Boltzmann constant | |
---|---|

Symbol: | k_{B}, k |

Value in joules per kelvin: | 1.380649×10^{−23} J⋅K^{−1}^{ [1] } |

The **Boltzmann constant** (*k*_{B} or k) is the proportionality factor that relates the average relative thermal energy of particles in a gas with the thermodynamic temperature of the gas.^{ [2] } 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, and is used in calculating thermal noise in resistors. The Boltzmann constant has dimensions of energy divided by temperature, the same as entropy. It is named after the Austrian scientist Ludwig Boltzmann.

- Roles of the Boltzmann constant
- Role in the equipartition of energy
- Role in Boltzmann factors
- Role in the statistical definition of entropy
- The thermal voltage
- History
- Value in different units
- Natural units
- See also
- Notes
- References
- External links

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 1.380649×10^{−23} J⋅K^{−1}.^{ [1] }

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 and absolute temperature T:

where R is the molar gas constant (8.31446261815324 J⋅K^{−1}⋅mol^{−1}).^{ [3] } Introducing the Boltzmann constant as the gas constant per molecule^{ [4] }`k` = `R`/`N`_{A} transforms the ideal gas law into an alternative form:

where N is the number of moles of gas.

Given a thermodynamic system at an absolute temperature T, the average thermal energy carried by each microscopic degree of freedom in the system is 1/2 `k``T` (i.e., about 2.07×10^{−21} J, or 0.013 eV , at room temperature). This is generally true only for classical systems with a large number of particles, and in which quantum effects are negligible.

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 3/2 `k``T` 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. 1/2 `k``T`.

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 temperature T have probability `P`_{i} 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 `k``T` that takes central importance.

Consequences of this include (in addition to the results for ideal gases above) the Arrhenius equation in chemical kinetics.

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.

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 `V`_{T}. The thermal voltage depends on absolute temperature T as

where q is the magnitude of the electrical charge on the electron with a value 1.602176634×10^{−19} C.^{ [5] } Equivalently,

At room temperature 300 K (27 °C; 80 °F), `V`_{T} is approximately 25.85 mV^{ [6] }^{ [7] } which can be derived by plugging in the values as follows:

At the standard state temperature of 298.15 K (25.00 °C; 77.00 °F), it is approximately 25.69 mV. 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.^{ [8] }^{ [9] }

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 k, 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 the law of black-body radiation in 1900–1901.^{ [10] } 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.^{ [11] }

In 1920, Planck wrote in his Nobel Prize lecture:^{ [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 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:^{ [12] }

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 of SI prior to the 2019 redefinition of the SI base units, the Boltzmann constant was a measured quantity rather than a fixed value. Its exact definition also varied over the years due to redefinitions of the kelvin (see Kelvin § History) and other SI base units (see Joule § 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.^{ [13] }^{ [14] } This decade-long effort was undertaken with different techniques by several laboratories;^{ [lower-alpha 1] } it is one of the cornerstones of the 2019 redefinition of SI base units. Based on these measurements, the CODATA recommended 1.380649×10^{−23} J/K to be the final fixed value of the Boltzmann constant to be used for the International System of Units.^{ [15] }

Values of k | Units | Comments |
---|---|---|

1.380649×10^{−23} | J/K | SI by definition, J/K = m^{2}⋅kg/(s^{2}⋅K) in SI base units |

8.617333262×10^{−5} | eV/K | † |

2.083661912×10^{10} | Hz/K | (k/h) † |

1.380649×10^{−16} | erg/K | CGS system, 1 erg = 1×10^{−7} J |

3.297623483×10^{−24} | cal/K | † 1 calorie = 4.1868 J |

1.832013046×10^{−24} | cal/°R | † |

5.657302466×10^{−24} | ft lb/°R | † |

0.695034800 | cm^{−1}/K | (k/(hc)) † |

3.166811563×10^{−6} | E_{h}/K | (E_{h} = hartree) |

1.987204259×10^{−3} | kcal/(mol⋅K) | (kN_{A}) † |

8.314462618×10^{−3} | kJ/(mol⋅K) | (kN_{A}) † |

−228.5991672 | dB(W/K/Hz) | 10 log_{10}(k/(1 W/K/Hz)),† used for thermal noise calculations |

1.536179187×10^{−40} | kg/K | k/c^{2}, where c is the speed of light ^{ [16] } |

†The value is exact but not expressible as a finite decimal; approximated to 9 decimal places only.

