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# Boltzmann constant

The Boltzmann constant ( or ), which is named after Ludwig Boltzmann, is a physical constant relating the average kinetic energy of particles in a gas with the temperature of the gas. It is the gas constant divided by the Avogadro constant : k = \frac{R}{N_\text{A}}.\, 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 brackets) is . Current measurements of the Boltzmann constant depend on the definition of the kelvin in terms of the triple point of water. In the proposed redefinition of SI base units scheduled{{Cite web |url=http://www.bipm.org/cc/TGFC/Allowed/Minutes/CODATA_Minutes_14-BIPM-public.pdf#page=7 |title=Report on the Meeting of the CODATA Task Group on Fundamental Constants |date=3–4 November 2014 |place= BIPM |first=B. |last=Wood |page=7 |quote= director Martin Milton responded to a question about what would happen if ... the CIPM or the CGPM voted not to move forward with the redefinition of the SI. He responded that he felt that by that time the decision to move forward should be seen as a foregone conclusion. }} for adoption at the 26th General Conference on Weights and Measures (CGPM) on 16 November 2018,{{cite conference |conference=SIM XXII General Assembly |location=Montevideo, Uruguay |url=http://www.sim-metrologia.org.br/docs/2016Presentations/BIPM%202016.pdf#page=10 |conference-url=http://www.sim-metrologia.org.br/docs/2016Presentations/ |page=10 |title=Highlights in the work of the BIPM in 2016 |first=Martin |last=Milton |date=14 November 2016 }} The conference runs from 13–16 November; the redefinition is scheduled for the morning of the last day. the definition of the kelvin will be 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.{{cite web |url=http://www.bipm.org/utils/en/pdf/si_brochure_draft_ch2.pdf |title=Draft Chapter 2 for SI Brochure, following redefinitions of the base units |first=Ian |last=Mills |publisher=CCU |date=29 September 2010 |accessdate=2011-01-01 }} The final value is not yet known, but will be based on the best measurements published by 1 July 2017.

## Bridge from macroscopic to microscopic physics

The Boltzmann constant, , is a bridge between macroscopic and microscopic physics. Macroscopically, the ideal gas law states that, for an ideal gas, the product of pressure and volume is proportional to the product of amount of substance (in moles) and absolute temperature : pV = nRT \, where is the gas constant (). Introducing the Boltzmann constant transforms the ideal gas law into an alternative form: p V = N k T , where is the number of molecules of gas. For = 1 mol}}, 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 , the average thermal energy carried by each microscopic degree of freedom in the system is on the order of magnitude of kT}} (i.e., about , or , at room temperature).

### Application to simple gas thermodynamics

In classical statistical mechanics, this average is predicted to hold exactly for homogeneous ideal gases. Monatomic ideal gases possess three degrees of freedom per atom, corresponding to the three spatial directions, which means a thermal energy of 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 for helium, down to for xenon. Kinetic theory gives the average pressure for an ideal gas as p = \frac{1}{3}\frac{N}{V} m \overline{v^2}. Combination with the ideal gas law p V = N k T shows that the average translational kinetic energy is \tfrac{1}{2}m \overline{v^2} = \tfrac{3}{2} k T. Considering that the translational motion velocity vector 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. 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 have probability of occupying a state with energy weighted by the corresponding Boltzmann factor: P_i \propto \frac{\exp\left(-\frac{E}{k T}\right)}{Z}, where is the partition function. Again, it is the energy-like quantity {{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

In statistical mechanics, the entropy of an isolated system at thermodynamic equilibrium is defined as the natural logarithm of , the number of distinct microscopic states available to the system given the macroscopic constraints (such as a fixed total energy ): S = k \,\ln W. This equation, which relates the microscopic details, or microstates, of the system (via ) to its macroscopic state (via the entropy ), is the central idea of statistical mechanics. Such is its importance that it is inscribed on Boltzmann's tombstone. The constant of proportionality 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}{k T}. This is a more natural form and this rescaled entropy exactly corresponds to Shannon's subsequent information entropy. The characteristic energy 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 . The thermal voltage depends on absolute temperature as V_\mathrm{T} = { k T \over q }, where is the magnitude of the electrical charge on the electron with a value and is the Boltzmann constant, . In electronvolts, the Boltzmann constant is , making it easy to calculate that at room temperature (≈ ), the value of the thermal voltage is approximately 25.85 millivolts ≈ . 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, it seems the relation was never expressed with a specific constant until Max Planck first introduced , and gave a precise value for it (, about 2.5% lower than today's figure), in his derivation of the law of black body radiation in 1900–1901.. English translation: 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 , and macroscopic energies for macroscopic quantities of the substance. The iconic terse form of the equation on Boltzmann's tombstone is in fact due to Planck, not Boltzmann. Planck actually introduced it in the same work as his eponymous In 1920, Planck wrote in his Nobel Prize lecture: 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: In 2013 the UK National Physical Laboratory used microwave and acoustic resonance measurements to determine the speed of sound of a monatomic gas in a triaxial ellipsoid chamber to determine a more accurate value for the constant as a part of the revision of the International System of Units. The new value was calculated as . In 2016 the CODATA accepted value was and is expected to be accepted by the International System of Units following a review.

## Value in different units

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