Boltzmann constant

# Boltzmann constant

Discussion

Encyclopedia
Values of k Units
J
Joule
The joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...

K
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...

−1
eV K−1
erg
Erg
An erg is the unit of energy and mechanical work in the centimetre-gram-second system of units, symbol "erg". Its name is derived from the Greek ergon, meaning "work"....

K−1
For details, see Value in different units below.

The Boltzmann constant (k or kB) is the physical constant
Physical constant
A physical constant is a physical quantity that is generally believed to be both universal in nature and constant in time. It can be contrasted with a mathematical constant, which is a fixed numerical value but does not directly involve any physical measurement.There are many physical constants in...

relating energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

at the individual particle level with temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

, which must necessarily be observed at the collective or bulk level. It is the gas constant
Gas constant
The gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...

R divided by the Avogadro constant NA:

It has the same units as entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

. It is named after the Austria
Austria
Austria , officially the Republic of Austria , is a landlocked country of roughly 8.4 million people in Central Europe. It is bordered by the Czech Republic and Germany to the north, Slovakia and Hungary to the east, Slovenia and Italy to the south, and Switzerland and Liechtenstein to the...

n physicist Ludwig Boltzmann
Ludwig Boltzmann
Ludwig Eduard Boltzmann was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics...

.

## Bridge from macroscopic to microscopic physics

Boltzmann's constant, k, is a bridge between macroscopic and microscopic physics, since temperature (T) makes sense only in the macroscopic world, while the quantity kT gives a quantity of energy which is on the order of, though rarely exactly the same as, the energy of a given atom in a substance with a temperature T.

Macroscopically, the ideal gas law
Ideal gas law
The ideal gas law is the equation of state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions, although it has several limitations. It was first stated by Émile Clapeyron in 1834 as a combination of Boyle's law and Charles's law...

states that, for an ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. 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.At normal conditions such as...

, the product of pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

P and volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

V is proportional to the product of amount of substance
Amount of substance
Amount of substance is a standards-defined quantity that measures the size of an ensemble of elementary entities, such as atoms, molecules, electrons, and other particles. It is sometimes referred to as chemical amount. The International System of Units defines the amount of substance to be...

n (in moles
Mole (unit)
The mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities as there are atoms in 12 grams of pure carbon-12 , the isotope of carbon with atomic weight 12. This corresponds to a value...

) and absolute temperature T:

where R is the gas constant
Gas constant
The gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...

. Introducing the Boltzmann constant transforms the ideal gas law into an equation about the microscopic properties of molecules,
where N is the number of molecules of gas. (For n = 1 i.e. for 1 mole
Mole (unit)
The mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities as there are atoms in 12 grams of pure carbon-12 , the isotope of carbon with atomic weight 12. This corresponds to a value...

, N is equal to NA, the Avogadro constant.) Thus, the left hand side of the equation is a macroscopic amount of pressure-volume work represented by the state of the bulk gas. The right hand side divides this energy into N units, one for each gas molecule, each of which represents kT amount of energy.

## Role in the equipartition of energy

Given a thermodynamic
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

system at an absolute temperature
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....

T, the thermal energy carried by each microscopic "degree of freedom" in the system is on the order of magnitude
Order of magnitude
An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. In its most common usage, the amount being scaled is 10 and the scale is the exponent being applied to this amount...

of kT/2 (i. e., about 2.07 J, or 0.013 eV, at room temperature).

### Application to simple gas thermodynamics

In classical
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

, this average is predicted to hold exactly for homogeneous ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. 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.At normal conditions such as...

es. Monatomic ideal gases possess three degrees of freedom
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

per atom, corresponding to the three spatial directions, which means a thermal energy of 1.5kT per atom (in the general case, DkT/2, where D is the number of spatial dimensions). As indicated in the article on heat capacity, this corresponds very well with experimental data. The thermal energy can be used to calculate the root mean square
Root mean square
In mathematics, the root mean square , also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. It is especially useful when variates are positive and negative, e.g., sinusoids...

speed of the atoms, which is inversely proportional to the square root of the atomic mass
Atomic mass
The atomic mass is the mass of a specific isotope, most often expressed in unified atomic mass units. The atomic mass is the total mass of protons, neutrons and electrons in a single atom....

