Boltzmann constant

Not to be confused with Stefan–Boltzmann constant.

The Boltzmann constant (k_B or k) is the proportionality factor that relates the average relative thermal energy of particles in a gas with the thermodynamic temperature of the gas.^[2] It occurs in the definitions of the kelvin (K) and the gas constant, and 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. It is named after the Austrian scientist Ludwig Boltzmann.

Boltzmann constant
Ludwig Boltzmann, the constant's namesake
Symbol:	kB, k
Value in joules per kelvin:	1.380649×10^−23 J⋅K^−1^[1]

As part of the 2019 redefinition of SI base units, the Boltzmann constant is one of the seven "defining constants" that have been given exact definitions. They are used in various combinations to define the seven SI base units. The Boltzmann constant is defined to be exactly 1.380649×10⁻²³ joules per kelvin.^[1]

Roles of the Boltzmann constant

 

Relationships between Boyle's, Charles's, Gay-Lussac's, Avogadro's, combined and ideal gas laws, with the Boltzmann constant k = R/N_A = n R/N (in each law, properties circled are variable and properties not circled are held constant)

IUPAC definition

Boltzmann constant: The Boltzmann constant, k, is one of seven fixed constants defining the International System of Units, the SI, with k = 1.380 649 x 10^-23 J K^-1. The Boltzmann constant is a proportionality constant between the quantities temperature (with unit kelvin) and energy (with unit joule). ^[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:

$${\displaystyle pV=nRT,}$$

 

where R is the molar gas constant (8.31446261815324 J⋅K⁻¹⋅mol⁻¹).^[4] Introducing the Boltzmann constant as the gas constant per molecule^[5] k = R/N_A (being N_A the Avogadro constant) transforms the ideal gas law into an alternative form:

$${\displaystyle pV=NkT,}$$

 

where N is the number of molecules of gas.

Role in the equipartition of energy

Main article: Equipartition of energy

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. Monatomic ideal gases (the six noble gases) possess three degrees of freedom per atom, corresponding to the three spatial directions. According to the equipartition of energy this means that there is a thermal energy of 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 1370 m/s for helium, down to 240 m/s for xenon.

Kinetic theory gives the average pressure p for an ideal gas as

$${\displaystyle p={\frac {1}{3}}{\frac {N}{V}}m{\overline {v^{2}}}.}$$

 

Combination with the ideal gas law

$${\displaystyle pV=NkT}$$

 

shows that the average translational kinetic energy is

$${\displaystyle {\tfrac {1}{2}}m{\overline {v^{2}}}={\tfrac {3}{2}}kT.}$$

 

Considering that the translational motion velocity vector 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. 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 T have probability P_i of occupying a state i with energy E weighted by the corresponding Boltzmann factor:

$${\displaystyle P_{i}\propto {\frac {\exp \left(-{\frac {E}{kT}}\right)}{Z}},}$$

 

where Z is the partition function. Again, it is the energy-like quantity 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

Further information: Entropy (statistical thermodynamics)

 

Boltzmann's grave in the Zentralfriedhof, Vienna, with bust and entropy formula.

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):

$${\displaystyle S=k\,\ln W.}$$

 

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. Such is its importance that it is inscribed on Boltzmann's tombstone.

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

$${\displaystyle \Delta S=\int {\frac {{\rm {d}}Q}{T}}.}$$

 

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

$${\displaystyle {S'=\ln W},\quad \Delta S'=\int {\frac {\mathrm {d} Q}{kT}}.}$$

 

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

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

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 by V_T. The thermal voltage depends on absolute temperature T as

$${\displaystyle V_{\mathrm {T} }={kT \over q}={RT \over F},}$$

 

where q is the magnitude of the electrical charge on the electron with a value 1.602176634×10⁻¹⁹ C.^[6] Equivalently,

$${\displaystyle {V_{\mathrm {T} } \over T}={k \over q}\approx 8.617333262\times 10^{-5}\ \mathrm {V/K} .}$$

 

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

$${\displaystyle V_{\mathrm {T} }={kT \over q}={\frac {1.38\times 10^{-23}\ \mathrm {J{\cdot }K^{-1}} \times 300\ \mathrm {K} }{1.6\times 10^{-19}\ \mathrm {C} }}\simeq 25.85\ \mathrm {mV} }$$

 

At the standard state temperature of 298.15 K (25.00 °C; 77.00 °F), it is approximately 25.69 mV. 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.^[9][10]

History

The Boltzmann constant is named after its 19th century Austrian discoverer, Ludwig Boltzmann. 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. The iconic terse form of the equation 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 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. 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:^[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 redefinition of the SI base units, the Boltzmann constant was a measured quantity rather than a fixed value. Its exact definition also varied over the years due to redefinitions of the kelvin (see Kelvin § History) and other SI base units (see Joule § History).

