Boltzmann constant

The Boltzmann constant (kB or k) is a physical constant named after its discoverer, Ludwig Boltzmann, which relates 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 is the gas constant R divided by the Avogadro constant NA:

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 1.38064852(79)×10−23 J/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 1.38064903(51)×10−23 J/K) is 1.380649×10−23 J/K.[6][7]

Values of k[1] Units
1.38064852(79)×10−23 J⋅K−1
8.6173303(50)×10−5 eV⋅K−1
1.38064852(79)×10−16 erg⋅K−1
For details, see § Value in different units below.

Bridge from macroscopic to microscopic physics

The Boltzmann constant, 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 p and volume V is proportional to the product of amount of substance n (in moles) and absolute temperature T:

where R is the gas constant (8.3144598(48) J⋅K−1⋅mol−1[1]). Introducing the Boltzmann constant transforms the ideal gas law into an alternative form:

where N is the number of molecules of gas. For n = 1 mol, 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 T, the average thermal energy carried by each microscopic degree of freedom in the system is 1/2kT (i.e., about 2.07×10−21 J, or 0.013 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 3/2kT 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

Combination with the ideal gas law

shows that the average translational kinetic energy is

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. 1/2kT.

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 Pi of occupying a state i with energy E weighted by the corresponding Boltzmann factor:

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

Zentralfriedhof Vienna - Boltzmann
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):

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:

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 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 VT. The thermal voltage depends on absolute temperature T as

where q is the magnitude of the electrical charge on the electron with a value 1.6021766208(98)×10−19 C.[1] Equivalently,

At room temperature (300 K), VT is approximately 25.85 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]


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 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.[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 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.[13]

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

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:[14]

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 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;[a] 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

Values of k Units Comments
1.38064852(79)×10−23 J/K SI units, 2014 CODATA value, J/K = m2⋅kg/(s2⋅K) in SI base units[1]
8.6173303(50)×10−5 eV/K 2014 CODATA value[1]
electronvolt = 1.6021766208(98)×10−19 J[1]

1/k = 11604.519(11) K/eV

2.0836612(12)×1010 Hz/K 2014 CODATA value[1]
1 Hzh = 6.626070040(81)×10−34 J[1]
3.1668114(29)×10−6 EH/K EH = 2Rhc = 4.359744650(54)×10−18 J[1] = 6.579683920729(33) Hzh[1]
1.0 Atomic units by definition
1.38064852(79)×10−16 erg/K CGS system, 1 erg = 1×10−7 J
3.2976230(30)×10−24 cal/K steam table calorie = 4.1868 J
1.8320128(17)×10−24 cal/°R degree Rankine = 5/9 K
5.6573016(51)×10−24 ft lb/°R foot-pound force = 1.3558179483314004 J
0.69503476(63) cm−1/K 2010 CODATA value[1]
1 cm−1 hc = 1.986445683(87)×10−23 J
0.0019872041(18) kcal/(mol⋅K) R noted kB, often used in statistical mechanics—using thermochemical calorie = 4.184 joule
0.0083144621(75) kJ/(mol⋅K) R noted kB, often used in statistical mechanics
4.10 pN⋅nm kT in piconewton nanometer at 24 °C, used in biophysics
−228.5991678(40) dBW/(K⋅Hz) in decibel watts, used in telecommunications (see Johnson–Nyquist noise)
1.442 695 041... Sh in shannons (logarithm base 2), used in information entropy (exact value 1/ln(2))
1 nat in nats (logarithm base e), used in information entropy (see § Planck units, below)

Since k is a physical constant of proportionality 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 °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 14 387.770 K, and also a relationship between voltage and temperature (multiplying the voltage by k in units of eV/K) with one volt being related to 11 604.519 K. The ratio of these two temperatures, 14 387.770 K/11 604.519 K ≈ 1.239842, is the numerical value of hc in units of eV⋅μm.

Planck units

The Boltzmann constant provides a mapping from this characteristic microscopic energy E to the macroscopic temperature scale T = E/k. In physics research another definition is often encountered in setting k to unity, resulting in the Planck units or natural units for temperature and energy. In this context temperature is measured effectively in units of energy and the Boltzmann constant is not explicitly needed.[18]

The equipartition formula for the energy associated with each classical degree of freedom then becomes

The use of natural units simplifies many physical relationships; in this form the definition of thermodynamic entropy coincides with the form of information entropy:

where Pi is the probability of each microstate.