. The root mean square speeds found at room temperature accurately reflect this, ranging from 1370 m/s for helium
Helium
Helium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas that heads the noble gas group in the periodic table...

, down to 240 m/s for xenon
Xenon
Xenon is a chemical element with the symbol Xe and atomic number 54. The element name is pronounced or . A colorless, heavy, odorless noble gas, xenon occurs in the Earth's atmosphere in trace amounts...

.

Kinetic theory gives the average pressure P for an ideal gas as
Substituting that the average translational kinetic energy is
gives
so the ideal gas equation is regained.

The ideal gas equation is also followed quite well for molecular gases; but the form for the heat capacity is more complicated, because the molecules possess new 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 seven degrees of simple freedom per molecule that are related to atomic motion (three translational, two rotational, and two 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 thermal energy available.

## Role in Boltzmann factors

More generally, systems in equilibrium at temperature T have probability p of occupying a state i with energy E weighted by the corresponding Boltzmann factor
Boltzmann factor
In physics, the Boltzmann factor is a weighting factor that determines the relative probability of a particle to be in a state i in a multi-state system in thermodynamic equilibrium at temperature T...

:
Where Z is the partition function
Partition function (statistical mechanics)
Partition functions describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...

.
Again, it is the energy-like quantity kT
KT (energy)
kT is the product of the Boltzmann constant, k, and the temperature, T. This product is used in physics as a scaling factor for energy values in molecular-scale systems , as the rates and frequencies of many processes and phenomena depend not on their energy alone, but on the ratio of that energy...

which takes central importance.

Consequences of this include (in addition to the results for ideal gases above) the Arrhenius equation
Arrhenius equation
The Arrhenius equation is a simple, but remarkably accurate, formula for the temperature dependence of the reaction rate constant, and therefore, rate of a chemical reaction. The equation was first proposed by the Dutch chemist J. H. van 't Hoff in 1884; five years later in 1889, the Swedish...

in chemical kinetics
Chemical kinetics
Chemical kinetics, also known as reaction kinetics, is the study of rates of chemical processes. Chemical kinetics includes investigations of how different experimental conditions can influence the speed of a chemical reaction and yield information about the reaction's mechanism and transition...

.

## Role in the statistical definition of entropy

In statistical mechanics, the entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

S of an isolated system
Isolated system
In the natural sciences an isolated system, as contrasted with an open system, is a physical system without any external exchange. If it has any surroundings, it does not interact with them. It obeys in particular the first of the conservation laws: its total energy - mass stays constant...

at thermodynamic equilibrium
Thermodynamic equilibrium
In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...

is defined as the natural logarithm
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

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 rather more natural form; and this rescaled entropy exactly corresponds to Shannon's subsequent information entropy
Information entropy
In information theory, entropy is a measure of the uncertainty associated with a random variable. In this context, the term usually refers to the Shannon entropy, which quantifies the expected value of the information contained in a message, usually in units such as bits...

.

The characteristic energy kT is thus the heat required to increase the rescaled entropy by one nat
Nat (information)
A nat is a logarithmic unit of information or entropy, based on natural logarithms and powers of e, rather than the powers of 2 and base 2 logarithms which define the bit. The nat is the natural unit for information entropy...

.

## Role in semiconductor physics: the thermal voltage

In semiconductors, the relationship between the flow of electrical current and the electrostatic potential across a p-n junction
P-n junction
A p–n junction is formed at the boundary between a P-type and N-type semiconductor created in a single crystal of semiconductor by doping, for example by ion implantation, diffusion of dopants, or by epitaxy .If two separate pieces of material were used, this would...

depends on a characteristic voltage called the thermal voltage, denoted VT. The thermal voltage depends on absolute temperature T as
where q is the magnitude of the electrical charge on the electron
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...

with a value and k is the Boltzmann constant in Joules/K. In electronvolt
Electronvolt
In physics, the electron volt is a unit of energy equal to approximately joule . By definition, it is equal to the amount of kinetic energy gained by a single unbound electron when it accelerates through an electric potential difference of one volt...

s, the Boltzmann constant is , making it easy to calculate that at room temperature
Room temperature
-Comfort levels:The American Society of Heating, Refrigerating and Air-Conditioning Engineers has listings for suggested temperatures and air flow rates in different types of buildings and different environmental circumstances. For example, a single office in a building has an occupancy ratio per...

