# 玻尔兹曼常数

Values of k Units
Values of Units

For details, see 模板:Section link below.

The first and third values are exact; the second is exactly equal to 模板:Sfrac. See the linked section for details.

J⋅K−1
eV⋅K−1
erg⋅K−1
For details, see below.

The first and third values are exact; the second is exactly equal to . See the linked section for details.

J⋅K−1
eV⋅K−1
erg⋅K−1
For details, see below.第一个和第三个值是精确的; 第二个值正好等于。有关详细信息，请参阅链接部分。|}

The Boltzmann constant (kB or k) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic 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 dimensions of energy divided by temperature, the same as entropy. It is named after the Austrian scientist Ludwig Boltzmann.

The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic 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 dimensions of energy divided by temperature, the same as entropy. It is named after the Austrian scientist Ludwig Boltzmann.

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 模板:Physical constants.

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 .

## Roles of the Boltzmann constant

$\displaystyle{ pV = nRT , }$

where R is the molar gas constant (模板:Val). Introducing the Boltzmann constant as the gas constant per molecule k = R/NA transforms the ideal gas law into an alternative form:

$\displaystyle{ p V = N k T , }$

where N is the number of molecules of gas. For n = 1 mol, N is equal to the number of particles in one mole (the Avogadro number).

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 molar gas constant (). Introducing the Boltzmann constant as the gas constant per molecule k = R/NA transforms the ideal gas law into an alternative form:

p V = N k T ,

where is the number of molecules of gas. For , is equal to the number of particles in one mole (the Avogadro number).

= = 波兹曼常数的作用 = = 宏观上，理想气体定律指出，对于理想气体，压力和体积的乘积与物质(摩尔)和绝对温度的乘积成正比: : pV = nRT，这里是气体常数()。引入波兹曼常数作为每个分子的气体常数 k = r/NA，将理想气体定律转换成另一种形式: : p v = n k t，这里是气体分子的数量。因为，等于一摩尔的粒子数(阿伏加德罗数)。

### Role in the equipartition of energy

Given a thermodynamic system at an absolute temperature T, the average thermal energy carried by each microscopic degree of freedom in the system is 模板:SfrackT (i.e., about 模板:Val, or 模板:Val, at room temperature).

Given a thermodynamic system at an absolute temperature , the average thermal energy carried by each microscopic degree of freedom in the system is (i.e., about , or , at room temperature).

= = = 在能量均分中的作用 = = = 在绝对温度下给定一个热力学系统，系统中每个微观自由度所携带的平均热能为(即在室温下约为，或在室温下约为)。

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 模板:SfrackT 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 模板:Val for helium, down to 模板:Val for xenon.

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 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 p for an ideal gas as

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

Combination with the ideal gas law

$\displaystyle{ p V = N k T }$

shows that the average translational kinetic energy is

$\displaystyle{ \tfrac{1}{2}m \overline{v^2} = \tfrac{3}{2} k T. }$

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. 模板:SfrackT.

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. .

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.

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 T have probability Pi of occupying a state i with energy E weighted by the corresponding Boltzmann factor:

$\displaystyle{ P_i \propto \frac{\exp\left(-\frac{E}{k T}\right)}{Z}, }$

where Z is the partition function. Again, it is the energy-like quantity kT that takes central importance.

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 that takes central importance.

= = = 在 Boltzmann 因子中的作用 = = = 更一般地说，在温度下处于平衡状态的系统有可能占据一个由相应的玻尔兹曼因子加权的状态: p i propto frac { exp left (- frac { e }{ k }{ t } right)}{ z } ，其中是配分函数。同样，类似能量的量具有核心的重要性。

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

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 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):

$\displaystyle{ S = k \,\ln W. }$

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.

= = = 在熵的统计定义中所起的作用 = = = = 在统计力学中，热力学平衡孤立系统的熵被定义为系统在给定宏观约束(例如一个固定的总能量)的情况下，可以获得的不同微观状态的自然对数: : s = k，ln w。

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.

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 k serves to make the statistical mechanical entropy equal to the classical thermodynamic entropy of Clausius:

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

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

$\displaystyle{ {S' = \ln W}, \quad \Delta S' = \int \frac{\mathrm{d}Q}{k T}. }$

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.

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.

