Table of thermodynamic equations

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

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	Q_L	J	ML²T⁻²

General derived quantities

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI unit	Dimension
Thermodynamic beta, inverse temperature	β	$\beta =1/k_{\text{B}}T$	J⁻¹	T²M⁻¹L⁻²
Thermodynamic temperature	τ	$\tau =k_{\text{B}}T$ $\tau =k_{\text{B}}\left(\partial U/\partial S\right)_{N}$ $1/\tau =1/k_{\text{B}}\left(\partial S/\partial U\right)_{N}$	J	ML²T⁻²
Entropy	S	$S=-k_{\text{B}}\sum _{i}p_{i}\ln p_{i}$ $S=-\left(\partial F/\partial T\right)_{V}$ , $S=-\left(\partial G/\partial T\right)_{N,P}$	J⋅K⁻¹	ML²T⁻²Θ⁻¹
Pressure	P	$P=-\left(\partial F/\partial V\right)_{T,N}$ $P=-\left(\partial U/\partial V\right)_{S,N}$	Pa	ML⁻¹T⁻²
Internal Energy	U	$U=\sum _{i}E_{i}$	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)	μ_i	$\mu _{i}=\left(\partial U/\partial N_{i}\right)_{N_{j\neq i},S,V}$ $\mu _{i}=\left(\partial F/\partial N_{i}\right)_{T,V}$ , where $F$ is not proportional to $N$ because $\mu _{i}$ 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 $\mu _{i}$ depends only on temperature and pressure and composition. $\mu _{i}/\tau =-1/k_{\text{B}}\left(\partial S/\partial N_{i}\right)_{U,V}$	J	ML²T⁻²
Helmholtz free energy	A, F	$F=U-TS$	J	ML²T⁻²
Landau potential, Landau free energy, Grand potential	Ω, Φ_G	$\Omega =U-TS-\mu N\$	J	ML²T⁻²
Massieu potential, Helmholtz free entropy	Φ	$\Phi =S-U/T$	J⋅K⁻¹	ML²T⁻²Θ⁻¹
Planck potential, Gibbs free entropy	Ξ	$\Xi =\Phi -pV/T$	J⋅K⁻¹	ML²T⁻²Θ⁻¹

Thermal properties of matter

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI unit	Dimension
General heat/thermal capacity	C	$C=\partial Q/\partial T$	J⋅K⁻¹	ML²T⁻²Θ⁻¹
Heat capacity (isobaric)	C_p	$C_{p}=\partial H/\partial T$	J⋅K⁻¹	ML²T⁻²Θ⁻¹
Specific heat capacity (isobaric)	C_mp	$C_{mp}=\partial ^{2}Q/\partial m\partial T$	J⋅kg⁻¹⋅K⁻¹	L²T⁻²Θ⁻¹
Molar specific heat capacity (isobaric)	C_np	$C_{np}=\partial ^{2}Q/\partial n\partial T$	J⋅K⁻¹⋅mol⁻¹	ML²T⁻²Θ⁻¹N⁻¹
Heat capacity (isochoric/volumetric)	C_V	$C_{V}=\partial U/\partial T$	J⋅K⁻¹	ML²T⁻²Θ⁻¹
Specific heat capacity (isochoric)	C_mV	$C_{mV}=\partial ^{2}Q/\partial m\partial T$	J⋅kg⁻¹⋅K⁻¹	L²T⁻²Θ⁻¹
Molar specific heat capacity (isochoric)	C_nV	$C_{nV}=\partial ^{2}Q/\partial n\partial T$	J⋅K⋅⁻¹ mol⁻¹	ML²T⁻²Θ⁻¹N⁻¹
Specific latent heat	L	$L=\partial Q/\partial m$	J⋅kg⁻¹	L²T⁻²
Ratio of isobaric to isochoric heat capacity, heat capacity ratio, adiabatic index, Laplace coefficient	γ	$\gamma =C_{p}/C_{V}=c_{p}/c_{V}=C_{mp}/C_{mV}$	1	1

Thermal transfer

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI unit	Dimension
Temperature gradient	No standard symbol	$\nabla T$	K⋅m⁻¹	ΘL⁻¹
Thermal conduction rate, thermal current, thermal/heat flux, thermal power transfer	P	$P=\mathrm {d} Q/\mathrm {d} t$	W	ML²T⁻³
Thermal intensity	I	$I=\mathrm {d} P/\mathrm {d} A$	W⋅m⁻²	MT⁻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	MT⁻³

Equations

The equations in this article are classified by subject.

