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

General derived quantities

Thermal properties of matter

Thermal transfer

Equations The equations in this article are classified by subject.

Thermodynamic processes

Kinetic theory

Ideal gas

Entropy

S = k B ln ⁡ Ω {\displaystyle S=k_{\mathrm {B} }\ln \Omega } , where kB is the Boltzmann constant, and Ω denotes the volume of macrostate in the phase space or otherwise called thermodynamic probability.

d S = δ Q T {\displaystyle dS={\frac {\delta Q}{T}}} , 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.

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:

d U = δ Q − δ W {\displaystyle dU=\delta Q-\delta W}

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:

Maxwell's relations The four most common Maxwell's relations are:

More relations include the following.

Other differential equations are:

Quantum properties

U = N k B T 2 ( ∂ ln ⁡ Z ∂ T ) V {\displaystyle U=Nk_{\text{B}}T^{2}\left({\frac {\partial \ln Z}{\partial T}}\right)_{V}}

S = U T + N k B ln ⁡ Z − N k ln ⁡ N + N k {\displaystyle S={\frac {U}{T}}+Nk_{\text{B}}\ln Z-Nk\ln N+Nk} 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:

Thermal properties of matter

Thermal transfer

Thermal efficiencies

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