalpy H $H=U+pV\,\!$  J [M][L]2[T]−2
Partition Function Z dimensionless dimensionless
Gibbs free energy G $G=H-TS\,\!$  J [M][L]2[T]−2
Chemical potential (of

component i in a mixture)

μi $\mu _{i}=\left(\partial U/\partial N_{i}\right)_{N_{j\neq i},S,V}\,\!$

$\mu _{i}=\left(\partial F/\partial N_{i}\right)_{T,V}\,\!$ , where F is not proportional to N because μi depends on pressure. $\mu _{i}=\left(\partial G/\partial N_{i}\right)_{T,P}\,\!$ , 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. $\mu _{i}/\tau =-1/k_{B}\left(\partial S/\partial N_{i}\right)_{U,V}\,\!$

J [M][L]2[T]−2
Helmholtz free energy A, F $F=U-TS\,\!$  J [M][L]2[T]−2
Landau potential, Landau Free Energy, Grand potential Ω, ΦG $\Omega =U-TS-\mu N\,\!$  J [M][L]2[T]−2
Massieu Potential, Helmholtz free entropy Φ $\Phi =S-U/T\,\!$  J K−1 [M][L]2[T]−2 [Θ]−1
Planck potential, Gibbs free entropy Ξ $\Xi =\Phi -pV/T\,\!$  J K−1 [M][L]2[T]−2 [Θ]−1

### Thermal properties of matter

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
General heat/thermal capacity C $C=\partial Q/\partial T\,\!$  J K −1 [M][L]2[T]−2 [Θ]−1
Heat capacity (isobaric) Cp $C_{p}=\partial H/\partial T\,\!$  J K −1 [M][L]2[T]−2 [Θ]−1
Specific heat capacity (isobaric) Cmp $C_{mp}=\partial ^{2}Q/\partial m\partial T\,\!$  J kg−1 K−1 [L]2[T]−2 [Θ]−1
Molar specific heat capacity (isobaric) Cnp $C_{np}=\partial ^{2}Q/\partial n\partial T\,\!$  J K −1 mol−1 [M][L]2[T]−2 [Θ]−1 [N]−1
Heat capacity (isochoric/volumetric) CV $C_{V}=\partial U/\partial T\,\!$  J K −1 [M][L]2[T]−2 [Θ]−1
Specific heat capacity (isochoric) CmV $C_{mV}=\partial ^{2}Q/\partial m\partial T\,\!$  J kg−1 K−1 [L]2[T]−2 [Θ]−1
Molar specific heat capacity (isochoric) CnV $C_{nV}=\partial ^{2}Q/\partial n\partial T\,\!$  J K −1 mol−1 [M][L]2[T]−2 [Θ]−1 [N]−1
Specific latent heat L $L=\partial Q/\partial m\,\!$  J kg−1 [L]2[T]−2
Ratio of isobaric to isochoric heat capacity, heat capacity ratio, adiabatic index γ $\gamma =C_{p}/C_{V}=c_{p}/c_{V}=C_{mp}/C_{mV}\,\!$  dimensionless dimensionless

### Thermal transfer

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Temperature gradient No standard symbol $\nabla T\,\!$  K m−1 [Θ][L]−1
Thermal conduction rate, thermal current, thermal/heat flux, thermal power transfer P $P=\mathrm {d} Q/\mathrm {d} t\,\!$  W = J s−1 [M] [L]2 [T]−3
Thermal intensity I $I=\mathrm {d} P/\mathrm {d} A$  W m−2 [M] [T]−3
Thermal/heat flux density (vector analogue of thermal intensity above) q $Q=\iint \mathbf {q} \cdot \mathrm {d} \mathbf {S} \mathrm {d} t\,\!$  W m−2 [M] [T]−3

## Equations

### Thermodynamic processes

Physical situation Equations
Isentropic process (adiabatic and reversible) $\Delta Q=0,\quad \Delta U=-\Delta W\,\!$

For an ideal gas
$p_{1}V_{1}^{\gamma }=p_{2}V_{2}^{\gamma }\,\!$
$T_{1}V_{1}^{\gamma -1}=T_{2}V_{2}^{\gamma -1}\,\!$
$p_{1}^{1-\gamma }T_{1}^{\gamma }=p_{2}^{1-\gamma }T_{2}^{\gamma }\,\!$

Isothermal process $\Delta U=0,\quad \Delta W=\Delta Q\,\!$

For an ideal gas
$W=kTN\ln(V_{2}/V_{1})\,\!$

Isobaric process p1 = p2, p = constant

$\Delta W=p\Delta V,\quad \Delta q=\Delta H+p\delta V\,\!$

Isochoric process V1 = V2, V = constant

$\Delta W=0,\quad \Delta Q=\Delta U\,\!$

Free expansion $\Delta U=0\,\!$
Work done by an expanding gas Process

$\Delta W=\int _{V_{1}}^{V_{2}}p\mathrm {d} V\,\!$

Net Work Done in Cyclic Processes
$\Delta W=\oint _{\mathrm {cycle} }p\mathrm {d} V\,\!$

### Kinetic theory

Ideal gas equations
Physical situation Nomenclature Equations
Ideal gas law
$pV=nRT=kTN\,\!$

${\frac {p_{1}V_{1}}{p_{2}V_{2}}}={\frac {n_{1}T_{1}}{n_{2}T_{2}}}={\frac {N_{1}T_{1}}{N_{2}T_{2}}}\,\!$

Pressure of an ideal gas
• m = mass of one molecule
• Mm = molar mass
$p={\frac {Nm\langle v^{2}\rangle }{3V}}={\frac {nM_{m}\langle v^{2}\rangle }{3V}}={\frac {1}{3}}\rho \langle v^{2}\rangle \,\!$

#### Ideal gas

Quantity General Equation Isobaric
Δp = 0
Isochoric
ΔV = 0
Isothermal
ΔT = 0
$Q=0$
Work
W
$\delta W=-pdV\;$  $-p\Delta V\;$  $0\;$  $-nRT\ln {\frac {V_{2}}{V_{1}}}\;$

$-nRT\ln {\frac {P_{1}}{P_{2}}}\;$

${\frac {PV^{\gamma }(V_{f}^{1-\gamma }-V_{i}^{1-\gamma })}{1-\gamma }}=C_{V}\left(T_{2}-T_{1}\right)$
Heat Capacity
C
(as for real gas) $C_{p}={\frac {5}{2}}nR\;$
(for monatomic ideal gas)

$C_{p}={\frac {7}{2}}nR\;$
(for diatomic ideal gas)

$C_{V}={\frac {3}{2}}nR\;$
(for monatomic ideal gas)

$C_{V}={\frac {5}{2}}nR\;$
(for diatomic ideal gas)

Internal Energy
ΔU
$\Delta U=C_{V}\Delta T\;$  $Q+W\;$

$Q_{p}-p\Delta V\;$
$Q\;$

$C_{V}\left(T_{2}-T_{1}\right)\;$
$0\;$

$Q=-W\;$
$W\;$

$C_{V}\left(T_{2}-T_{1}\right)\;$
Enthalpy
ΔH
$H=U+pV\;$  $C_{p}\left(T_{2}-T_{1}\right)\;$  $Q_{V}+V\Delta p\;$  $0\;$  $C_{p}\left(T_{2}-T_{1}\right)\;$
Entropy
Δs
$\Delta S=C_{V}\ln {T_{2} \over T_{1}}+nR\ln {V_{2} \over V_{1}}$
$\Delta S=C_{p}\ln {T_{2} \over T_{1}}-nR\ln {p_{2} \over p_{1}}$ 
$C_{p}\ln {\frac {T_{2}}{T_{1}}}\;$  $C_{V}\ln {\frac {T_{2}}{T_{1}}}\;$  $nR\ln {\frac {V_{2}}{V_{1}}}\;$
${\frac {Q}{T}}\;$
$C_{p}\ln {\frac {V_{2}}{V_{1}}}+C_{V}\ln {\frac {p_{2}}{p_{1}}}=0\;$
Constant $\;$  ${\frac {V}{T}}\;$  ${\frac {p}{T}}\;$  $pV\;$  $pV^{\gamma }\;$

### Entropy

• $S=k_{B}(\ln \Omega )$ , where kB is the Boltzmann constant, and Ω denotes the volume of macrostate in the phase space or otherwise called thermodynamic probability.
• $dS={\frac {\delta Q}{T}}$ , 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:$\theta =k_{B}T/mc^{2}\,\!$

K2 is the Modified Bessel function of the second kind.