The value chosen for a unit of the Planck temperature is that corresponding to the energy of the Planck mass or 1.416808(33)×1032 K.[1]

See also


  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).


  1. ^ a b c d e f g h i j k l Barry N. Taylor of the Data Center in close collaboration with Peter J. Mohr of the Physical Measurement Laboratory's Atomic Physics Division, Termed the "2014 CODATA recommended values", they are generally recognized worldwide for use in all fields of science and technology. The values became available on 25 June 2015 and replaced the 2010 CODATA set. They are based on all of the data available through 31 December 2014. Available:
  2. ^ Richard Feynman (1970). The Feynman Lectures on Physics Vol I. Addison Wesley Longman. ISBN 978-0-201-02115-8.
  3. ^ Shultis, J. Kenneth; Faw, Richard E. (2016). Fundamentals of Nuclear Science and Engineering (3rd ed.). CRC Press. p. 6. ISBN 978-1498769303.
  4. ^ Milton, Martin (14 November 2016). Highlights in the work of the BIPM in 2016 (PDF). SIM XXII General Assembly. Montevideo, Uruguay. p. 10. The conference runs from 13–16 November; the redefinition is scheduled for the morning of the last day.
  5. ^ Mills, Ian (29 September 2010). "Draft Chapter 2 for SI Brochure, following redefinitions of the base units" (PDF). CCU. Retrieved 2011-01-01.
  6. ^ Newell, David B.; Cabiati, F.; Fischer, J.; Fujii, K.; Karshenboim, S. G.; Margolis, H. S.; de Mirandés, E.; Mohr, P. J.; Nez, F.; Pachucki, K.; Quinn, T. J.; Taylor, B. N.; Wang, M.; Wood, B. M.; Zhang, Z.; et al. (Committee on Data for Science and Technology (CODATA) Task Group on Fundamental Constants (TGFC)) (20 October 2017). "The CODATA 2017 Values of h, e, k, and NA for the Revision of the SI". Metrologia. 55 (1): L13–L16. Bibcode:2018Metro..55L..13N. doi:10.1088/1681-7575/aa950a.
  7. ^ Proceedings of the 106th meeting (PDF). International Committee for Weights and Measures. Sèvres. 16 October 2017. pp. 17–23.
  8. ^ Rashid, Muhammad H. (2016). Microelectronic circuits : analysis and design (Third ed.). Cengage Learning. pp. 183–184. ISBN 9781305635166.
  9. ^ Cataldo, Enrico; Lieto, Alberto Di; 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].
  10. ^ Kirby, Brian J. (2009). Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices. Cambridge University Press. ISBN 978-0-521-11903-0.
  11. ^ Tabeling, Patrick (2006). Introduction to Microfluidics. Oxford University Press. ISBN 978-0-19-856864-3.
  12. ^ Planck, Max (1901), "Ueber das Gesetz der Energieverteilung im Normalspectrum" (PDF), Ann. Phys., 309 (3): 553–63, Bibcode:1901AnP...309..553P, doi:10.1002/andp.19013090310, archived from the original (PDF) on 10 June 2012. English translation: "On the Law of Distribution of Energy in the Normal Spectrum". Archived from the original on 2008-12-17.
  13. ^ Duplantier, Bertrand (2005). "Le mouvement brownien, 'divers et ondoyant'" [Brownian motion, 'diverse and undulating'] (PDF). Séminaire Poincaré 1 (in French): 155–212.
  14. ^ a b Planck, Max (2 June 1920), The Genesis and Present State of Development of the Quantum Theory (Nobel Lecture)
  15. ^ 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). "New measurement of the Boltzmann constant by acoustic thermometry of helium-4 gas". Metrologia. 54 (6): 856–873. Bibcode:2017Metro..54..856P. doi:10.1088/1681-7575/aa7bf5.
  16. ^ 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.
  17. ^ 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.
  18. ^ Kalinin, M; Kononogov, S (2005), "Boltzmann's Constant, the Energy Meaning of Temperature, and Thermodynamic Irreversibility", Measurement Techniques, 48 (7): 632–36, doi:10.1007/s11018-005-0195-9

External links

Bjerrum length

The Bjerrum length (after Danish chemist Niels Bjerrum 1879–1958 ) is the separation at which the electrostatic interaction between two elementary charges is comparable in magnitude to the thermal energy scale, , where is the Boltzmann constant and is the absolute temperature in kelvins. This length scale arises naturally in discussions of electrostatic, electrodynamic and electrokinetic phenomena in electrolytes, polyelectrolytes and colloidal dispersions.

