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 QL J ML2T−2

General derived quantities

Quantity (common name/s) (Common) 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 (of component i in a mixture) μi , where is not proportional to because depends on pressure. , where is proportional to (as long as the molar ratio composition of the system remains the same) 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, ΦG 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 (common name/s) (Common) symbol/s Defining equation SI unit Dimension General heat/thermal capacity C J⋅K−1 ML2T−2Θ−1 Heat capacity (isobaric) Cp J⋅K−1 ML2T−2Θ−1 Specific heat capacity (isobaric) Cmp J⋅kg−1⋅K−1 L2T−2Θ−1 Molar specific heat capacity (isobaric) Cnp J⋅K−1⋅mol−1 ML2T−2Θ−1N−1 Heat capacity (isochoric/volumetric) CV J⋅K−1 ML2T−2Θ−1 Specific heat capacity (isochoric) CmV J⋅kg−1⋅K−1 L2T−2Θ−1 Molar specific heat capacity (isochoric) CnV 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 (common name/s) (Common) 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 (vector analogue of thermal intensity above) q 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)

For an ideal gasIsothermal processFor an ideal gasIsobaric processp1 = 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

+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 Pressure of an ideal gas m = mass of one molecule Mm = molar mass

Ideal gas

+ Quantity General Equation IsobaricΔp = 0 IsochoricΔV = 0 IsothermalΔT = 0 Adiabatic Work W Heat Capacity C (as for real gas) (for monatomic ideal gas) (for diatomic ideal gas) (for monatomic ideal gas) (for diatomic ideal gas) Internal Energy ΔU Enthalpy ΔH Entropy Δs Keenan, Thermodynamics, Wiley, New York, 1947 Constant

Entropy

, where kB 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 (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:

K2 is the modified Bessel function of the second kind. Non-relativistic speeds

Relativistic speeds (Maxwell–Jüttner distribution) Entropy Logarithm of the density of states Pi = probability of system in microstate i Ω = total number of microstates

where: Entropy change Entropic force Equipartition theorem df = 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-section n = 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"Physical chemistry, P.W. Atkins, Oxford University Press, 1978, 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 (heteronuclear molecules) σ = 2 (homonuclear)

Thermal properties of matter

Coefficients Equation Joule-Thomson coefficient Compressibility (constant temperature) Coefficient of thermal expansion (constant pressure) Heat capacity (constant pressure) Heat capacity (constant volume)

Derivation of heat capacity (constant pressure)Since

Derivation of heat capacity (constant volume)Since

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

Thermal transfer

Physical situation Nomenclature Equations Net intensity emission/absorption Texternal = external temperature (outside of system) Tsystem = internal temperature (inside system) ε = emissivity Internal energy of a substance CV = isovolumetric heat capacity of substance ΔT = temperature change of substance Meyer's equation Cp = isobaric heat capacity CV = isovolumetric heat capacity n = amount of substance Effective thermal conductivities λi = thermal conductivity of substance i λnet = equivalent thermal conductivity Series

Parallel

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:

Carnot engine efficiency: Refrigeration K = coefficient of refrigeration performance Refrigeration performance

Carnot refrigeration performance

Table of thermodynamic equations

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