# Boltzmann constant
The Boltzmann constant, symbol \( k \) or \( k_{\mathrm{B}} \), is the proportionality constant that relates thermodynamic temperature to energy: the thermal energy scale of a system at temperature \( T \) is \( kT \). Since the 2018 revision of the [International System of Units](https://www.edgepedia.org/international-system-of-units) (SI), it has had the exact fixed value \( k = 1.380\,649 \times 10^{-23} \ \mathrm{J\ K^{-1}} \), and it serves as one of the seven defining constants of the SI, fixing the kelvin.[1](https://www.bipm.org/en/si-base-units/kelvin) • [3](https://www.nist.gov/si-redefinition/meet-constants)
| Key fact | Detail |
|---|---|
| Exact value | \( k = 1.380\,649 \times 10^{-23} \ \mathrm{J\ K^{-1}} \), fixed with no uncertainty in the revised SI[1](https://www.bipm.org/en/si-base-units/kelvin) |
| Physical meaning | Proportionality constant between temperature (kelvin) and energy (joule); one kelvin corresponds to a change in thermal energy \( kT \) of \( 1.380\,649 \times 10^{-23} \) J[1](https://www.bipm.org/en/si-base-units/kelvin) • [7](https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf/2d2b50bf-f2b4-9661-f402-5f9d66e4b507?download=true&t=1632138671693&version=1.10) |
| Relation to other constants | \( k = R/N_{\mathrm{A}} \), the universal gas constant divided by the Avogadro constant; sometimes called the gas constant per molecule[4](https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node56.html) |
| Role in entropy | In statistical physics, \( S = k \ln \Omega \), connecting entropy \( S \) with the number \( \Omega \) of quantum-mechanically accessible states[7](https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf/2d2b50bf-f2b4-9661-f402-5f9d66e4b507?download=true&t=1632138671693&version=1.10) • [5](https://www.phys.unsw.edu.au/~jw/entropy.html) |
| Typical scale | At room temperature, \( k_{\mathrm{B}}T \) is about 25 meV; each degree of freedom carries average kinetic energy \( \tfrac{1}{2}k_{\mathrm{B}}T \)[10](https://www.rug.nl/education/courses/other-education/radiation-protection/a-z/a-z-data/energy-thermal?lang=en) |
| Defining role | The kelvin is defined by taking the fixed numerical value of \( k \) to be \( 1.380\,649 \times 10^{-23} \) when expressed in \( \mathrm{J\ K^{-1}} \)[1](https://www.bipm.org/en/si-base-units/kelvin) |
| Measurement history | The final pre-redefinition value combined acoustic gas thermometry, dielectric-constant gas thermometry, and Johnson noise thermometry to a relative standard uncertainty of \( 3.7 \times 10^{-7} \)[9](https://pmc.ncbi.nlm.nih.gov/articles/PMC6508687/) |
## Physical meaning
The Boltzmann constant converts between the two scales in which energy at the molecular level is expressed. The total kinetic energy \( E \) of a gas, in joules, is related to its temperature \( T \) in kelvins through \( E = k_{\mathrm{B}}T \) up to a numerical factor set by the number of degrees of freedom.[2](https://www.nist.gov/si-redefinition/kelvin/kelvin-boltzmann-constant) More precisely, each degree of freedom of motion carries an average kinetic energy of \( \tfrac{1}{2}k_{\mathrm{B}}T \), a result associated with Ludwig Eduard Boltzmann (1844–1906).[10](https://www.rug.nl/education/courses/other-education/radiation-protection/a-z/a-z-data/energy-thermal?lang=en) At room temperature the product \( k_{\mathrm{B}}T \) is about 25 meV, which is the energy scale used, for example, to characterize thermal neutrons in reactor physics.[10](https://www.rug.nl/education/courses/other-education/radiation-protection/a-z/a-z-data/energy-thermal?lang=en)
The constant also enters the statistical definition of entropy. The SI Brochure describes \( k \) as a proportionality constant between the quantities temperature and energy, whose numerical value comes from historical specifications of the temperature scale, and notes that in statistical physics it connects entropy \( S \) with the number \( \Omega \) of quantum-mechanically accessible states through \( S = k \ln \Omega \).[7](https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf/2d2b50bf-f2b4-9661-f402-5f9d66e4b507?download=true&t=1632138671693&version=1.10) This is the famous Boltzmann entropy formula, \( S = k_{\mathrm{B}} \ln W \), where \( W \) is the number of different possible configurations of a system; the equation is carved on Boltzmann's tombstone.[5](https://www.phys.unsw.edu.au/~jw/entropy.html)