Non-relativistic speeds

$P\left(v\right)=4\pi \left({\frac {m}{2\pi k_{B}T}}\right)^{3/2}v^{2}e^{-mv^{2}/2k_{B}T}\,\!$

Relativistic speeds (Maxwell-Jüttner distribution)
$f(p)={\frac {1}{4\pi m^{3}c^{3}\theta K_{2}(1/\theta )}}e^{-\gamma (p)/\theta }$

Entropy Logarithm of the density of states
• Pi = probability of system in microstate i
• Ω = total number of microstates
$S=-k_{B}\sum _{i}P_{i}\ln P_{i}=k_{\mathrm {B} }\ln \Omega \,\!$

where:
$P_{i}=1/\Omega \,\!$

Entropy change $\Delta S=\int _{Q_{1}}^{Q_{2}}{\frac {\mathrm {d} Q}{T}}\,\!$

$\Delta S=k_{B}N\ln {\frac {V_{2}}{V_{1}}}+NC_{V}\ln {\frac {T_{2}}{T_{1}}}\,\!$

Entropic force $\mathbf {F} _{\mathrm {S} }=-T\nabla S\,\!$
Equipartition theorem
• df = degree of freedom
Average kinetic energy per degree of freedom

$\langle E_{\mathrm {k} }\rangle ={\frac {1}{2}}kT\,\!$

Internal energy $U=d_{f}\langle E_{\mathrm {k} }\rangle ={\frac {d_{f}}{2}}kT\,\!$

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

Physical situation Nomenclature Equations
Mean speed $\langle v\rangle ={\sqrt {\frac {8k_{B}T}{\pi m}}}\,\!$
Root mean square speed $v_{\mathrm {rms} }={\sqrt {\langle v^{2}\rangle }}={\sqrt {\frac {3k_{B}T}{m}}}\,\!$
Modal speed $v_{\mathrm {mode} }={\sqrt {\frac {2k_{B}T}{m}}}\,\!$
Mean free path
• σ = Effective cross-section
• n = Volume density of number of target particles
• = Mean free path
$\ell =1/{\sqrt {2}}n\sigma \,\!$

### Quasi-static and reversible processes

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

$dU=\delta Q-\delta W$

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

### Thermodynamic potentials

The following energies are called the thermodynamic potentials,

Name Symbol Formula Natural variables
Internal energy $U$  $\int (T{\text{d}}S-p{\text{d}}V+\sum _{i}\mu _{i}{\text{d}}N_{i})$  $S,V,\{N_{i}\}$
Helmholtz free energy $F$  $U-TS$  $T,V,\{N_{i}\}$
Enthalpy $H$  $U+pV$  $S,p,\{N_{i}\}$
Gibbs free energy $G$  $U+pV-TS$  $T,p,\{N_{i}\}$
Landau Potential (Grand potential) $\Omega$ , $\Phi _{\text{G}}$  $U-TS-$ $\sum _{i}\,$ $\mu _{i}N_{i}$  $T,V,\{\mu _{i}\}$

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

Potential Differential
Internal energy $dU\left(S,V,{N_{i}}\right)=TdS-pdV+\sum _{i}\mu _{i}dN_{i}$
Enthalpy $dH\left(S,p,{N_{i}}\right)=TdS+Vdp+\sum _{i}\mu _{i}dN_{i}$
Helmholtz free energy $dF\left(T,V,{N_{i}}\right)=-SdT-pdV+\sum _{i}\mu _{i}dN_{i}$
Gibbs free energy $dG\left(T,p,{N_{i}}\right)=-SdT+Vdp+\sum _{i}\mu _{i}dN_{i}$

### 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
$\left({\frac {\partial T}{\partial V}}\right)_{S}=-\left({\frac {\partial P}{\partial S}}\right)_{V}={\frac {\partial ^{2}U}{\partial S\partial V}}$

$\left({\frac {\partial T}{\partial P}}\right)_{S}=+\left({\frac {\partial V}{\partial S}}\right)_{P}={\frac {\partial ^{2}H}{\partial S\partial P}}$

$+\left({\frac {\partial S}{\partial V}}\right)_{T}=\left({\frac {\partial P}{\partial T}}\right)_{V}=-{\frac {\partial ^{2}F}{\partial T\partial V}}$

$-\left({\frac {\partial S}{\partial P}}\right)_{T}=\left({\frac {\partial V}{\partial T}}\right)_{P}={\frac {\partial ^{2}G}{\partial T\partial P}}$

More relations include the following.