In standard units, the Bjerrum length is given by

where is the elementary charge, is the relative dielectric constant of the medium and is the vacuum permittivity. For water at room temperature (), , so that .

In Gaussian units, and the Bjerrum length has the simpler form

Boltzmann's entropy formula

In statistical mechanics, Boltzmann's equation is a probability equation relating the entropy S of an ideal gas to the quantity W,

the number of real microstates corresponding to the gas' macrostate:

where kB is the Boltzmann constant (also written as simply k) and equal to 1.38065 × 10−23 J/K.

In short, the Boltzmann formula shows the relationship between entropy and the number of ways the atoms or molecules of a thermodynamic system can be arranged.

Carrier-to-noise-density ratio

In satellite communications, carrier-to-noise-density ratio (C/N0) is the ratio of the carrier power C to the noise power density N0, expressed in dB-Hz.

When considering only the receiver as a source of noise, it is called carrier-to-receiver-noise-density ratio.

It determines whether a receiver can lock on to the carrier and if the information encoded in the signal can be retrieved, given the amount of noise present in the received signal. The carrier-to-receiver noise density ratio is usually expressed in dBHz.

The noise power density, N0=kT, is the receiver noise power per hertz, which can be written in terms of the Boltzmann constant k (in joules per kelvin) and the noise temperature T (in kelvins).

Entropy of activation

In chemical kinetics, the entropy of activation'of a reaction is one of the two parameters (along with the enthalpy of activation) which are typically obtained from the temperature dependence of a reaction rate constant, when these data are analyzed using the Eyring equation. The standard entropy of activation is symbolized ΔS‡ and equals the change in entropy when the reactants change from their initial state to the activated complex or transition state (Δ = change, S = entropy, ‡ = activation). It determines the preexponential factor A of the Arrhenius equation for temperature dependence of reaction rates. The relationship depends on the molecularity of the reaction: for reactions in solution and unimolecular gas reactions A = (ekBT/h) exp(ΔS‡/R), while for bimolecular gas reactions A = (e2kBT/h) (RT/p) exp(ΔS‡/R). In these equations e is the base of natural logarithms, h is the Planck constant, kB is the Boltzmann constant and T the absolute temperature. R' is the ideal gas constant in units of (bar·L)/(mol·K). The factor is needed because of the pressure dependence of the reaction rate. R' = 8.3145 × 10-2 (bar·L)/(mol·K).

The value of ΔS‡ provides clues about the molecularity of the rate determining step in a reaction, i.e. whether the reactants are bonded to each other, or not. Positive values suggest that entropy increases upon achieving the transition state, which often indicates a dissociative mechanism in which the activated complex is loosely bound and about to dissociate. Negative values for ΔS‡ indicate that entropy decreases on forming the transition state, which often indicates an associative mechanism in which two reaction partners form a single activated complex.

Gas constant

The gas constant is also known as the molar, universal, or ideal gas constant, denoted by the symbol R or R and is equivalent to the Boltzmann constant, but expressed in units of energy per temperature increment per mole, i.e. the pressure–volume product, rather than energy per temperature increment per particle. The constant is also a combination of the constants from Boyle's law, Charles's law, Avogadro's law, and Gay-Lussac's law. It is a physical constant that is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation.

Physically, the gas constant is the constant of proportionality that happens to relate the energy scale in physics to the temperature scale, when a mole of particles at the stated temperature is being considered. Thus, the value of the gas constant ultimately derives from historical decisions and accidents in the setting of the energy and temperature scales, plus similar historical setting of the value of the molar scale used for the counting of particles. The last factor is not a consideration in the value of the Boltzmann constant, which does a similar job of equating linear energy and temperature scales.