Because temperature in kelvins is proportional to average molecular kinetic energy, temperature itself could in principle be measured in joules. In a completely rational system of units, entropy would be a pure number, and so would \( k_{\mathrm{B}} \) and the gas constant \( R \); measuring them in joules per kelvin is a historical accident, and \( k_{\mathrm{B}} \) is a conversion factor between units rather than a statement about the universe in the way that, for example, the gravitational constant is.[5](https://www.phys.unsw.edu.au/~jw/entropy.html) Some fields in fact work in energy units for temperature, quoting plasma temperatures in electronvolts.[5](https://www.phys.unsw.edu.au/~jw/entropy.html)
## Relation to the gas constant and Avogadro constant
The Boltzmann constant is the per-molecule version of the molar gas constant: \( k = R/N_{\mathrm{A}} \), where \( R \) is the universal gas constant, \( 8.3143 \ \mathrm{J/(mol \cdot K)} \), and \( N_{\mathrm{A}} \) is Avogadro's number, \( 6.02 \times 10^{23} \) molecules per mole.[4](https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node56.html) With this value, the statistical definition of entropy coincides with the macroscopic definition inherited from classical thermodynamics.[4](https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node56.html) In the revised SI both \( k \) and \( N_{\mathrm{A}} \) are fixed defining constants, \( N_{\mathrm{A}} \) being exactly \( 6.022\,140\,76 \times 10^{23} \ \mathrm{mol^{-1}} \).[3](https://www.nist.gov/si-redefinition/meet-constants)
## Role in the redefined kelvin
Before 2018 the kelvin was defined through the triple point of a particular isotopic mixture of water, set equal to 273.16 K. Because two exactly identical formulations of that water mixture cannot be prepared, kelvin measurements differed slightly from setup to setup.[2](https://www.nist.gov/si-redefinition/kelvin/kelvin-boltzmann-constant) In November 2018 the kelvin was redefined, and the Boltzmann constant was given its exact fixed value in joules per kelvin.[2](https://www.nist.gov/si-redefinition/kelvin/kelvin-boltzmann-constant) The BIPM states the definition formally: the kelvin is defined by taking the fixed numerical value of \( k \) to be \( 1.380\,649 \times 10^{-23} \) when expressed in \( \mathrm{J\ K^{-1}} \), which equals \( \mathrm{kg\ m^2\ s^{-2}\ K^{-1}} \), with the kilogram, meter, and second defined in terms of \( h \), \( c \), and \( \Delta\nu_{\mathrm{Cs}} \).[1](https://www.bipm.org/en/si-base-units/kelvin) The practical effect is that one kelvin equals the change of thermodynamic temperature that changes the thermal energy \( kT \) by \( 1.380\,649 \times 10^{-23} \) J.[1](https://www.bipm.org/en/si-base-units/kelvin)
The metrological community set explicit conditions before fixing the value: at least one measurement method for \( k_{\mathrm{B}} \) had to reach an uncertainty below 1 part in a million, and at least one fundamentally different type of measurement had to yield similar values with uncertainties below 3 parts per million.[2](https://www.nist.gov/si-redefinition/kelvin/kelvin-boltzmann-constant) These conditions were met in 2017, and [acoustic thermometry](https://www.edgepedia.org/acoustic-thermometry), dielectric-constant gas thermometry (DCGT), and Johnson noise thermometry (JNT) results from various groups were used in the final determination.[2](https://www.nist.gov/si-redefinition/kelvin/kelvin-boltzmann-constant)
## How the constant was measured
The most accurate pre-redefinition values came from acoustic gas thermometry, which exploits the fact that the speed of sound in a gas depends directly on its temperature; NIST has a long history with this method, and Michael Moldover and colleagues developed acoustic resonators for some of the most accurate measurements of the constant.[2](https://www.nist.gov/si-redefinition/kelvin/kelvin-boltzmann-constant) A review of determinations of \( k \) describes the main experimental techniques as acoustic thermometry, Johnson noise thermometry, dielectric constant gas thermometry, and Doppler-broadened molecular spectroscopy, and highlights the decisive role of ab initio calculations of the thermophysical properties of gases, especially helium-4.[8](https://www.sciencedirect.com/science/article/pii/S1631070518301348)
