Table of thermodynamic equations
The classical Carnot heat engine
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.
Many of the definitions below are also used in the thermodynamics of chemical reactions.
General basic quantities
General derived quantities
Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension Thermodynamic beta, Inverse temperature β J−1 [T]2[M]−1[L]−2 Thermodynamic temperature τ J [M] [L]2 [T]−2 Entropy S
J K−1 [M][L]2[T]−2 [Θ]−1 Pressure P Pa M L−1T−2 Internal Energy U J [M][L]2[T]−2 Enthalpy H J [M][L]2[T]−2 Partition Function Z dimensionless dimensionless Gibbs free energy G J [M][L]2[T]−2 Chemical potential (of
component i in a mixture)
, where F is not proportional to N because μi depends on pressure. , 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.
J [M][L]2[T]−2 Helmholtz free energy A, F J [M][L]2[T]−2 Landau potential, Landau Free Energy, Grand potential Ω, ΦG J [M][L]2[T]−2 Massieu Potential, Helmholtz free entropy Φ J K−1 [M][L]2[T]−2 [Θ]−1 Planck potential, Gibbs free entropy Ξ 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 J K −1 [M][L]2[T]−2 [Θ]−1 Heat capacity (isobaric) Cp J K −1 [M][L]2[T]−2 [Θ]−1 Specific heat capacity (isobaric) Cmp J kg−1 K−1 [L]2[T]−2 [Θ]−1 Molar specific heat capacity (isobaric) Cnp J K −1 mol−1 [M][L]2[T]−2 [Θ]−1 [N]−1 Heat capacity (isochoric/volumetric) CV J K −1 [M][L]2[T]−2 [Θ]−1 Specific heat capacity (isochoric) CmV J kg−1 K−1 [L]2[T]−2 [Θ]−1 Molar specific heat capacity (isochoric) CnV J K −1 mol−1 [M][L]2[T]−2 [Θ]−1 [N]−1 Specific latent heat L J kg−1 [L]2[T]−2 Ratio of isobaric to isochoric heat capacity, heat capacity ratio, adiabatic index γ dimensionless dimensionless
Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension Temperature gradient No standard symbol K m−1 [Θ][L]−1 Thermal conduction rate, thermal current, thermal/heat flux, thermal power transfer P W = J s−1 [M] [L]2 [T]−3 Thermal intensity I W m−2 [M] [T]−3 Thermal/heat flux density (vector analogue of thermal intensity above) q W m−2 [M] [T]−3
The equations in this article are classified by subject.
Physical situation Equations Isentropic process (adiabatic and reversible)
For an ideal gas
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
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
Pressure of an ideal gas
- m = mass of one molecule
- Mm = molar mass
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.
Relativistic speeds (Maxwell-Jüttner distribution)
Entropy Logarithm of the density of states
- Pi = probability of system in microstate i
- Ω = total number of microstates
Entropic force Equipartition theorem
- df = degree of freedom
Average kinetic energy per degree of freedom
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
- ℓ = Mean free path
Quasi-static and reversible processes
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,
|Helmholtz free energy|
|Gibbs free energy|
|Landau Potential (Grand potential)||,|Potential Differential Internal energy Enthalpy Helmholtz free energy Gibbs free energy
The four most common Maxwell's relations are:
Physical situation Nomenclature Equations Thermodynamic potentials as functions of their natural variables
More relations include the following.
Other differential equations are:
Name H U G Gibbs–Helmholtz equation
- Indistinguishable Particles
Degree of freedom Partition function Translation Vibration Rotation
- 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)
Derivation of heat capacity (constant volume)
(where δWrev is the work done by the system),
Physical situation Nomenclature Equations Net intensity emission/absorption
- Texternal = external temperature (outside of system)
- Tsystem = internal temperature (inside system)
- ε = emmisivity
Internal energy of a substance
- CV = isovolumetric heat capacity of substance
- ΔT = temperature change of substance
- Cp = isobaric heat capacity
- CV = isovolumetric heat capacity
- n = number of moles
Effective thermal conductivities
- λi = thermal conductivity of substance i
- λnet = equivalent thermal conductivity
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
Carnot engine efficiency:
- K = coefficient of refrigeration performance
Carnot refrigeration performance
- Antoine equation
- Bejan number
- Bowen ratio
- Bridgman's equations
- Clausius–Clapeyron relation
- Departure functions
- Duhem–Margules equation
- Ehrenfest equations
- Gibbs–Helmholtz equation
- Gibbs' phase rule
- Kopp's law
- Kopp–Neumann law
- Noro–Frenkel law of corresponding states
- Onsager reciprocal relations
- Stefan number
- Triple product rule
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- Chapters 1–10, Part 1: "Equilibrium".
- Bridgman, P.W., Phys. Rev., 3, 273 (1914).
- 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.
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