Boltzmann constant

Values of Template:Mvar	Units
Template:Val	J⋅K−1
Template:Val	eV⋅K−1
Template:Val	erg⋅K−1
For details, see Template:Section link below.

The Boltzmann constant (Template:Math or Template:Mvar), named after its discoverer, Ludwig Boltzmann, is a physical constant that relates the average relative kinetic energy of particles in a gas with the temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, and in Planck's law of black-body radiation and Boltzmann's entropy formula. The Boltzmann constant has the dimension energy divided by temperature, the same as entropy.

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 Template:Val.

This definition allows the kelvin to be defined in terms of the Boltzmann constant, the metre, the second, and the kilogram. Before 2019, its value in SI units was a measured quantity. Measurements of the Boltzmann constant depended on the definition of the kelvin in terms of the triple point of water. The measured values were used to determine the quantity that is used in the 2019 definition, to make the definition's value for the kelvin identical to the old value to within the limits of experimental accuracy at the time of the definition.

Bridge from macroscopic to microscopic physics

Template:Ideal gas law relationships.svg The Boltzmann constant, Template:Mvar, 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 Template:Mvar and volume Template:Mvar is proportional to the product of amount of substance Template:Mvar (in moles) and absolute temperature Template:Mvar:

$pV=nRT,$

where Template:Mvar is the gas constant (Template:Val). Introducing the Boltzmann constant transforms the ideal gas law into an alternative form:

$pV=NkT,$

where Template:Mvar is the number of molecules of gas. For Template:Nowrap, Template:Mvar is equal to the number of particles in one mole (Avogadro's number).

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).

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, which means 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

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

Combination with the ideal gas law

$pV=NkT$

shows that the average translational kinetic energy is

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

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:Mvar of occupying a state Template:Mvar with energy Template:Mvar weighted by the corresponding Boltzmann factor:

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

where Template:Mvar is the partition function. Again, it is the energy-like quantity [[kT (energy)|Template:Mvar]] 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

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):

$S=k\,\ln W.$

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:

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

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

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

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

The characteristic energy Template:Mvar 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 Template:Math. The thermal voltage depends on absolute temperature Template:Mvar as

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

where Template:Mvar is the magnitude of the electrical charge on the electron with a value Template:Val. Equivalently,

${V_{\mathrm {T} } \over T}={k \over q}\approx 8.61733034\times 10^{-5}\ \mathrm {V/K} .$

At room temperature (Template:Val), Template:Math is approximately Template:Val. 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.

History

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 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. 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]].

In 1920, Planck wrote in his Nobel Prize lecture: Template:Quotation

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: Template:Quotation

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. This decade-long effort was undertaken with different techniques by several laboratories;Template:Efn 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.

Value in different units

Values of Template:Mvar	Units	Comments
Template:Val	J/K	SI by definition, J/K = m2⋅kg/(s2⋅K) in SI base units
Template:Val	eV/K	electronvolt = Template:Val Template:Math = Template:Val
Template:Val	Hz/K	2014 CODATA value Template:Val⋅[[Planck constant|Template:Mvar]] = Template:Val
Template:Val	[[Hartree|Template:Math]]/K	Template:Math = Template:Val = Template:Val⋅h
Template:Val	Atomic units	by definition
Template:Val	erg/K	CGS system, 1 erg = Template:Val
Template:Val	cal/K	1 steam table calorie = Template:Val
Template:Val	cal/°R	1 degree Rankine = Template:Sfrac K
Template:Val	ft lb/°R	1 foot-pound force = Template:Val
Template:Val	cm−1/K	2010 CODATA value 1 cm−1 ⋅Template:Mvar = Template:Val
Template:Val	kcal/(mol⋅K)	R noted kB, often used in statistical mechanics—using thermochemical calorie = 4.184 joule
Template:Val	kJ/(mol⋅K)	R noted kB, often used in statistical mechanics
4.10	pN⋅nm	Template:Mvar in piconewton nanometer at Template:Val, used in biophysics
Template:Val	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 Template:Sfrac)
1	nat	in nats (logarithm base Template:Mvar), used in information entropy (see Template:Section link, below)

Since Template:Mvar 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 Template:Val is defined to be the same as a change of Template:Val. The characteristic energy Template:Mvar 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 Template:Mvar to the macroscopic temperature scale Template:Math. In physics research another definition is often encountered in setting Template:Mvar 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.

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

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

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

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

where Template:Mvar 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 Template:Val.

See also

• CODATA 2018
• Thermodynamic beta

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