The gas constant value is

8.3144598(48) J⋅mol−1⋅K−1

The two digits in parentheses are the uncertainty (standard deviation) in the last two digits of the value. The relative uncertainty is 5.7×10−7. Some have suggested that it might be appropriate to name the symbol R the Regnault constant in honour of the French chemist Henri Victor Regnault, whose accurate experimental data were used to calculate the early value of the constant; however, the exact reason for the original representation of the constant by the letter R is elusive.

The gas constant occurs in the ideal gas law, as follows:

where P is the absolute pressure (SI unit pascals), V is the volume of gas (SI unit cubic metres), n is the amount of gas (SI unit moles), m is the mass (SI unit kilograms) contained in V, and T is the thermodynamic temperature (SI unit kelvins). Rspecific is the molar-weight-specific gas constant, discussed below. The gas constant is expressed in the same physical units as molar entropy and molar heat capacity.

Isothermal–isobaric ensemble

The isothermal–isobaric ensemble (constant temperature and constant pressure ensemble) is a statistical mechanical ensemble that maintains constant temperature and constant pressure applied. It is also called the -ensemble, where the number of particles is also kept as a constant. This ensemble plays an important role in chemistry as chemical reactions are usually carried out under constant pressure condition. The partition function can be written as the weighted sum of the partition function of canonical ensemble, .

where ( is the Boltzmann constant), and is volume of the system.

There are several candidates for the normalization factor , e.g., , or . These choices make the partition function a nondimensional quantity. The differences vanish in the thermodynamic limit, i.e., in the limit of infinite number of particles.

The characteristic state function of this ensemble is the Gibbs free energy,

This thermodynamic potential is related to the Helmholtz free energy (logarithm of the canonical partition function), , in the following way:

KT (energy)

kT (also written as kBT) is the product of the Boltzmann constant, k (or kB), and the temperature, T. This product is used in physics as a scale factor for energy values in molecular-scale systems (sometimes it is used as a unit of energy), as the rates and frequencies of many processes and phenomena depend not on their energy alone, but on the ratio of that energy and kT, that is, on E / kT (see Arrhenius equation, Boltzmann factor). For a system in equilibrium in canonical ensemble, the probability of the system being in state with energy E is proportional to e−ΔE / kT.

More fundamentally, kT is the amount of heat required to increase the thermodynamic entropy of a system, in natural units, by one nat. E / kT therefore represents an amount of entropy per molecule, measured in natural units.

In macroscopic scale systems, with large numbers of molecules, RT value is commonly used; its SI units are joules per mole (J/mol): (RT = kT ⋅ NA).


The Kelvin scale is an absolute thermodynamic temperature scale using as its null point absolute zero, the temperature at which all thermal motion ceases in the classical description of thermodynamics. The kelvin (symbol: K) is the base unit of temperature in the International System of Units (SI).

Until 2018, the kelvin was defined as the fraction 1/273.16 of the thermodynamic temperature of the triple point of water (exactly 0.01 °C or 32.018 °F). In other words, it was defined such that the triple point of water is exactly 273.16 K.

On 16 November 2018, a new definition was adopted, in terms of a fixed value of the Boltzmann constant. For legal metrology purposes, the new definition will officially come into force on 20 May 2019 (the 130th anniversary of the Metre Convention).The Kelvin scale is named after the Belfast-born, Glasgow University engineer and physicist William Thomson, 1st Baron Kelvin (1824–1907), who wrote of the need for an "absolute thermometric scale". Unlike the degree Fahrenheit and degree Celsius, the kelvin is not referred to or written as a degree. The kelvin is the primary unit of temperature measurement in the physical sciences, but is often used in conjunction with the degree Celsius, which has the same magnitude. The definition implies that absolute zero (0 K) is equivalent to −273.15 °C (−459.67 °F).

List of scientific constants named after people

This is a list of physical and mathematical constants named after people.Eponymous constants and their influence on scientific citations have been discussed in the literature.

