[2] It occurs in the definitions of the kelvin (K) and the gas constant, in Planck's law of black-body radiation and Boltzmann's entropy formula, and is used in calculating thermal noise in resistors.

The Boltzmann constant has dimensions of energy divided by temperature, the same as entropy and heat capacity.

[1] Correspondingly, the SI units for temperature and energy are calibrated to one another so that kB kelvin = 1.380649×10−23 joules.

[3] Macroscopically, the ideal gas law states that, for an ideal gas, the product of pressure p and volume V is proportional to the product of amount of substance n and absolute temperature T:

Given a thermodynamic system at an absolute temperature T, the average thermal energy carried by each microscopic degree of freedom in the system is ⁠1/2⁠ kT (i.e., about 2.07×10−21 J, or 0.013 eV, at room temperature).

This is generally true only for classical systems with a large number of particles, and in which quantum effects are negligible.

In classical statistical mechanics, this average is predicted to hold exactly for homogeneous ideal gases.

The root mean square speeds found at room temperature accurately reflect this, ranging from 1370 m/s for helium, down to 240 m/s for xenon.

The ideal gas equation is also obeyed closely by molecular gases; but the form for the heat capacity is more complicated, because the molecules possess additional internal degrees of freedom, as well as the three degrees of freedom for movement of the molecule as a whole.

Diatomic gases, for example, possess a total of six degrees of simple freedom per molecule that are related to atomic motion (three translational, two rotational, and one vibrational).

At lower temperatures, not all these degrees of freedom may fully participate in the gas heat capacity, due to quantum mechanical limits on the availability of excited states at the relevant thermal energy per molecule.

More generally, systems in equilibrium at temperature T have probability Pi of occupying a state i with energy E weighted by the corresponding Boltzmann factor:

Consequences of this include (in addition to the results for ideal gases above) the Arrhenius equation in chemical kinetics.

In statistical mechanics, the entropy S of an isolated system at thermodynamic equilibrium is defined as the natural logarithm of W, the number of distinct microscopic states available to the system given the macroscopic constraints (such as a fixed total energy E):

This equation, which relates the microscopic details, or microstates, of the system (via W) to its macroscopic state (via the entropy S), is the central idea of statistical mechanics.

In semiconductors, the Shockley diode equation—the relationship between the flow of electric current and the electrostatic potential across a p–n junction—depends on a characteristic voltage called the thermal voltage, denoted by VT.

At room temperature 300 K (27 °C; 80 °F), VT is approximately 25.85 mV[7][8] which can be derived by plugging in the values as follows:

The thermal voltage is also important in plasmas and electrolyte solutions (e.g. the Nernst equation); in both cases it provides a measure of how much the spatial distribution of electrons or ions is affected by a boundary held at a fixed voltage.

Although Boltzmann first linked entropy and probability in 1877, the relation was never expressed with a specific constant until Max Planck first introduced k, and gave a more precise value for it (1.346×10−23 J/K, about 2.5% lower than today's figure), in his derivation of the law of black-body radiation in 1900–1901.

[11] Before 1900, equations involving Boltzmann factors were not written using the energies per molecule and the Boltzmann constant, but rather using a form of the gas constant R, and macroscopic energies for macroscopic quantities of the substance.

Planck actually introduced it in the same work as his eponymous h.[12] In 1920, Planck wrote in his Nobel Prize lecture:[13] This constant is often referred to as Boltzmann's constant, although, to my knowledge, Boltzmann himself never introduced it—a peculiar state of affairs, which can be explained by the fact that Boltzmann, as appears from his occasional utterances, never gave thought to the possibility of carrying out an exact measurement of the constant.This "peculiar state of affairs" is illustrated by reference to one of the great scientific debates of the time.

There was considerable disagreement in the second half of the nineteenth century as to whether atoms and molecules were real or whether they were simply a heuristic tool for solving problems.

Planck's 1920 lecture continued:[13] Nothing can better illustrate the positive and hectic pace of progress which the art of experimenters has made over the past twenty years, than the fact that since that time, not only one, but a great number of methods have been discovered for measuring the mass of a molecule with practically the same accuracy as that attained for a planet.In versions of SI prior to the 2019 revision of the SI, the Boltzmann constant was a measured quantity rather than having a fixed numerical value.

In 2017, the most accurate measures of the Boltzmann constant were obtained by acoustic gas thermometry, which determines the speed of sound of a monatomic gas in a triaxial ellipsoid chamber using microwave and acoustic resonances.

[14][15][16] This decade-long effort was undertaken with different techniques by several laboratories;[a] it is one of the cornerstones of the 2019 revision of the SI.

Based on these measurements, the CODATA recommended 1.380649×10−23 J/K to be the final fixed value of the Boltzmann constant to be used for the International System of Units.

The Boltzmann constant provides a mapping from the characteristic microscopic energy E to the macroscopic temperature scale T = ⁠E/k⁠.

In fundamental physics, this mapping is often simplified by using the natural units of setting k to unity.

[22][23] In particular, the SI unit kelvin becomes superfluous, being defined in terms of joules as 1 K = 1.380649×10−23 J.

[24] With this convention, temperature is always given in units of energy, and the Boltzmann constant is not explicitly needed in formulas.