Boltzmann constant

Template:Infobox The Boltzmann constant (Template:Math or Template:Mvar) is the proportionality factor that relates the average relative thermal energy of particles in a gas with the thermodynamic temperature of the gas.<ref name="Feynman1Ch39-10">Template:Cite book</ref> It occurs in the definitions of the kelvin (K) and the molar gas constant, 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 and heat capacity. It is named after the Austrian scientist Ludwig Boltzmann.

As part of the 2019 revision of the SI, the Boltzmann constant is one of the seven "defining constants" that have been defined so as to have exact finite 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 exactly Template:Val joules per kelvin,<ref name="SI2019"/> with the effect of defining the SI unit kelvin.

Roles of the Boltzmann constant

Template:Ideal gas law relationships.svg Template:Quote box Macroscopically, the ideal gas law states that, for an ideal gas, the product of pressure Template:Mvar and volume Template:Mvar is proportional to the product of amount of substance Template:Mvar and absolute temperature Template:Mvar: <math display="block">pV = nRT ,</math> where Template:Mvar is the molar gas constant (Template:Val).<ref>Template:Cite web</ref> Introducing the Boltzmann constant as the gas constant per molecule<ref>Template:Cite book</ref> Template:Math (Template:Math being the Avogadro constant) transforms the ideal gas law into an alternative form: <math display="block">p V = N k T ,</math> where Template:Mvar is the number of molecules of gas.

Role in the equipartition of energy

Template:Main Given a thermodynamic system at an absolute temperature Template:Mvar, the average thermal energy carried by each microscopic degree of freedom in the system is Template:Math (i.e., about Template:Val, or Template:Val, at room temperature). This is generally true only for classical systems with a large number of particles.

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 Template:Math 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 Template:Val for helium, down to Template:Val for xenon.

Kinetic theory gives the average pressure Template:Mvar for an ideal gas as <math display="block"> p = \frac{1}{3}\frac{N}{V} m \overline{v^2}.</math>

Combination with the ideal gas law <math display="block">p V = N k T</math> shows that the average translational kinetic energy is <math display="block"> \tfrac{1}{2}m \overline{v^2} = \tfrac{3}{2} k T.</math>

Considering that the translational motion velocity vector Template:Math 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. Template:Math.

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 Template:Mvar have probability Template:Math of occupying a state Template:Mvar with energy Template:Mvar weighted by the corresponding Boltzmann factor: <math display="block">P_i \propto \frac{\exp\left(-\frac{E}{k T}\right)}{Z},</math> where Template:Mvar is the partition function. Again, it is the energy-like quantity Template:Math 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

File:Zentralfriedhof Vienna - Boltzmann.JPG

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

In statistical mechanics, the entropy Template:Mvar of an isolated system at thermodynamic equilibrium is defined as the natural logarithm of Template:Mvar, the number of distinct microscopic states available to the system given the macroscopic constraints (such as a fixed total energy Template:Mvar): <math display="block">S = k \,\ln W.</math>

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

The constant of proportionality Template:Mvar serves to make the statistical mechanical entropy equal to the classical thermodynamic entropy of Clausius: <math display="block">\Delta S = \int \frac{{\rm d}Q}{T}.</math>

One could choose instead a rescaled dimensionless entropy in microscopic terms such that <math display="block">{S' = \ln W}, \quad \Delta S' = \int \frac{\mathrm{d}Q}{k T}.</math>

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

The characteristic energy Template:Mvar is thus the energy required to increase the rescaled entropy by one nat.

