Table of thermodynamic equations

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

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Units	Dimension
Thermodynamic beta, Inverse temperature	β	${\displaystyle \beta =1/k_{B}T\,\!}$	J−1	[T]2[M]−1[L]−2
Thermodynamic temperature	τ	${\displaystyle \tau =k_{B}T\,\!}$ ${\displaystyle \tau =k_{B}\left(\partial U/\partial S\right)_{N}\,\!}$ ${\displaystyle 1/\tau =1/k_{B}\left(\partial S/\partial U\right)_{N}\,\!}$	J	[M] [L]2 [T]−2
Entropy	S	${\displaystyle S=-k_{B}\sum _{i}p_{i}\ln p_{i}}$ ${\displaystyle S=-\left(\partial F/\partial T\right)_{V}\,\!}$ , ${\displaystyle S=-\left(\partial G/\partial T\right)_{N,P}\,\!}$	J K−1	[M][L]2[T]−2 [Θ]−1
Pressure	P	${\displaystyle P=-\left(\partial F/\partial V\right)_{T,N}\,\!}$ ${\displaystyle P=-\left(\partial U/\partial V\right)_{S,N}\,\!}$	Pa	M L−1T−2
Internal Energy	U	${\displaystyle U=\sum _{i}E_{i}\!}$	J	[M][L]2[T]−2
Enthalpy	H	${\displaystyle H=U+pV\,\!}$	J	[M][L]2[T]−2
Partition Function	Z		dimensionless	dimensionless
Gibbs free energy	G	${\displaystyle G=H-TS\,\!}$	J	[M][L]2[T]−2
Chemical potential (of component i in a mixture)	μi	${\displaystyle \mu _{i}=\left(\partial U/\partial N_{i}\right)_{N_{j\neq i},S,V}\,\!}$ ${\displaystyle \mu _{i}=\left(\partial F/\partial N_{i}\right)_{T,V}\,\!}$, where F is not proportional to N because μi depends on pressure. ${\displaystyle \mu _{i}=\left(\partial G/\partial N_{i}\right)_{T,P}\,\!}$, 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. ${\displaystyle \mu _{i}/\tau =-1/k_{B}\left(\partial S/\partial N_{i}\right)_{U,V}\,\!}$	J	[M][L]2[T]−2
Helmholtz free energy	A, F	${\displaystyle F=U-TS\,\!}$	J	[M][L]2[T]−2
Landau potential, Landau Free Energy, Grand potential	Ω, ΦG	${\displaystyle \Omega =U-TS-\mu N\,\!}$	J	[M][L]2[T]−2
Massieu Potential, Helmholtz free entropy	Φ	${\displaystyle \Phi =S-U/T\,\!}$	J K−1	[M][L]2[T]−2 [Θ]−1
Planck potential, Gibbs free entropy	Ξ	${\displaystyle \Xi =\Phi -pV/T\,\!}$	J K−1	[M][L]2[T]−2 [Θ]−1

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

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

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

Ideal gas equations

Physical situation	Nomenclature	Equations
Ideal gas law	p = pressure V = volume of container T = temperature n = number of moles R = Gas constant N = number of molecules k = Boltzmann's constant	${\displaystyle pV=nRT=kTN\,\!}$ ${\displaystyle {\frac {p_{1}V_{1}}{p_{2}V_{2}}}={\frac {n_{1}T_{1}}{n_{2}T_{2}}}={\frac {N_{1}T_{1}}{N_{2}T_{2}}}\,\!}$
Pressure of an ideal gas	m = mass of one molecule Mm = molar mass	${\displaystyle p={\frac {Nm\langle v^{2}\rangle }{3V}}={\frac {nM_{m}\langle v^{2}\rangle }{3V}}={\frac {1}{3}}\rho \langle v^{2}\rangle \,\!}$

