As part of the2019 revision of the SI, the Boltzmann constant is one of the seven "defining constants" that have been defined so as to have exactfinite decimal values in SI units. They are used in various combinations to define the seven SI base units. The Boltzmann constant is defined to be exactly1.380649×10−23joules per kelvin,[1] with the effect of defining the SI unit kelvin.
Boltzmann constant: The Boltzmann constant,k, is one of seven fixed constants defining the International System of Units, the SI, withk =1.380649×10−23 J K−1. The Boltzmann constant is a proportionality constant between the quantities temperature (with unit kelvin) and energy (with unit joule).[3]
Inclassicalstatistical mechanics, this average is predicted to hold exactly for homogeneousideal 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 of3/2kT per atom. This corresponds very well with experimental data. The thermal energy can be used to calculate theroot-mean-square speed of the atoms, which turns out to be inversely proportional to the square root of theatomic mass. The root mean square speeds found at room temperature accurately reflect this, ranging from1370 m/s forhelium, down to240 m/s forxenon.
Kinetic theory gives the average pressurep for an ideal gas as
Combination with the ideal gas lawshows that the average translational kinetic energy is
Considering that the translational motion velocity vectorv 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.
More generally, systems in equilibrium at temperatureT have probabilityPi of occupying a statei with energyE weighted by the correspondingBoltzmann factor:whereZ is thepartition function. Again, it is the energy-like quantitykT that takes central importance.
Boltzmann's grave in theZentralfriedhof, Vienna, with bust and entropy formula.
In statistical mechanics, theentropyS of anisolated system atthermodynamic equilibrium is defined as thenatural logarithm ofW, the number of distinct microscopic states available to the system given the macroscopic constraints (such as a fixed total energyE):
This equation, which relates the microscopic details, or microstates, of the system (viaW) to its macroscopic state (via the entropyS), is the central idea of statistical mechanics. Such is its importance that it is inscribed on Boltzmann's tombstone.
The constant of proportionalityk serves to make the statistical mechanical entropy equal to the classical thermodynamic entropy ofClausius:
One could choose instead a rescaleddimensionless entropy in microscopic terms such that
This is a more natural form and this rescaled entropy corresponds exactly to Shannon'sinformation entropy.
The characteristic energykT is thus the energy required to increase the rescaled entropy by onenat.
Atroom temperature 300 K (27 °C; 80 °F),VT is approximately25.85 mV,[7][8] which can be derived by plugging in the values as follows:
At thestandard state temperature of 298.15 K (25.00 °C; 77.00 °F), it is approximately25.69 mV. The thermal voltage is also important in plasmas and electrolyte solutions (e.g. theNernst 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]
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 untilMax Planck first introducedk, 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 thelaw 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 thegas constantR, and macroscopic energies for macroscopic quantities of the substance. The iconic terse form of the equationS =k lnW on Boltzmann's tombstone is in fact due to Planck, not Boltzmann. Planck actually introduced it in the same work as hiseponymoush.[12]
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 aheuristic tool for solving problems. There was no agreement whetherchemical molecules, as measured byatomic weights, were the same asphysical molecules, as measured bykinetic 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 ofSI prior to the2019 revision of the SI, the Boltzmann constant was a measured quantity rather than having a fixed numerical value. Its exact definition also varied over the years due to redefinitions of the kelvin (seeKelvin § History) and other SI base units (seeJoule § 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][16] This decade-long effort was undertaken with different techniques by several laboratories;[a] it is one of the cornerstones of the revision of the SI. Based on these measurements, the value of1.380649×10−23 J/K was recommended as the final fixed value of the Boltzmann constant to be used for the 2019 revision of the SI.[17]
As a precondition for redefining the Boltzmann constant, there must be one experimental value with a relative uncertainty below 1ppm, and at least one measurement from a second technique with a relative uncertainty below 3 ppm. The acoustic gas thermometry reached 0.2 ppm, and Johnson noise thermometry reached 2.8 ppm.[18]
Sincek is aproportionality constant 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 inSI units means a change in temperature by1 K changes a particle's energy by only a small amount. A change of1 °C is defined to be the same as a change of1 K. The characteristic energykT is a term encountered in many physical relationships.
The Boltzmann constant sets up a relationship between wavelength and temperature (dividinghc/k by a wavelength gives a temperature) with1000 nm being related to14387.777 K, and also a relationship between voltage and temperature, with one volt corresponding to11604.518 K.
The Boltzmann constant provides a mapping from the characteristic microscopic energyE to the macroscopic temperature scaleT =E/k. In fundamental physics, this mapping is often simplified by using thenatural units of settingk to unity. This convention means that temperature and energy quantities have the samedimensions.[22][23] In particular, the SI unit kelvin becomes superfluous, being defined in terms of joules as1 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.[22]
This convention simplifies many physical relationships and formulas. For example, the equipartition formula for the energy associated with each classical degree of freedom becomes
As another example, the definition of thermodynamic entropy coincides with the form ofinformation entropy:wherePi is the probability of eachmicrostate.
^Petrucci, Ralph H.; Harwood, William S.; Herring, F. Geoffrey (2002).General Chemistry: Principles and Modern Applications (8th ed.). Prentice Hall. p. 785.ISBN0-13-014329-4.
^Rashid, Muhammad H. (2016).Microelectronic circuits: analysis and design (3rd ed.). Cengage Learning. pp. 183–184.ISBN9781305635166.
^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].
^Kittel, Charles; Kroemer, Herbert (1980).Thermal physics (2nd ed.). San Francisco: W. H. Freeman. p. 41.ISBN0716710889.We prefer to use a more natural temperature scale ... the fundamental temperature has the units of energy.