Thermodynamic processes

Physical situation	Equations
Isentropic process (adiabatic and reversible)	$Q=0,\quad \Delta U=-W$ 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	$\Delta U=0,\quad W=Q$ For an ideal gas $W=kTN\ln(V_{2}/V_{1})$ $W=nRT\ln(V_{2}/V_{1})$
Isobaric process	p₁ = p₂, p = constant $W=p\Delta V,\quad Q=\Delta U+p\delta V$
Isochoric process	V₁ = V₂, V = constant $W=0,\quad Q=\Delta U$
Free expansion	$\Delta U=0$
Work done by an expanding gas	Process $W=\int _{V_{1}}^{V_{2}}p\mathrm {d} V$ Net work done in cyclic processes $W=\oint _{\mathrm {cycle} }p\mathrm {d} V=\oint _{\mathrm {cycle} }\Delta Q$

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	$pV=nRT=kTN$ ${\frac {p_{1}V_{1}}{p_{2}V_{2}}}={\frac {n_{1}T_{1}}{n_{2}T_{2}}}={\frac {N_{1}T_{1}}{N_{2}T_{2}}}$
Pressure of an ideal gas	m = mass of one molecule M_m = 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	Adiabatic $Q=0$
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	$\Delta S=C_{V}\ln {T_{2} \over T_{1}}+nR\ln {V_{2} \over V_{1}}$ $\Delta S=C_{p}\ln {T_{2} \over T_{1}}-nR\ln {p_{2} \over p_{1}}$ ^[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

$S=k_{\mathrm {B} }\ln \Omega$ , where k_B 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: $\theta =k_{\text{B}}T/mc^{2}$ K₂ is the modified Bessel function of the second kind.	Non-relativistic speeds $P\left(v\right)=4\pi \left({\frac {m}{2\pi k_{\text{B}}T}}\right)^{3/2}v^{2}e^{-mv^{2}/2k_{\text{B}}T}$ 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	P_i = probability of system in microstate i Ω = total number of microstates	$S=-k_{\text{B}}\sum _{i}P_{i}\ln P_{i}=k_{\mathrm {B} }\ln \Omega$ where: $P_{i}=1/\Omega$
Entropy change		$\Delta S=\int _{Q_{1}}^{Q_{2}}{\frac {\mathrm {d} Q}{T}}$ $\Delta S=k_{\text{B}}N\ln {\frac {V_{2}}{V_{1}}}+NC_{V}\ln {\frac {T_{2}}{T_{1}}}$
Entropic force		$\mathbf {F} _{\mathrm {S} }=-T\nabla S$
Equipartition theorem	d_f = degree of freedom	Average kinetic energy per degree of freedom $\langle E_{\mathrm {k} }\rangle ={\frac {1}{2}}kT$ Internal energy $U=d_{\text{f}}\langle E_{\mathrm {k} }\rangle ={\frac {d_{\text{f}}}{2}}kT$

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

Physical situation	Nomenclature	Equations
Mean speed		$\langle v\rangle ={\sqrt {\frac {8k_{\text{B}}T}{\pi m}}}$
Root mean square speed		$v_{\mathrm {rms} }={\sqrt {\langle v^{2}\rangle }}={\sqrt {\frac {3k_{\text{B}}T}{m}}}$
Modal speed		$v_{\mathrm {mode} }={\sqrt {\frac {2k_{\text{B}}T}{m}}}$
Mean free path	σ = effective cross-section n = volume density of number of target particles ℓ = 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	$U$	$\int \left(T\,\mathrm {d} S-p\,\mathrm {d} V+\sum _{i}\mu _{i}\mathrm {d} N_{i}\right)$	$S,V,\{N_{i}\}$
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, or 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"^[2] 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

$U=Nk_{\text{B}}T^{2}\left({\frac {\partial \ln Z}{\partial T}}\right)_{V}$
$S={\frac {U}{T}}+Nk_{\text{B}}\ln Z-Nk\ln N+Nk$ 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	$Z_{t}={\frac {(2\pi mk_{\text{B}}T)^{\frac {3}{2}}V}{h^{3}}}$
Vibration	$Z_{v}={\frac {1}{1-e^{\frac {-h\omega }{2\pi k_{\text{B}}T}}}}$
Rotation	$Z_{r}={\frac {2Ik_{\text{B}}T}{\sigma ({\frac {h}{2\pi }})^{2}}}$ where: σ = 1 (heteronuclear molecules) σ = 2 (homonuclear)