Quantity	General Equation	Isobaric Δp = 0	Isochoric ΔV = 0	Isothermal ΔT = 0	Adiabatic ${\displaystyle Q=0}$
Work W	${\displaystyle \delta W=-pdV\;}$	${\displaystyle -p\Delta V\;}$	${\displaystyle 0\;}$	${\displaystyle -nRT\ln {\frac {V_{2}}{V_{1}}}\;}$ ${\displaystyle -nRT\ln {\frac {P_{1}}{P_{2}}}\;}$	${\displaystyle {\frac {PV^{\gamma }(V_{f}^{1-\gamma }-V_{i}^{1-\gamma })}{1-\gamma }}=C_{V}\left(T_{2}-T_{1}\right)}$
Heat Capacity C	(as for real gas)	${\displaystyle C_{p}={\frac {5}{2}}nR\;}$ (for monatomic ideal gas) ${\displaystyle C_{p}={\frac {7}{2}}nR\;}$ (for diatomic ideal gas)	${\displaystyle C_{V}={\frac {3}{2}}nR\;}$ (for monatomic ideal gas) ${\displaystyle C_{V}={\frac {5}{2}}nR\;}$ (for diatomic ideal gas)
Internal Energy ΔU	${\displaystyle \Delta U=C_{V}\Delta T\;}$	${\displaystyle Q+W\;}$ ${\displaystyle Q_{p}-p\Delta V\;}$	${\displaystyle Q\;}$ ${\displaystyle C_{V}\left(T_{2}-T_{1}\right)\;}$	${\displaystyle 0\;}$ ${\displaystyle Q=-W\;}$	${\displaystyle W\;}$ ${\displaystyle C_{V}\left(T_{2}-T_{1}\right)\;}$
Enthalpy ΔH	${\displaystyle H=U+pV\;}$	${\displaystyle C_{p}\left(T_{2}-T_{1}\right)\;}$	${\displaystyle Q_{V}+V\Delta p\;}$	${\displaystyle 0\;}$	${\displaystyle C_{p}\left(T_{2}-T_{1}\right)\;}$
Entropy Δs	${\displaystyle \Delta S=C_{V}\ln {T_{2} \over T_{1}}+nR\ln {V_{2} \over V_{1}}}$ ${\displaystyle \Delta S=C_{p}\ln {T_{2} \over T_{1}}-nR\ln {p_{2} \over p_{1}}}$[1]	${\displaystyle C_{p}\ln {\frac {T_{2}}{T_{1}}}\;}$	${\displaystyle C_{V}\ln {\frac {T_{2}}{T_{1}}}\;}$	${\displaystyle nR\ln {\frac {V_{2}}{V_{1}}}\;}$ ${\displaystyle {\frac {Q}{T}}\;}$	${\displaystyle C_{p}\ln {\frac {V_{2}}{V_{1}}}+C_{V}\ln {\frac {p_{2}}{p_{1}}}=0\;}$
Constant	${\displaystyle \;}$	${\displaystyle {\frac {V}{T}}\;}$	${\displaystyle {\frac {p}{T}}\;}$	${\displaystyle pV\;}$	${\displaystyle pV^{\gamma }\;}$

• ${\displaystyle S=k_{\mathrm {B} }\ln \Omega }$, where k_B is the Boltzmann constant, and Ω denotes the volume of macrostate in the phase space or otherwise called thermodynamic probability.
• ${\displaystyle dS={\frac {\delta Q}{T}}}$, for reversible processes only