Since k is a proportionality factor 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 1 °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 14387.777 K, and also a relationship between voltage and temperature (*kT* in units of eV corresponds to a voltage) with one volt being related to 11604.518 K. The ratio of these two temperatures, 14387.777 K / 11604.518 K ≈ 1.239842, is the numerical value of *hc* in units of eV⋅μm.

The Boltzmann constant provides a mapping from the characteristic microscopic energy E to the macroscopic temperature scale `T` = `E`/`k`. In fundamental physics, this mapping is often simplified by using the natural units of setting k to unity. This convention means that temperature and energy quantities have the same dimensions.^{ [17] }^{ [18] } In particular, the SI unit kelvin becomes superfluous, being defined in terms of joules as 1 K = 1.380649×10^{−23} J.^{ [19] } With this convention, temperature is always given in units of energy, and the Boltzmann constant is not explicitly needed in formulas.^{ [17] }

This convention simplifies many physical relationships and formulas. For example, the equipartition formula for the energy associated with each classical degree of freedom ( above) becomes

As another example, the definition of thermodynamic entropy coincides with the form of information entropy:

where `P`_{i} is the probability of each microstate.

**Entropy** is a scientific concept that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the microscopic description of nature in statistical physics, and to the principles of information theory. It has found far-ranging applications in chemistry and physics, in biological systems and their relation to life, in cosmology, economics, sociology, weather science, climate change, and information systems including the transmission of information in telecommunication.

In thermodynamics, the **specific heat capacity** of a substance is the amount of heat that must be added to one unit of mass of the substance in order to cause an increase of one unit in temperature. It is also referred to as **massic heat capacity** or as the **specific heat.** More formally it is the heat capacity of a sample of the substance divided by the mass of the sample. The SI unit of specific heat capacity is joule per kelvin per kilogram, J⋅kg^{−1}⋅K^{−1}. For example, the heat required to raise the temperature of 1 kg of water by 1 K is 4184 joules, so the specific heat capacity of water is 4184 J⋅kg^{−1}⋅K^{−1}.

**Thermodynamic temperature** is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.

The **Stefan–Boltzmann law**, also known as *Stefan's law*, describes the intensity of the thermal radiation emitted by matter in terms of that matter's temperature. It is named for Josef Stefan, who empirically derived the relationship, and Ludwig Boltzmann who derived the law theoretically.

The **molar gas constant** is denoted by the symbol *R* or *R*. It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per amount of substance, 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, the Arrhenius equation, and the Nernst equation.

An **ideal gas** is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics. The requirement of zero interaction can often be relaxed if, for example, the interaction is perfectly elastic or regarded as point-like collisions.

In physics, **Planck's law** describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature T, when there is no net flow of matter or energy between the body and its environment.

The **internal energy** of a thermodynamic system is the energy contained within it, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accounting for the gains and losses of energy due to changes in its internal state, including such quantities as magnetization. It excludes the kinetic energy of motion of the system as a whole and the potential energy of position of the system as a whole, with respect to its surroundings and external force fields. It includes the thermal energy, *i.e.*, the constituent particles' kinetic energies of motion relative to the motion of the system as a whole. The internal energy of an isolated system cannot change, as expressed in the law of conservation of energy, a foundation of the first law of thermodynamics.

In thermodynamics, the **Onsager reciprocal relations** express the equality of certain ratios between flows and forces in thermodynamic systems out of equilibrium, but where a notion of local equilibrium exists.