Johnson noise thermometry is an electronic approach: the thermal noise power generated by a sensing resistor immersed in a triple-point-of-water cell is compared with the noise power of a quantum-accurate pseudo-random noise waveform. A 2017 NIST measurement using a 100 Ω resistor, matched beryllium-copper coaxial leads, and a 100-day integration period gave \( k = 1.3806497(38) \times 10^{-23} \ \mathrm{J/K} \), a relative standard uncertainty of \( 2.8 \times 10^{-6} \), and a relative offset of \( 0.89 \times 10^{-6} \) from the CODATA 2014 recommended value.[6](https://www.nist.gov/publications/improved-electronic-measurement-boltzmann-constant-johnson-noise-thermometry-0) Earlier acoustic gas thermometry results had reached relative uncertainty below \( 10^{-6} \), supported by a DCGT result at \( 1.9 \times 10^{-6} \), together meeting the [International Committee for Weights and Measures](https://www.edgepedia.org/international-committee-for-weights-and-measures)' requirements for the 2018 redefinition.[6](https://www.nist.gov/publications/improved-electronic-measurement-boltzmann-constant-johnson-noise-thermometry-0)
The final adjusted value adopted for the redefinition carried a relative standard uncertainty of \( 3.7 \times 10^{-7} \), based on experiments with the three methods of acoustic gas thermometry, dielectric-constant gas thermometry, and Johnson noise thermometry.[9](https://pmc.ncbi.nlm.nih.gov/articles/PMC6508687/) After the redefinition, the LNE-Cnam laboratory determined \( k \) by acoustic thermometry with helium-4 gas in a 3-liter quasi-spherical resonator, achieving the lowest uncertainty recorded for this measurement, an uncertainty that will remain unmatched now that the numerical value of \( k \) is fixed.[11](https://doi.org/10.59161/VYPX3641)
## History of the name
The name honors Boltzmann's statistical interpretation of entropy, of which \( S = k \ln W \) became the emblem.[5](https://www.phys.unsw.edu.au/~jw/entropy.html)
## References
1. [SI base unit: kelvin (K), BIPM](https://www.bipm.org/en/si-base-units/kelvin)
2. [Kelvin: Boltzmann Constant, NIST SI Redefinition](https://www.nist.gov/si-redefinition/kelvin/kelvin-boltzmann-constant)
3. [Meet the Constants, NIST SI Redefinition](https://www.nist.gov/si-redefinition/meet-constants)
4. [A Statistical Definition of Entropy, MIT Unified Engineering Thermodynamics notes](https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node56.html)
5. [Entropy and disorder. Does entropy have dimensions?, Joe Wolfe, UNSW Sydney](https://www.phys.unsw.edu.au/~jw/entropy.html)
6. [Qu, J. et al. (2017). An improved electronic measurement of the Boltzmann constant by Johnson noise thermometry. Metrologia 54(4), NIST](https://www.nist.gov/publications/improved-electronic-measurement-boltzmann-constant-johnson-noise-thermometry-0)
7. [The International System of Units (SI Brochure, 9th edition), BIPM](https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf/2d2b50bf-f2b4-9661-f402-5f9d66e4b507?download=true&t=1632138671693&version=1.10)
8. [Determinations of the Boltzmann constant, Comptes Rendus Physique (ScienceDirect)](https://www.sciencedirect.com/science/article/pii/S1631070518301348)
9. [The Boltzmann project, Metrologia (PMC)](https://pmc.ncbi.nlm.nih.gov/articles/PMC6508687/)
10. [Energy (thermal), Groningen Academy for Radiation Protection, University of Groningen](https://www.rug.nl/education/courses/other-education/radiation-protection/a-z/a-z-data/energy-thermal?lang=en)
11. [Pitre, L. et al. (2025). Redefining Exactitude at LNE-Cnam: The Last and Most Accurate Boltzmann Constant Measurement. BIPM](https://doi.org/10.59161/VYPX3641)
12. [Robotti, N. and Badino, M. Max Planck and the 'Constants of Nature', University of Pavia](https://fisica.unipv.it/percorsi/pdf/robotti-badino.pdf)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Thermodynamics › Statistical mechanics and kinetic theory › Entropy, microstates, and information-theoretic links*
*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026; Oct 21, 2026 · Last review: Oct 15, 2026*
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