  • Reduced Planck constant or Dirac constant (-bar, ħ) – Max Planck, Paul Dirac
Nat (unit)

The natural unit of information (symbol: nat), sometimes also nit or nepit, is a unit of information or entropy, based on natural logarithms and powers of e, rather than the powers of 2 and base 2 logarithms, which define the bit. This unit is also known by its unit symbol, the nat. The nat is the coherent unit for information entropy. The International System of Units, by assigning the same units (joule per kelvin) both to heat capacity and to thermodynamic entropy implicitly treats information entropy as a quantity of dimension one, with 1 nat = 1. Physical systems of natural units that normalize the Boltzmann constant to 1 are effectively measuring thermodynamic entropy in nats.

When the Shannon entropy is written using a natural logarithm,

it is implicitly giving a number measured in nats.

One nat is equal to 1/ln 2 shannons (or bits) ≈ 1.44 Sh or, equivalently, 1/ln 10 hartleys ≈ 0.434 Hart. The factors 1.44 and 0.434 arise from the relationships

, and

One nat is the information content of an event if the probability of that event occurring is 1/e.

Particle number

The particle number (or number of particles) of a thermodynamic system, conventionally indicated with the letter N, is the number of constituent particles in that system. The particle number is a fundamental parameter in thermodynamics which is conjugate to the chemical potential. Unlike most physical quantities, particle number is a dimensionless quantity. It is an extensive parameter, as it is directly proportional to the size of the system under consideration, and thus meaningful only for closed systems.

A constituent particle is one that cannot be broken into smaller pieces at the scale of energy k·T involved in the process (where k is the Boltzmann constant and T is the temperature). For example, for a thermodynamic system consisting of a piston containing water vapour, the particle number is the number of water molecules in the system. The meaning of constituent particle, and thereby of particle number, is thus temperature-dependent.

Recoil temperature

In laser cooling, the Boltzmann constant times the recoil temperature is equal to the recoil energy deposited in a single atom initially at rest by the spontaneous emission of a single photon. The recoil temperature is


since the photon's momentum is (here is the wavevector of the light, is the mass of an atom, is Boltzmann's constant and is Planck's constant). The recoil temperature for the D2 lines of alkali atoms is typically on the order of 1 μK, and thus lower than the Doppler temperature. An example of a process where the recoil temperature can be reached is Sisyphus cooling.

Rotational temperature

The characteristic rotational temperatureR or θrot) is commonly used in statistical thermodynamics, to simplify the expression of the rotational partition function and the rotational contribution to molecular thermodynamic properties. It has units of temperature and is defined as


where is the rotational constant, and is a molecular moment of inertia. Also h is the Planck constant, c is the speed of light, ħ = h/2π is the reduced Planck constant and kB is the Boltzmann constant.

The physical meaning of θR is as an estimate of the temperature at which thermal energy (of the order of kBT) is comparable to the spacing between rotational energy levels (of the order of hcB). At about this temperature the population of excited rotational levels becomes important. Some typical values are given in the table. In each case the value refers to the most common isotopic species.

Stefan–Boltzmann constant

The Stefan–Boltzmann constant (also Stefan's constant), a physical constant denoted by the Greek letter σ (sigma), is the constant of proportionality in the Stefan–Boltzmann law: "the total intensity radiated over all wavelengths increases as the temperature increases", of a black body which is proportional to the fourth power of the thermodynamic temperature. The theory of thermal radiation lays down the theory of quantum mechanics, by using physics to relate to molecular, atomic and sub-atomic levels. Slovenian physicist Josef Stefan formulated the constant in 1879, and it was later derived in 1884 by Austrian physicist Ludwig Boltzmann. The equation can also be derived from Planck's law, by integrating over all wavelengths at a given temperature, which will represent a small flat black body box. "The amount of thermal radiation emitted increases rapidly and the principal frequency of the radiation becomes higher with increasing temperatures". The Stefan–Boltzmann constant can be used to measure the amount of heat that is emitted by a blackbody, which absorbs all of the radiant energy that hits it, and will emit all the radiant energy. Furthermore, the Stefan–Boltzmann constant allows for temperature (K) to be converted to units for intensity (W⋅m−2), which is power per unit area.

The value of the Stefan–Boltzmann constant is given in SI units by

σ = 5.670367(13)×10−8 W⋅m−2⋅K−4.

In cgs units the Stefan–Boltzmann constant is:

σ5.6704×10−5 erg⋅cm−2⋅s−1⋅K−4.

In thermochemistry the Stefan–Boltzmann constant is often expressed in cal⋅cm−2⋅day−1⋅K−4:

σ11.7×10−8 cal cm−2⋅day−1⋅K−4.

In US customary units the Stefan–Boltzmann constant is:

σ1.714×10−9 BTU⋅hr−1⋅ft−2⋅°R−4.

The value of the Stefan–Boltzmann constant is derivable as well as experimentally determinable; see Stefan–Boltzmann law for details. It can be defined in terms of the Boltzmann constant as


The CODATA recommended value is calculated from the measured value of the gas constant:


Dimensional formula: M1T−3Θ−4

A related constant is the radiation constant (or radiation density constant) a which is given by:

Stefan–Boltzmann law

The Stefan–Boltzmann law describes the power radiated from a black body in terms of its temperature. Specifically, the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time (also known as the black-body radiant emittance) is directly proportional to the fourth power of the black body's thermodynamic temperature T:

The constant of proportionality σ, called the Stefan–Boltzmann constant, is derived from other known physical constants. The value of the constant is

where k is the Boltzmann constant, h is Planck's constant, and c is the speed of light in a vacuum. The radiance (watts per square metre per steradian) is given by

A body that does not absorb all incident radiation (sometimes known as a grey body) emits less total energy than a black body and is characterized by an emissivity, :

The radiant emittance has dimensions of energy flux (energy per time per area), and the SI units of measure are joules per second per square metre, or equivalently, watts per square metre. The SI unit for absolute temperature T is the kelvin. is the emissivity of the grey body; if it is a perfect blackbody, . In the still more general (and realistic) case, the emissivity depends on the wavelength, .

To find the total power radiated from an object, multiply by its surface area, :

Wavelength- and subwavelength-scale particles, metamaterials, and other nanostructures are not subject to ray-optical limits and may be designed to exceed the Stefan–Boltzmann law.

Thermal energy

Thermal energy can refer to several distinct thermodynamic quantities, such as the internal energy of a system; heat or sensible heat, which are defined as types of transfer of energy (as is work); or for the characteristic energy of a degree of freedom in a thermal system , where is temperature and is the Boltzmann constant.

Thermodynamic beta

In statistical thermodynamics, thermodynamic beta, also known as coldness, is the reciprocal of the thermodynamic temperature of a system:

It was originally introduced in 1971 (as Kältefunktion "coldness function") by Ingo Müller [de], one of the proponents of the rational thermodynamics school of thought, based on earlier proposals for a "reciprocal temperature" function.

Thermodynamic beta has units reciprocal to that of energy (in SI units, ). In non-thermal units, it can also be measured in byte per joule, or more conveniently, gigabyte per nanojoule; 1 K−1 is equivalent to about 13,062 gigabytes per nanojoule; at room temperature: T = 300K, β ≈ 44 GB/nJ39 eV−12.4×1020 J−1.

Vacancy defect

In crystallography, a vacancy is a type of point defect in a crystal. Crystals inherently possess imperfections, sometimes referred to as crystalline defects. A defect in which an atom is missing from one of the lattice sites is known as a "vacancy" defect. It is also known as a Schottky defect, although in ionic crystals the concepts are not identical.

Vacancies occur naturally in all crystalline materials. At any given temperature, up to the melting point of the material, there is an equilibrium concentration (ratio of vacant lattice sites to those containing atoms). At the melting point of some metals the ratio can be approximately 1:1000. This temperature dependence can be modelled by

where Nv is the vacancy concentration, Qv is the energy required for vacancy formation, kB is the Boltzmann constant, T is the absolute temperature, and N is the concentration of atomic sites i.e.

where ρ is density, NA Avogadro constant, and A the atomic mass.

It is the simplest point defect. In this system, an atom is missing from its regular atomic site. Vacancies are formed during solidification due to vibration of atoms, local rearrangement of atoms, plastic deformation and ionic bombardments.

The creation of a vacancy can be simply modeled by considering the energy required to break the bonds between an atom inside the crystal and its nearest neighbor atoms. Once that atom is removed from the lattice site, it is put back on the surface of the crystal and some energy is retrieved because new bonds are established with other atoms on the surface. However, there is a net input of energy because there are fewer bonds between surface atoms than between atoms in the interior of the crystal.

Mole concepts
Physical quantities

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