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

### The thermal voltage模板:Anchor

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 VT. The thermal voltage depends on absolute temperature T as $\displaystyle{ V_\mathrm{T} = { k T \over q }, }$ where q is the magnitude of the electrical charge on the electron with a value 模板:Physical constants Equivalently, $\displaystyle{ { V_\mathrm{T} \over T } = { k \over q } \approx 8.61733034 \times 10^{-5}\ \mathrm{V/K}. }$

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 . 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 Equivalently,

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


At room temperature 模板:Convert, VT is approximately 模板:Val which can be derived by plugging in the values as follows:

At room temperature , is approximately which can be derived by plugging in the values as follows:

$\displaystyle{ V_\mathrm{T}={kT \over q} =\frac{1.38\times 10^{-23} \mathrm{J}\cdot k^{-1} \times 300 \mathrm{K}}{1.6 \times 10^{-19} \mathrm{C}} \simeq 25.85 \mathrm{mV} }$

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}

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}

At the standard state temperature of 模板:Convert, it is approximately 模板: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.

At the standard state temperature of , it is approximately . 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.

## 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 k, and gave a more precise value for it (模板: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 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.

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 , and gave a more 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 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.

In 1920, Planck wrote in his Nobel Prize lecture:

1920年，普朗克在他的诺贝尔奖演讲中写道:

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:

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.

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 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 模板:Section link) and other SI base units (see 模板:Section link).

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 ) and other SI base units (see ).

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;模板: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.

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; 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.

2017年，声学气体温度测量法获得了波兹曼常数的最精确测量结果，声学气体温度测量法利用微波和声学共振测量三轴椭球形腔中单原子气体的声速。几个实验室用不同的技术进行了这项长达十年的努力; 这是2019年重新定义国际单位制基础单位的基石之一。根据这些测量结果，CODATA 建议将1.380649 × 10-23 j 作为国际单位制波兹曼常数的最终固定值。

## Value in different units

J/K SI by definition, J/K = m2⋅kg/(s2⋅K) in SI base units
eV/K
Hz/K ()
erg/K CGS system, 1 erg =
cal/K 1 calorie =
cal/°R
ft lb/°R
cm−1/K ()
Eh/K (Eh = Hartree)
kcal/(mol⋅K) ()
kJ/(mol⋅K) ()
dB(W/K/Hz) , used for thermal noise calculations

= = = 不同单位的值 = = { | class = “ wikable”|-！价值观！Units

- 按照定义，j/k = m 2 · kg/(s 2 · k)以 SI 为基本单位 | -
()
| 1 erg = | - | 1卡路里 = | - |
| ()
(Eh = Hartree) | - | ()
| ()
，用于热噪声计算 | }

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 模板:Val is defined to be the same as a change of 模板:Val. The characteristic energy kT is a term encountered in many physical relationships.

Since 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 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 模板:Val, and also a relationship between voltage and temperature (multiplying the voltage by k in units of eV/K) with one volt being related to 模板:Val. The ratio of these two temperatures, 模板:Val / 模板:Val ≈ 1.239842, is the numerical value of hc in units of eV⋅μm.

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 , and also a relationship between voltage and temperature (multiplying the voltage by k in units of eV/K) with one volt being related to . The ratio of these two temperatures,  /  ≈ 1.239842, is the numerical value of hc in units of eV⋅μm.

### Natural units

The Boltzmann constant provides a mapping from this characteristic microscopic energy E to the macroscopic temperature scale T = 模板:Sfrac. In physics research another definition is often encountered in setting k to unity, resulting in temperature and energy quantities of the same type. In this context temperature is measured effectively in units of energy and the Boltzmann constant is not explicitly needed.

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 temperature and energy quantities of the same type. In this context temperature is measured effectively in units of energy and the Boltzmann constant is not explicitly needed.

= = = = 自然单位 = = = 波兹曼常数提供了从这种特征的微观能量到宏观温度尺度的映射。在物理学研究中，经常遇到另一种定义，即在设定单位时，导致同一类型的温度和能量量。在这种情况下，温度是以能量为单位有效地测量的，并不明确需要波兹曼常数。

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

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

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

$\displaystyle{ S = - \sum_i P_i \ln P_i. }$

where Pi is the probability of each microstate.

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_i P_i \ln P_i.

where is the probability of each microstate.