The equations in this article are classified by subject.
Thermodynamic processes
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Physical situation |
Equations |
Isentropic process (adiabatic and reversible) |
For an ideal gas


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Isothermal process |
For an ideal gas
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Isobaric process |
p1 = p2, p = constant

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Isochoric process |
V1 = V2, V = constant

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Free expansion |
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Work done by an expanding gas |
Process

Net work done in cyclic processes
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Kinetic theory
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Ideal gas
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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.
More information Ratio of thermal to rest mass-energy of each molecule:
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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)
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Entropy Logarithm of the density of states |
- Pi = probability of system in microstate i
- Ω = total number of microstates
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where:
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Entropy change |
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Entropic force |
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Equipartition theorem |
df = degree of freedom |
Average kinetic energy per degree of freedom

Internal energy
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Corollaries of the non-relativistic Maxwell–Boltzmann distribution are below.
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,
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Name |
Symbol |
Formula |
Natural variables |
Internal energy |
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Helmholtz free energy |
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Enthalpy |
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Gibbs free energy |
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Landau potential, or grand potential |
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and the corresponding fundamental thermodynamic relations or "master equations"[2] are:
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Potential |
Differential |
Internal energy |
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Enthalpy |
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Helmholtz free energy |
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Gibbs free energy |
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Maxwell's relations
The four most common Maxwell's relations are:
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Physical situation |
Nomenclature |
Equations |
Thermodynamic potentials as functions of their natural variables |
= Internal energy
= Enthalpy
= Helmholtz free energy
= Gibbs free energy
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More relations include the following.
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Other differential equations are:
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: