玻尔兹曼常数
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Values of k^{[1]} | Units
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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 (k_{B} 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.^{[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. 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. 波兹曼常数(或)是比例因子，它将气体中粒子的平均相对动能与气体的热力学温度相联系。它出现在开尔文和气体常数的定义中，以及普朗克黑体辐射定律和玻尔兹曼熵公式中。波兹曼常数的能量维数除以温度，和熵一样。它是以奥地利科学家路德维希·玻尔兹曼命名的。 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 . 作为2019年国际单位制基本单位重新定义的一部分，波兹曼常数单位是7个给出了精确定义的“定义常数”之一。它们被用于不同的组合来定义七个 SI 基本单位。波兹曼常数的定义是。 Roles of the Boltzmann constant模板:Ideal gas law relationships.svg 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 (in moles) and absolute temperature T:
where R is the molar gas constant (模板:Val).^{[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 molecules of gas. For n = 1 mol, N is equal to the number of particles in one mole (the Avogadro number).
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:
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 energyGiven 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).
= = = 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. 在经典统计力学理论中，这个平均值被精确地预测为均匀理想气体的平均值。单原子理想气体(六种惰性气体)每个原子具有三个自由度，对应于三个空间方向。根据能量均分，这意味着每个原子的热能为。这与实验数据非常吻合。热能可以用来计算原子的均方根速度，这个值与原子质量的平方根成反比。在室温下测得的平方平均数速度精确地反映了这一点，从氦到氙。 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. 模板:SfrackT. Kinetic theory gives the average pressure 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 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. . 动力学理论给出理想气体的平均压强为: p = frac {1}{3} frac { n }{ v } m 上线{ v ^ 2}。结合理想气体定律 p v = nkt，得到平均平动能为: tfrac {1}{2} m overline { v ^ 2} = tfrac {3}{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. 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. 理想气体方程也被分子气体严格遵守，但是热容的形式更加复杂，因为分子具有额外的内部自由度，以及分子作为一个整体运动的三个自由度。例如，双原子气体，每个与原子运动有关的分子总共有六个简单自由度(三个平动，两个转动，一个振动)。在较低的温度下，并非所有这些自由度都可以完全参与气体热容，这是由于在每个分子相应的热能处激发态的可用性受到量子力学的限制。 Role in Boltzmann factorsMore 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 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:
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. 这样做的结果包括(除了上述理想气体的结果之外)位于阿伦尼乌斯方程的化学动力学。 Role in the statistical definition of entropyIn 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):
= = = 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. 这个方程，将系统的微观细节或微观状态(通过)与其宏观状态(通过熵)联系起来，是统计力学的中心思想。由于它的重要性，它被刻在波尔兹曼的墓碑上。 The constant of proportionality k serves to make the statistical mechanical entropy equal to the classical thermodynamic entropy of Clausius:
The constant of proportionality serves to make the statistical mechanical entropy equal to the classical thermodynamic entropy of Clausius:
比例常数使得统计力学熵等于克劳修斯的经典熵: : Delta s = int frac { rm d } q }{ t }。 One could choose instead a rescaled dimensionless entropy in microscopic terms such that
One could choose instead a rescaled dimensionless entropy in microscopic terms such that
我们可以选择微观条件下的重新标度的无量纲熵: { s’= ln w } ，四个 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. 这是一种更自然的形式，这种重新调整的熵恰好对应于香农随后的熵。 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. 因此，特征能量就是增加重标熵所需要的能量。 The thermal voltage模板:AnchorIn 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 [math]\displaystyle{ V_\mathrm{T} = { k T \over q }, }[/math] where q is the magnitude of the electrical charge on the electron with a value 模板:Physical constants Equivalently, [math]\displaystyle{ { V_\mathrm{T} \over T } = { k \over q } \approx 8.61733034 \times 10^{-5}\ \mathrm{V/K}. }[/math]
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}. 在半导体中，肖克利二极管方程ーー电流流动与 p-n 结上的静电势之间的关系ーー取决于一个称为热电压的特征电压。热电压取决于绝对温度为 v _ mathrm { t } = { k t over q } ，其中电子上电荷的大小与 t } = { k over q }约8.61733034乘以10 ^ {-5} mathrm { v/k }。 At room temperature 模板:Convert, V_{T} is approximately 模板:Val^{[5]}^{[6]} 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: 在室温下，可通过插入下列数值得出: [math]\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} }[/math] 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.^{[7]}^{[8]} 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. 在标准状态温度，它大约是。热电压在等离子体和电解质溶液中也很重要。能斯特方程) ，在这两种情况下，它都提供了电子或离子的空间分布在多大程度上受到在固定电压下保持的边界的影响。 HistoryThe 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.^{[9]} 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.^{[10]} 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 . 波兹曼常数是以19世纪发现它的奥地利人路德维希·玻尔兹曼命名的。虽然玻尔兹曼在1877年首次将熵和概率联系起来，但直到马克斯 · 普朗克在1900-1901年推导黑体辐射定律时首次提出并给出了一个更精确的数值(比今天的数值低2.5%)之前，这种关系从未用一个特定的常数来表示。.在1900年之前，包含玻耳兹曼因子的方程式并不是用单个分子的能量和波兹曼常数来表示的，而是用气体常数的形式来表示，用宏观能量表示物质的宏观量。波尔兹曼墓碑上方程式的标志性简洁形式实际上是由于普朗克，而不是波尔兹曼。事实上，普朗克在他的同名作品中引入了这个概念。 In 1920, Planck wrote in his Nobel Prize lecture:^{[11]}
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:^{[11]}
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 ). 在2019年国际单位制基本单位重新定义之前的国际单位制版本中，波兹曼常数是一个测量量而不是一个固定值。由于开尔文(见)和其他国际单位制基本单位的重新定义(见) ，它的确切定义也随着时间的推移而变化。 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.^{[12]}^{[13]} 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.^{[14]} 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
= = = 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 | - | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| 1 erg = | - | | 1卡路里 = | - | | | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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，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. 由于是温度和能量之间的比例因子，其数值取决于能量和温度单位的选择。以 SI 单位表示的波兹曼常数的小数值意味着温度变化1 k 只会使粒子的能量发生很小的变化。变化的定义与变化的定义相同。特征能量是许多物理关系中都会遇到的一个术语。 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. 波兹曼常数建立了波长和温度的关系(hc/k 除以波长就是温度) ，与一微米相关，与电压和温度的关系(电压乘以 k 以 eV/k 为单位) ，与一伏特相关。这两个温度的比值,/≈1.239842，是 hc 的数值，单位为 eV · μm。 Natural unitsThe 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.^{[15]} 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
The use of natural units simplifies many physical relationships; in this form the definition of thermodynamic entropy coincides with the form of information entropy:
where P_{i} is the probability of each microstate. The equipartition formula for the energy associated with each classical degree of freedom then becomes
The use of natural units simplifies many physical relationships; in this form the definition of thermodynamic entropy coincides with the form of information entropy:
where is the probability of each microstate. 使用自然单位简化了许多物理关系; 在这种形式下，熵的定义与熵的形式一致: s =-sum i p i ln pi。每个微观状态的概率是多少。 See also
2018年3月18日，热力学beta NotesReferences
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