1. Bridge from macroscopic to microscopic physics
    Role in the equipartition of energy  Role in Boltzmann factors  Role in the statistical definition of entropy  Role in semiconductor physics: the thermal voltage 

2. History

3. Value in different units
    Planck units 

4. See also

5. Notes

6. References

7. External links

The Boltzmann constant ({{math|k_B}} or {{mvar|k}}) is a physical constant relating the average relative kinetic energy of particles in a gas with the temperature of the gas^[2] and occurs in Planck's law of black-body radiation and in Boltzmann's entropy formula. It was introduced by Max Planck, but named after Ludwig Boltzmann.

It is the gas constant {{mvar|R}} divided by the Avogadro constant {{math|N_A}}:

The Boltzmann constant has the dimension energy divided by temperature, the same as entropy.

As of 2017, its value in SI units is a measured quantity. The recommended value (as of 2015, with standard uncertainty in parentheses) is {{val|1.38064852|(79)|e=-23|u=J|up=K}}.^[3]

Historically, measurements of the Boltzmann constant depended on the definition of the kelvin in terms of the triple point of water. However, in the redefinition of SI base units adopted at the 26th General Conference on Weights and Measures (CGPM) on 16 November 2018,^[4] the definition of the kelvin was changed to one based on a fixed, exact numerical value of the Boltzmann constant, similar to the way that the speed of light was given an exact numerical value at the 17th CGPM in 1983.^[5] The final value (based on the 2017 CODATA adjusted value of {{val|1.38064903|(51)|e=-23|u=J|up=K}}) is {{val|1.380649|e=-23|u=J|up=K}}.^[6][7]

Bridge from macroscopic to microscopic physics

The Boltzmann constant, {{mvar|k}}, is a scaling factor between macroscopic (thermodynamic temperature) and microscopic (thermal energy) physics. Macroscopically, the ideal gas law states that, for an ideal gas, the product of pressure {{mvar|p}} and volume {{mvar|V}} is proportional to the product of amount of substance {{mvar|n}} (in moles) and absolute temperature {{mvar|T}}:

where {{mvar|R}} is the gas constant ({{val|8.3144598|(48)|u=J⋅K⁻¹⋅mol⁻¹}}^[1]). Introducing the Boltzmann constant transforms the ideal gas law into an alternative form:

where {{mvar|N}} is the number of molecules of gas. For {{nowrap|1={{mvar|n}} = 1 mol}}, {{mvar|N}} is equal to the number of particles in one mole (Avogadro's number).

Role in the equipartition of energy

Given a thermodynamic system at an absolute temperature {{mvar|T}}, the average thermal energy carried by each microscopic degree of freedom in the system is {{math|{{sfrac|1|2}}kT}} (i.e., about {{val|2.07|e=−21|u=J}}, or {{val|0.013|ul=eV}}, at room temperature).

In classical statistical mechanics, this average is predicted to hold exactly for homogeneous ideal gases. Monatomic ideal gases (the 6 noble gases) possess three degrees of freedom per atom, corresponding to the three spatial directions, which means a thermal energy of {{math|{{sfrac|3|2}}kT}} per atom. This corresponds very well with experimental data. The thermal energy can be used to calculate the root-mean-square speed of the atoms, which turns out to be inversely proportional to the square root of the atomic mass. The root mean square speeds found at room temperature accurately reflect this, ranging from {{val|1370|u=m/s}} for helium, down to {{val|240|u=m/s}} for xenon.

Kinetic theory gives the average pressure {{mvar|p}} for an ideal gas as

Combination with the ideal gas law

shows that the average translational kinetic energy is

Considering that the translational motion velocity vector {{math|v}} has three degrees of freedom (one for each dimension) gives the average energy per degree of freedom equal to one third of that, i.e. {{math|{{sfrac|1|2}}kT}}.

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.

Role in Boltzmann factors

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

where {{mvar|Z}} is the partition function. Again, it is the energy-like quantity {{mvar|kT}} that takes central importance.

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

Role in the statistical definition of entropy

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

This equation, which relates the microscopic details, or microstates, of the system (via {{mvar|W}}) to its macroscopic state (via the entropy {{mvar|S}}), is the central idea of statistical mechanics. Such is its importance that it is inscribed on Boltzmann's tombstone.

The constant of proportionality {{mvar|k}} serves to make the statistical mechanical entropy equal to the classical thermodynamic entropy of Clausius:

One could choose instead a rescaled dimensionless entropy in microscopic terms such that

This is a more natural form and this rescaled entropy exactly corresponds to Shannon's subsequent information entropy.

The characteristic energy {{mvar|kT}} is thus the energy required to increase the rescaled entropy by one nat.

Role in semiconductor physics: the thermal voltage

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 {{math|V_T}}. The thermal voltage depends on absolute temperature {{mvar|T}} as

where {{mvar|q}} is the magnitude of the electrical charge on the electron with a value {{val|1.6021766208|(98)|e=−19|ul=C}}.^[1] Equivalently,

At room temperature ({{val|300|u=K}}), {{math|V_T}} is approximately {{val|25.85|u=mV}}.^[8][9] The thermal voltage is also important in plasmas and electrolyte solutions; 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.^[10][11]

History

Although Boltzmann first linked entropy and probability in 1877, the relation was never expressed with a specific constant until Max Planck first introduced {{mvar|k}}, and gave a precise value for it ({{val|1.346|e=−23|u=J/K}}, about 2.5% lower than today's figure), in his derivation of the law of black body radiation in 1900–1901.^[12] 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 {{mvar|R}}, and macroscopic energies for macroscopic quantities of the substance. The iconic terse form of the equation {{math|1=S = k ln W}} on Boltzmann's tombstone is in fact due to Planck, not Boltzmann. Planck actually introduced it in the same work as his eponymous {{mvar|h}}.^[13]

In 1920, Planck wrote in his Nobel Prize lecture:^[14]

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. There was no agreement whether chemical molecules, as measured by atomic weights, were the same as physical molecules, as measured by kinetic theory. Planck's 1920 lecture continued:^[14]

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.^[15][16] This decade-long effort was undertaken with different techniques by several laboratories;{{efn|Independent techniques exploited : acoustic gas thermometry, dielectric constant gas thermometry, johnson noise thermometry. Involved laboratories cited by CODATA in 2017 : LNE-Cnam (France), NPL (UK), [https://www.inrim.it/ INRIM] (Italy), PTB (Germany), NIST (USA), NIM (China).}} it is one of the cornerstones of the 2019 redefinition of SI base units. Based on these measurements, the CODATA recommended 1.380 649 × 10⁻²³ J⋅K⁻¹ to be the final fixed value of the Boltzmann constant to be used for the International System of Units.^[17]

Value in different units