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)	$C_{V}=\left({\partial Q_{rev} \over \partial T}\right)_{V}=\left({\partial U \over \partial T}\right)_{V}=T\left({\partial S \over \partial T}\right)_{V}$

Derivation of heat capacity (constant pressure)

Since

\left({\frac {\partial T}{\partial p}}\right)_{H}\left({\frac {\partial p}{\partial H}}\right)_{T}\left({\frac {\partial H}{\partial T}}\right)_{p}=-1

{\begin{aligned}\left({\frac {\partial T}{\partial p}}\right)_{H}&=-\left({\frac {\partial H}{\partial p}}\right)_{T}\left({\frac {\partial T}{\partial H}}\right)_{p}\\&={\frac {-1}{\left({\frac {\partial H}{\partial T}}\right)_{p}}}\left({\frac {\partial H}{\partial p}}\right)_{T}\end{aligned}}

C_{p}=\left({\frac {\partial H}{\partial T}}\right)_{p}

\Rightarrow \left({\frac {\partial T}{\partial p}}\right)_{H}=-{\frac {1}{C_{p}}}\left({\frac {\partial H}{\partial p}}\right)_{T}

Derivation of heat capacity (constant volume)

Since

dU=\delta Q_{rev}-\delta W_{rev},

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

\delta S={\frac {\delta Q_{rev}}{T}},\delta W_{rev}=p\delta V

dU=T\delta S-p\delta V

\left({\frac {\partial U}{\partial T}}\right)_{V}=T\left({\frac {\partial S}{\partial T}}\right)_{V}-p\left({\frac {\partial V}{\partial T}}\right)_{V};C_{V}=\left({\frac {\partial U}{\partial T}}\right)_{V}

\Rightarrow C_{V}=T\left({\frac {\partial S}{\partial T}}\right)_{V}

Thermal transfer

Physical situation	Nomenclature	Equations
Net intensity emission/absorption	T_external = external temperature (outside of system) T_system = internal temperature (inside system) ε = emissivity	$I=\sigma \epsilon \left(T_{\mathrm {external} }^{4}-T_{\mathrm {system} }^{4}\right)$
Internal energy of a substance	C_V = isovolumetric heat capacity of substance ΔT = temperature change of substance	$\Delta U=NC_{V}\Delta T$
Meyer's equation	C_p = isobaric heat capacity C_V = isovolumetric heat capacity n = amount of substance	$C_{p}-C_{V}=nR$
Effective thermal conductivities	λ_i = thermal conductivity of substance i λ_net = 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 Q_H = heat energy in higher temperature reservoir Q_L = heat energy in lower temperature reservoir T_H = temperature of higher temp. reservoir T_L = temperature of lower temp. reservoir	Thermodynamic engine: $\eta =\left\|{\frac {W}{Q_{\text{H}}}}\right\|$ Carnot engine efficiency: $\eta _{\text{c}}=1-\left\|{\frac {Q_{\text{L}}}{Q_{\text{H}}}}\right\|=1-{\frac {T_{\text{L}}}{T_{\text{H}}}}$
Refrigeration	K = coefficient of refrigeration performance	Refrigeration performance $K=\left\|{\frac {Q_{\text{L}}}{W}}\right\|$ Carnot refrigeration performance $K_{\text{C}}={\frac {\|Q_{\text{L}}\|}{\|Q_{\text{H}}\|-\|Q_{\text{L}}\|}}={\frac {T_{\text{L}}}{T_{\text{H}}-T_{\text{L}}}}$

References

^ Keenan, Thermodynamics, Wiley, New York, 1947
^ 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. 3 (4). American Physical Society (APS): 273–281. doi:10.1103/physrev.3.273. ISSN 0031-899X.
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.

External links

Thermodynamic equation calculator

[1] Keenan, Thermodynamics, Wiley, New York, 1947

[2] Physical chemistry, P.W. Atkins, Oxford University Press, 1978, ISBN 0 19 855148 7

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Definitions

General basic quantities

General derived quantities

Thermal properties of matter

Thermal transfer

Equations

Thermodynamic processes

Kinetic theory

Ideal gas

Entropy

Statistical physics

Quasi-static and reversible processes

Thermodynamic potentials

Maxwell's relations

Quantum properties

Thermal properties of matter

Thermal transfer

Thermal efficiencies

See also

References

External links