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

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

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

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

${\displaystyle dU=\delta Q-\delta W}$

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

The following energies are called the thermodynamic potentials,

Name	Symbol	Formula	Natural variables
Internal energy	${\displaystyle U}$	${\displaystyle \int \left(T\,dS-p\,dV+\sum _{i}\mu _{i}dN_{i}\right)}$	${\displaystyle S,V,\{N_{i}\}}$
Helmholtz free energy	${\displaystyle F}$	${\displaystyle U-TS}$	${\displaystyle T,V,\{N_{i}\}}$
Enthalpy	${\displaystyle H}$	${\displaystyle U+pV}$	${\displaystyle S,p,\{N_{i}\}}$
Gibbs free energy	${\displaystyle G}$	${\displaystyle U+pV-TS}$	${\displaystyle T,p,\{N_{i}\}}$
Landau potential, or grand potential	${\displaystyle \Omega }$, ${\displaystyle \Phi _{\text{G}}}$	${\displaystyle U-TS-}$${\displaystyle \sum _{i}\,}$${\displaystyle \mu _{i}N_{i}}$	${\displaystyle T,V,\{\mu _{i}\}}$

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

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

The four most common Maxwell's relations are:

Physical situation	Nomenclature	Equations
Thermodynamic potentials as functions of their natural variables	${\displaystyle U(S,V)\,}$ = Internal energy ${\displaystyle H(S,P)\,}$ = Enthalpy ${\displaystyle F(T,V)\,}$ = Helmholtz free energy ${\displaystyle G(T,P)\,}$ = Gibbs free energy	${\displaystyle \left({\frac {\partial T}{\partial V}}\right)_{S}=-\left({\frac {\partial P}{\partial S}}\right)_{V}={\frac {\partial ^{2}U}{\partial S\partial V}}}$ ${\displaystyle \left({\frac {\partial T}{\partial P}}\right)_{S}=+\left({\frac {\partial V}{\partial S}}\right)_{P}={\frac {\partial ^{2}H}{\partial S\partial P}}}$ ${\displaystyle +\left({\frac {\partial S}{\partial V}}\right)_{T}=\left({\frac {\partial P}{\partial T}}\right)_{V}=-{\frac {\partial ^{2}F}{\partial T\partial V}}}$ ${\displaystyle -\left({\frac {\partial S}{\partial P}}\right)_{T}=\left({\frac {\partial V}{\partial T}}\right)_{P}={\frac {\partial ^{2}G}{\partial T\partial P}}}$

More relations include the following.

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

Other differential equations are:

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

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

Physical situation	Nomenclature	Equations
Net intensity emission/absorption	Texternal = external temperature (outside of system) Tsystem = internal temperature (inside system) ε = emmisivity	${\displaystyle I=\sigma \epsilon \left(T_{\mathrm {external} }^{4}-T_{\mathrm {system} }^{4}\right)\,\!}$
Internal energy of a substance	CV = isovolumetric heat capacity of substance ΔT = temperature change of substance	${\displaystyle \Delta U=NC_{V}\Delta T\,\!}$
Meyer's equation	Cp = isobaric heat capacity CV = isovolumetric heat capacity n = number of moles	${\displaystyle C_{p}-C_{V}=nR\,\!}$
Effective thermal conductivities	λi = thermal conductivity of substance i λnet = equivalent thermal conductivity	Series ${\displaystyle \lambda _{\mathrm {net} }=\sum _{j}\lambda _{j}\,\!}$ Parallel ${\displaystyle {\frac {1}{\lambda }}_{\mathrm {net} }=\sum _{j}\left({\frac {1}{\lambda }}_{j}\right)\,\!}$

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: ${\displaystyle \eta =\left|{\frac {W}{Q_{H}}}\right|\,\!}$ Carnot engine efficiency: ${\displaystyle \eta _{c}=1-\left|{\frac {Q_{L}}{Q_{H}}}\right|=1-{\frac {T_{L}}{T_{H}}}\,\!}$
Refrigeration	K = coefficient of refrigeration performance	Refrigeration performance ${\displaystyle K=\left|{\frac {Q_{L}}{W}}\right|\,\!}$ Carnot refrigeration performance ${\displaystyle K_{C}={\frac {|Q_{L}|}{|Q_{H}|-|Q_{L}|}}={\frac {T_{L}}{T_{H}-T_{L}}}\,\!}$