 $\left({\partial S \over \partial U}\right)_{V,N}={1 \over T}$ $\left({\partial S \over \partial V}\right)_{N,U}={p \over T}$ $\left({\partial S \over \partial N}\right)_{V,U}=-{\mu \over T}$ $\left({\partial T \over \partial S}\right)_{V}={T \over C_{V}}$ $\left({\partial T \over \partial S}\right)_{P}={T \over C_{P}}$ $-\left({\partial p \over \partial V}\right)_{T}={1 \over {VK_{T}}}$ Other differential equations are:

Name H U G
Gibbs–Helmholtz equation $H=-T^{2}\left({\frac {\partial \left(G/T\right)}{\partial T}}\right)_{p}$  $U=-T^{2}\left({\frac {\partial \left(F/T\right)}{\partial T}}\right)_{V}$  $G=-V^{2}\left({\frac {\partial \left(F/V\right)}{\partial V}}\right)_{T}$
$\left({\frac {\partial H}{\partial p}}\right)_{T}=V-T\left({\frac {\partial V}{\partial T}}\right)_{P}$  $\left({\frac {\partial U}{\partial V}}\right)_{T}=T\left({\frac {\partial P}{\partial T}}\right)_{V}-P$

### Quantum properties

• $U=Nk_{B}T^{2}\left({\frac {\partial \ln Z}{\partial T}}\right)_{V}~$
• $S={\frac {U}{T}}+N~$
• $S={\frac {U}{T}}+Nk_{B}\ln Z-Nk\ln N+Nk~$  Indistinguishable Particles

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

Degree of freedom Partition function
Translation $Z_{t}={\frac {(2\pi mk_{B}T)^{\frac {3}{2}}V}{h^{3}}}$
Vibration $Z_{v}={\frac {1}{1-e^{\frac {-h\omega }{2\pi k_{B}T}}}}$
Rotation $Z_{r}={\frac {2Ik_{B}T}{\sigma ({\frac {h}{2\pi }})^{2}}}$

## Thermal properties of matter

Coefficients Equation
Joule-Thomson coefficient $\mu _{JT}=\left({\frac {\partial T}{\partial p}}\right)_{H}$
Compressibility (constant temperature) $K_{T}=-{1 \over V}\left({\partial V \over \partial p}\right)_{T,N}$
Coefficient of thermal expansion (constant pressure) $\alpha _{p}={\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{p}$
Heat capacity (constant pressure) $C_{p}=\left({\partial Q_{rev} \over \partial T}\right)_{p}=\left({\partial U \over \partial T}\right)_{p}+p\left({\partial V \over \partial T}\right)_{p}=\left({\partial H \over \partial T}\right)_{p}=T\left({\partial S \over \partial T}\right)_{p}$
Heat capacity (constant volume) $C_{V}=\left({\partial Q_{rev} \over \partial T}\right)_{V}=\left({\partial U \over \partial T}\right)_{V}=T\left({\partial S \over \partial T}\right)_{V}$

### Thermal transfer

Physical situation Nomenclature Equations
Net intensity emission/absorption
• Texternal = external temperature (outside of system)
• Tsystem = internal temperature (inside system)
• ε = emmisivity
$I=\sigma \epsilon \left(T_{\mathrm {external} }^{4}-T_{\mathrm {system} }^{4}\right)\,\!$
Internal energy of a substance
• CV = isovolumetric heat capacity of substance
• ΔT = temperature change of substance
$\Delta U=NC_{V}\Delta T\,\!$
Meyer's equation
• Cp = isobaric heat capacity
• CV = isovolumetric heat capacity
• n = number of moles
$C_{p}-C_{V}=nR\,\!$
Effective thermal conductivities
• λi = thermal conductivity of substance i
• λnet = equivalent thermal conductivity
Series

$\lambda _{\mathrm {net} }=\sum _{j}\lambda _{j}\,\!$

Parallel ${\frac {1}{\lambda }}_{\mathrm {net} }=\sum _{j}\left({\frac {1}{\lambda }}_{j}\right)\,\!$

### 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:

$\eta =\left|{\frac {W}{Q_{H}}}\right|\,\!$

Carnot engine efficiency:
$\eta _{c}=1-\left|{\frac {Q_{L}}{Q_{H}}}\right|=1-{\frac {T_{L}}{T_{H}}}\,\!$

Refrigeration
• K = coefficient of refrigeration performance
Refrigeration performance

$K=\left|{\frac {Q_{L}}{W}}\right|\,\!$

Carnot refrigeration performance $K_{C}={\frac {|Q_{L}|}{|Q_{H}|-|Q_{L}|}}={\frac {T_{L}}{T_{H}-T_{L}}}\,\!$