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 Template:Math. The thermal voltage depends on absolute temperature Template:Mvar as <math display="block"> V_\mathrm{T} = { k T \over q } = { R T \over F },</math> where Template:Mvar is the magnitude of the electrical charge on the electron with a value Template:Physconst Equivalently, <math display="block"> { V_\mathrm{T} \over T } = { k \over q } \approx 8.617333262 \times 10^{-5}\ \mathrm{V/K}.</math>

At room temperature Template:Convert, Template:Math is approximately Template:Val,<ref>Template:Cite book</ref><ref>Template:Cite arXiv</ref> which can be derived by plugging in the values as follows: <math display="block">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}</math>

At the standard state temperature of Template:Convert, it is approximately Template:Val. 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.<ref name="Kirby">Template:Cite book</ref><ref name="Tabeling">Template:Cite book</ref>

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 Template:Mvar, and gave a more precise value for it (Template:Val, about 2.5% lower than today's figure), in his derivation of the law of black-body radiation in 1900–1901.<ref name="Planck01">Template:Cite journal. English translation: Template:Cite web</ref> 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 Template:Mvar, and macroscopic energies for macroscopic quantities of the substance. The iconic terse form of the equation Template:Math on Boltzmann's tombstone is in fact due to Planck, not Boltzmann. Planck actually introduced it in the same work as his [[Planck constant|eponymous Template:Mvar]].<ref>Template:Cite journal</ref>

In 1920, Planck wrote in his Nobel Prize lecture:<ref name="PlanckNobel">Template:Cite web</ref> Template:Blockquote

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:<ref name="PlanckNobel" /> Template:Blockquote

In versions of SI prior to the 2019 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 (see Template:Section link) and other SI base units (see Template:Section link).

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.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref><ref>Template:Cite journal</ref> This decade-long effort was undertaken with different techniques by several laboratories;Template:Efn it is one of the cornerstones of the revision of the SI. Based on these measurements, the value of Template:Val was recommended as the final fixed value of the Boltzmann constant to be used for the 2019 revision of the SI.<ref>Template:Cite journal</ref>

As a precondition for redefining the Boltzmann constant, there must be one experimental value with a relative uncertainty below 1 ppm, 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.<ref>Template:Cite journal</ref>

Value in different units

Values of Template:Mvar	Comments
Template:Physconst	SI definition
Template:ValTemplate:Px2<ref>Template:Cite web</ref>
Template:Val	(Template:Math)
Template:Val	CGS, 1 erg = Template:Val
Template:Val	1 calorie = Template:Val
Template:Val
Template:Val
Template:Val	(Template:Math)
Template:Val
Template:Val	(Template:Math)
Template:Val	(Template:Math)
Template:Val	Template:Math, used for thermal noise calculations
Template:ValTemplate:Px2<ref>Template:Cite web</ref>	(Template:Math)

Since Template:Mvar is a proportionality 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 in SI units means a change in temperature by 1 K changes a particle's energy by only a small amount. A change of Template:Val is defined to be the same as a change of Template:Val. The characteristic energy Template:Math is a term encountered in many physical relationships.

The Boltzmann constant sets up a relationship between wavelength and temperature (dividing Template:Math by a wavelength gives a temperature) with Template:Val being related to Template:Val, and also a relationship between voltage and temperature, with one volt corresponding to Template:Val. The ratio of these two temperatures, Template:Val / Template:Val ≈ 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 Template:Mvar to the macroscopic temperature scale Template:Math. In fundamental physics, this mapping is often simplified by using the natural units of setting Template:Mvar to unity. This convention means that temperature and energy quantities have the same dimensions.<ref name=Kalinin/><ref>Template:Cite book</ref> In particular, the SI unit kelvin becomes superfluous, being defined in terms of joules as Template:Nowrap.<ref>Template:Cite journal</ref> With this convention, temperature is always given in units of energy, and the Boltzmann constant is not explicitly needed in formulas.<ref name=Kalinin>Template:Cite journal</ref>

This convention simplifies many physical relationships and formulas. For example, the equipartition formula for the energy associated with each classical degree of freedom becomes <math display="block">E_{\mathrm{dof}} = \tfrac{1}{2} T </math>

As another example, the definition of thermodynamic entropy coincides with the form of information entropy: <math display="block"> S = - \sum_i P_i \ln P_i.</math> where Template:Math 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

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