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] This decade-long effort was undertaken with different techniques by several laboratories;^[a] it is one of the cornerstones of the 2019 redefinition of SI base units. 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.^[16]

Value in different units

Values of k	Units	Comments
1.380649×10^−23	J/K	SI by definition, J/K = m^2⋅kg/(s^2⋅K) in SI base units
8.617333262×10^−5	eV/K	†
2.083661912×10^10	Hz/K	(k/h) †
1.380649×10^−16	erg/K	CGS system, 1 erg = 1×10^−7 J
3.297623483×10^−24	cal/K	† 1 calorie = 4.1868 J
1.832013046×10^−24	cal/°R	†
5.657302466×10^−24	ft lb/°R	†
0.695034800	cm^−1/K	(k/(hc)) †
3.166811563×10^−6	Eh/K	(Eh = hartree)
1.987204259×10^−3	kcal/(mol⋅K)	(kNA) †
8.314462618×10^−3	kJ/(mol⋅K)	(kNA) †
−228.5991672	dB(W/K/Hz)	10 log10(k/(1 W/K/Hz)),† used for thermal noise calculations
1.536179187×10^−40	kg/K	k/c^2, where c is the speed of light^[17]

†The value is exact but not expressible as a finite decimal; approximated to 9 decimal places only.

Since k is a proportionality factor between temperature and energy, its numerical value depends on the choice of units for energy and temperature. The small numerical value of the Boltzmann constant in SI units means a change in temperature by 1 K only changes a particle's energy by a small amount. A change of 1 °C is defined to be the same as a change of 1 K. The characteristic energy kT is a term encountered in many physical relationships.

The Boltzmann constant sets up a relationship between wavelength and temperature (dividing hc/k by a wavelength gives a temperature) with one micrometer being related to 14387.777 K, and also a relationship between voltage and temperature (kT in units of eV corresponds to a voltage) with one volt being related to 11604.518 K. The ratio of these two temperatures, 14387.777 K / 11604.518 K ≈ 1.239842, is the numerical value of hc in units of eV⋅μm.

Natural 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. This convention means that temperature and energy quantities have the same dimensions.^[18][19] In particular, the SI unit kelvin becomes superfluous, being defined in terms of joules as 1 K = 1.380649×10⁻²³ J.^[20] With this convention, temperature is always given in units of energy, and the Boltzmann constant is not explicitly needed in formulas.^[18]

This convention simplifies many physical relationships and formulas. For example, the equipartition formula for the energy associated with each classical degree of freedom (${\displaystyle {\tfrac {1}{2}}kT}$  above) becomes

$${\displaystyle E_{\mathrm {dof} }={\tfrac {1}{2}}T}$$

 

As another example, the definition of thermodynamic entropy coincides with the form of information entropy:

$${\displaystyle S=-\sum _{i}P_{i}\ln P_{i}.}$$

 

where P_i is the probability of each microstate.

See also

• Committee on Data of the International Science Council
• Thermodynamic beta
• List of scientists whose names are used in physical constants

Notes

1. 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), INRIM (Italy), PTB (Germany), NIST (USA), NIM (China).