**Ludwig Eduard Boltzmann** was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of thermodynamics. In 1877 he provided the current definition of entropy, , where Ω is the number of microstates whose energy equals the system's energy, interpreted as a measure of the statistical disorder of a system. Max Planck named the constant *k*_{B} the Boltzmann constant.

The **Sackur–Tetrode equation** is an expression for the entropy of a monatomic ideal gas.

The **Joule expansion** is an irreversible process in thermodynamics in which a volume of gas is kept in one side of a thermally isolated container, with the other side of the container being evacuated. The partition between the two parts of the container is then opened, and the gas fills the whole container.

In chemical kinetics, the **entropy of activation** of a reaction is one of the two parameters which are typically obtained from the temperature dependence of a reaction rate constant, when these data are analyzed using the Eyring equation of the transition state theory. The standard entropy of activation is symbolized Δ*S*^{‡} and equals the change in entropy when the reactants change from their initial state to the activated complex or transition state.

**Wien's approximation** is a law of physics used to describe the spectrum of thermal radiation. This law was first derived by Wilhelm Wien in 1896. The equation does accurately describe the short-wavelength (high-frequency) spectrum of thermal emission from objects, but it fails to accurately fit the experimental data for long-wavelength (low-frequency) emission.

In classical thermodynamics, **entropy** is a property of a thermodynamic system that expresses the direction or outcome of spontaneous changes in the system. The term was introduced by Rudolf Clausius in the mid-19th century to explain the relationship of the internal energy that is available or unavailable for transformations in form of heat and work. Entropy predicts that certain processes are irreversible or impossible, despite not violating the conservation of energy. The definition of entropy is central to the establishment of the second law of thermodynamics, which states that the entropy of isolated systems cannot decrease with time, as they always tend to arrive at a state of thermodynamic equilibrium, where the entropy is highest. Entropy is therefore also considered to be a measure of disorder in the system.

The concept **entropy** was first developed by German physicist Rudolf Clausius in the mid-nineteenth century as a thermodynamic property that predicts that certain spontaneous processes are irreversible or impossible. In statistical mechanics, entropy is formulated as a statistical property using probability theory. The **statistical entropy** perspective was introduced in 1870 by Austrian physicist Ludwig Boltzmann, who established a new field of physics that provided the descriptive linkage between the macroscopic observation of nature and the microscopic view based on the rigorous treatment of large ensembles of microscopic states that constitute thermodynamic systems.

In thermodynamics, entropy is a numerical quantity that shows that many physical processes can go in only one direction in time. For example, cream and coffee can be mixed together, but cannot be "unmixed"; a piece of wood can be burned, but cannot be "unburned". The word 'entropy' has entered popular usage to refer a lack of order or predictability, or of a gradual decline into disorder. A more physical interpretation of thermodynamic entropy refers to spread of energy or matter, or to extent and diversity of microscopic motion.

In statistical mechanics, **Boltzmann's equation** is a probability equation relating the entropy , also written as , of an ideal gas to the multiplicity, the number of real microstates corresponding to the gas's macrostate:

**Temperature** is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making up a substance.

**Entropy production** is the amount of entropy which is produced during heat process to evaluate the efficiency of the process.

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- 1 2 Kalinin, M.; Kononogov, S. (2005). "Boltzmann's Constant, the Energy Meaning of Temperature, and Thermodynamic Irreversibility".
*Measurement Techniques*.**48**(7): 632–636. doi:10.1007/s11018-005-0195-9. S2CID 118726162. - ↑ Kittel, Charles; Kroemer, Herbert (1980).
*Thermal physics*(2nd ed.). San Francisco: W. H. Freeman. p. 41. ISBN 0716710889.We prefer to use a more natural temperature scale ... the fundamental temperature has the units of energy.

- ↑ Mohr, Peter J.; Shirley, Eric L.; Phillips, William D.; Trott, Michael (1 October 2022). "On the dimension of angles and their units".
*Metrologia*.**59**(5): 053001. arXiv: 2203.12392 . Bibcode:2022Metro..59e3001M. doi: 10.1088/1681-7575/ac7bc2 .

- 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

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