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	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	symbol/s	Defining equation	SI unit	Dimension
Thermodynamic beta, inverse temperature	β		J⁻¹	T²M⁻¹L⁻²
Thermodynamic temperature	τ		J	ML²T⁻²
Entropy	S	,	J⋅K⁻¹	ML²T⁻²Θ⁻¹
Pressure	P		Pa	ML⁻¹T⁻²
Internal Energy	U		J	ML²T⁻²
Enthalpy	H		J	ML²T⁻²
Partition Function	Z		1	1
Gibbs free energy	G		J	ML²T⁻²
Chemical potential	μ_i	, where is not proportional to because depends on pressure. , where is proportional to because depends only on temperature and pressure and composition.	J	ML²T⁻²
Helmholtz free energy	A, F		J	ML²T⁻²
Landau potential, Landau free energy, Grand potential	Ω, Φ_G		J	ML²T⁻²
Massieu potential, Helmholtz free entropy	Φ		J⋅K⁻¹	ML²T⁻²Θ⁻¹
Planck potential, Gibbs free entropy	Ξ		J⋅K⁻¹	ML²T⁻²Θ⁻¹

Thermal properties of matter

Quantity	symbol/s	Defining equation	SI unit	Dimension
General heat/thermal capacity	C		J⋅K⁻¹	ML²T⁻²Θ⁻¹
Heat capacity	C_p		J⋅K⁻¹	ML²T⁻²Θ⁻¹
Specific heat capacity	C_mp		J⋅kg⁻¹⋅K⁻¹	L²T⁻²Θ⁻¹
Molar specific heat capacity	C_np		J⋅K⁻¹⋅mol⁻¹	ML²T⁻²Θ⁻¹N⁻¹
Heat capacity	C_V		J⋅K⁻¹	ML²T⁻²Θ⁻¹
Specific heat capacity	C_mV		J⋅kg⁻¹⋅K⁻¹	L²T⁻²Θ⁻¹
Molar specific heat capacity	C_nV		J⋅K⋅⁻¹ mol⁻¹	ML²T⁻²Θ⁻¹N⁻¹
Specific latent heat	L		J⋅kg⁻¹	L²T⁻²
Ratio of isobaric to isochoric heat capacity, heat capacity ratio, adiabatic index, Laplace coefficient	γ		1	1

Thermal transfer

Quantity	symbol/s	Defining equation	SI unit	Dimension
Temperature gradient	No standard symbol		K⋅m⁻¹	ΘL⁻¹
Thermal conduction rate, thermal current, thermal/heat flux, thermal power transfer	P		W	ML²T⁻³
Thermal intensity	I		W⋅m⁻²	MT⁻³
Thermal/heat flux density	q		W⋅m⁻²	MT⁻³

Equations

The equations in this article are classified by subject.

Thermodynamic processes

Physical situation	Equations
Isentropic process	For an ideal gas
Isothermal process	For an ideal gas
Isobaric process	p₁ = p₂, p = constant
Isochoric process	V₁ = V₂, V = constant
Free expansion
Work done by an expanding gas	Process Net work done in cyclic processes

Kinetic theory

Physical situation	Nomenclature	Equations
Ideal gas law	p = pressureV = volume of containerT = temperature n = amount of substanceR = gas constantN = number of moleculesk = Boltzmann constant
Pressure of an ideal gas	m = mass of one moleculeM_m = molar mass

Ideal gas

Quantity	General Equation	Isobaric Δp = 0	Isochoric ΔV = 0	Isothermal ΔT = 0	Adiabatic
Work W
Heat Capacity C
Internal Energy ΔU
Enthalpy ΔH
Entropy Δs
Constant

Entropy

, where k_B is the Boltzmann constant, and Ω denotes the volume of macrostate in the phase space or otherwise called thermodynamic probability.
, 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,γ = Lorentz factor as function of momentum Ratio of thermal to rest mass-energy of each molecule: K₂ is the modified Bessel function of the second kind.	Non-relativistic speeds Relativistic speeds
Entropy Logarithm of the density of states	P_i = probability of system in microstate i Ω = total number of microstates	where:
Entropy change
Entropic force
Equipartition theorem	d_f = degree of freedom	Average kinetic energy per degree of freedom Internal energy

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

Physical situation	Nomenclature	Equations
Mean speed
Root mean square speed
Modal speed
Mean free path	σ = effective cross-sectionn = volume density of number of target particles = mean free path

Quasi-static and reversible processes

For quasi-static and reversible processes, the first law of thermodynamics is:
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,
and the corresponding fundamental thermodynamic relations or "master equations" are:

Potential	Differential
Internal energy
Enthalpy
Helmholtz free energy
Gibbs free energy

Maxwell's relations

The four most common Maxwell's relations are:

Physical situation	Nomenclature	Equations
Thermodynamic potentials as functions of their natural variables	= Internal energy = Enthalpy = Helmholtz free energy = Gibbs free energy

More relations include the following.
Other differential equations are:

Name	H	U	G
Gibbs–Helmholtz equation

Quantum properties

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
Vibration
Rotation	where:σ = 1 σ = 2

Thermal properties of matter

Coefficients	Equation
Joule-Thomson coefficient
Compressibility
Coefficient of thermal expansion
Heat capacity
Heat capacity

Derivation of heat capacity

Since

Derivation of heat capacity

Since
,

Thermal transfer

Physical situation	Nomenclature	Equations
Net intensity emission/absorption	T_external = external temperature T_system = internal temperature ε = emissivity
Internal energy of a substance	C_V = isovolumetric heat capacity of substance ΔT = temperature change of substance
Meyer's equation	C_p = isobaric heat capacityC_V = isovolumetric heat capacityn = amount of substance
Effective thermal conductivities	λ_i = thermal conductivity of substance iλ_net = equivalent thermal conductivity	Series Parallel

Thermal efficiencies

Physical situation	Nomenclature	Equations
Thermodynamic engines	η = efficiencyW = work done by engineQ_H = heat energy in higher temperature reservoirQ_L = heat energy in lower temperature reservoirT_H = temperature of higher temp. reservoirT_L = temperature of lower temp. reservoir	Thermodynamic engine: Carnot engine efficiency:
Refrigeration	K = coefficient of refrigeration performance	Refrigeration performance Carnot refrigeration performance