References

1. Newell, David B.; Tiesinga, Eite (2019). The International System of Units (SI). NIST Special Publication 330. Gaithersburg, Maryland: National Institute of Standards and Technology. doi:10.6028/nist.sp.330-2019. S2CID 242934226.
2. Richard Feynman (1970). The Feynman Lectures on Physics Vol I. Addison Wesley Longman. ISBN 978-0-201-02115-8.
3. "Boltzmann constant". Gold Book. IUPAC. 2020. doi:10.1351/goldbook.B00695. Retrieved 1 April 2024.
4. "Proceedings of the 106th meeting" (PDF). 16–20 October 2017.
5. Petrucci, Ralph H.; Harwood, William S.; Herring, F. Geoffrey (2002). General Chemistry: Principles and Modern Applications (8th ed.). Prentice Hall. p. 785. ISBN 0-13-014329-4.
6. "2018 CODATA Value: elementary charge". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.
7. Rashid, Muhammad H. (2016). Microelectronic circuits: analysis and design (3rd ed.). Cengage Learning. pp. 183–184. ISBN 9781305635166.
8. Cataldo, Enrico; Di Lieto, Alberto; Maccarrone, Francesco; Paffuti, Giampiero (18 August 2016). "Measurements and analysis of current-voltage characteristic of a pn diode for an undergraduate physics laboratory". arXiv:1608.05638v1 [physics.ed-ph].
9. Kirby, Brian J. (2009). Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices (PDF). Cambridge University Press. ISBN 978-0-521-11903-0.
10. Tabeling, Patrick (2006). Introduction to Microfluidics. Oxford University Press. ISBN 978-0-19-856864-3.
11. Planck, Max (1901). "Ueber das Gesetz der Energieverteilung im Normalspectrum". Annalen der Physik. 309 (3): 553–63. Bibcode:1901AnP...309..553P. doi:10.1002/andp.19013090310.. English translation: . Archived from the original on 17 December 2008.
12. Gearhart, Clayton A. (2002). "Planck, the Quantum, and the Historians". Physics in Perspective. 4 (2): 177. Bibcode:2002PhP.....4..170G. doi:10.1007/s00016-002-8363-7. ISSN 1422-6944. S2CID 26918826.
13. Planck, Max (2 June 1920). "The Genesis and Present State of Development of the Quantum Theory". Nobel Lectures, Physics 1901-1921. Elsevier Publishing Company, Amsterdam (published 1967).
14. Pitre, L; Sparasci, F; Risegari, L; Guianvarc’h, C; Martin, C; Himbert, M E; Plimmer, M D; Allard, A; Marty, B; Giuliano Albo, P A; Gao, B; Moldover, M R; Mehl, J B (1 December 2017). (PDF). Metrologia. 54 (6): 856–873. Bibcode:2017Metro..54..856P. doi:10.1088/1681-7575/aa7bf5. hdl:11696/57295. S2CID 53680647. Archived from the original (PDF) on 5 March 2019.
15. de Podesta, Michael; Mark, Darren F; Dymock, Ross C; Underwood, Robin; Bacquart, Thomas; Sutton, Gavin; Davidson, Stuart; Machin, Graham (1 October 2017). "Re-estimation of argon isotope ratios leading to a revised estimate of the Boltzmann constant" (PDF). Metrologia. 54 (5): 683–692. Bibcode:2017Metro..54..683D. doi:10.1088/1681-7575/aa7880. S2CID 125912713.
16. Newell, D. B.; Cabiati, F.; Fischer, J.; Fujii, K.; Karshenboim, S. G.; Margolis, H. S.; Mirandés, E. de; Mohr, P. J.; Nez, F. (2018). "The CODATA 2017 values of h, e, k, and N A for the revision of the SI". Metrologia. 55 (1): L13. Bibcode:2018Metro..55L..13N. doi:10.1088/1681-7575/aa950a. ISSN 0026-1394.
17. "CODATA Value: Kelvin-kilogram relationship".
18. Kalinin, M.; Kononogov, S. (2005). "Boltzmann's Constant, the Energy Meaning of Temperature, and Thermodynamic Irreversibility". Measurement Techniques. 48 (7): 632–636. doi:10.1007/s11018-005-0195-9. S2CID 118726162.
19. Kittel, Charles; Kroemer, Herbert (1980). Thermal physics (2nd ed.). San Francisco: W. H. Freeman. p. 41. ISBN 0716710889. We prefer to use a more natural temperature scale ... the fundamental temperature has the units of energy.
20. Mohr, Peter J.; Shirley, Eric L.; Phillips, William D.; Trott, Michael (1 October 2022). "On the dimension of angles and their units". Metrologia. 59 (5): 053001. arXiv:2203.12392. Bibcode:2022Metro..59e3001M. doi:10.1088/1681-7575/ac7bc2.

External links

• Draft Chapter 2 for SI Brochure, following redefinitions of the base units (prepared by the Consultative Committee for Units)
• Big step towards redefining the kelvin: Scientists find new way to determine Boltzmann constant