(≈ 300 K), the value of the thermal voltage is approximately 25.85 millivolts ≈ 26 mV http://www.google.com/search?hl=en&q=300+kelvin+*+k+%2F+elementary+charge+in+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
Max Planck
Max Karl Ernst Ludwig Planck, ForMemRS, was a German physicist who actualized the quantum physics, initiating a revolution in natural science and philosophy. He is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.-Life and career:Planck came...

first introduced k , and gave an accurate value for it (1.346 J/K, 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 Boltzmann's constant, but rather using a form of the gas constant
Gas constant
The gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...

R, and macroscopic energies for macroscopic quantities of the substance. The iconic terse form of the equation S = k log W on Boltzmann's tombstone is in fact due to Planck, not Boltzmann. Planck actually introduced it in the same work as his h.

As Planck wrote in his Nobel Prize
Nobel Prize
The Nobel Prizes are annual international awards bestowed by Scandinavian committees in recognition of cultural and scientific advances. The will of the Swedish chemist Alfred Nobel, the inventor of dynamite, established the prizes in 1895...

lecture in 1920,
This "peculiar state of affairs" can be understood 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
Heuristic
Heuristic refers to experience-based techniques for problem solving, learning, and discovery. Heuristic methods are used to speed up the process of finding a satisfactory solution, where an exhaustive search is impractical...

, a useful tool for solving problems. Nor was there agreement as to whether "chemical molecules" (as measured by atomic weight
Atomic weight
Atomic weight is a dimensionless physical quantity, the ratio of the average mass of atoms of an element to 1/12 of the mass of an atom of carbon-12...

s) were the same as "physical molecules" (as measured by kinetic theory
Kinetic theory
The kinetic theory of gases describes a gas as a large number of small particles , all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container...

). To continue the quotation from Planck's 1920 lecture:

## Value in different units

1.380 6488(13) J
Joule
The joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...

/K
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...

SI
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...

units, 2010 CODATA value
8.617 3324(78) eV
Electronvolt
In physics, the electron volt is a unit of energy equal to approximately joule . By definition, it is equal to the amount of kinetic energy gained by a single unbound electron when it accelerates through an electric potential difference of one volt...

/K
2010 CODATA value
electronvolt
Electronvolt
In physics, the electron volt is a unit of energy equal to approximately joule . By definition, it is equal to the amount of kinetic energy gained by a single unbound electron when it accelerates through an electric potential difference of one volt...

= 1.602 176 565(35)
Values of k Units
{{gaps|1.380|648|8(13)}}{{e|−23}} J
Joule
The joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...

K
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...

−1
{{gaps|8.617|332|4(78)}}{{e|−5}} eV K−1
{{gaps|1.380|648|8(13)}}{{e|−16}} erg
Erg
An erg is the unit of energy and mechanical work in the centimetre-gram-second system of units, symbol "erg". Its name is derived from the Greek ergon, meaning "work"....

K−1
For details, see Value in different units below.

The Boltzmann constant (k or kB) is the physical constant
Physical constant
A physical constant is a physical quantity that is generally believed to be both universal in nature and constant in time. It can be contrasted with a mathematical constant, which is a fixed numerical value but does not directly involve any physical measurement.There are many physical constants in...

relating energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

at the individual particle level with temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

, which must necessarily be observed at the collective or bulk level. It is the gas constant
Gas constant
The gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...

R divided by the Avogadro constant NA:

It has the same units as entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

. It is named after the Austria
Austria
Austria , officially the Republic of Austria , is a landlocked country of roughly 8.4 million people in Central Europe. It is bordered by the Czech Republic and Germany to the north, Slovakia and Hungary to the east, Slovenia and Italy to the south, and Switzerland and Liechtenstein to the...

n physicist Ludwig Boltzmann
Ludwig Boltzmann
Ludwig Eduard Boltzmann was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics...

.

## Bridge from macroscopic to microscopic physics

Boltzmann's constant, k, is a bridge between macroscopic and microscopic physics, since temperature (T) makes sense only in the macroscopic world, while the quantity kT gives a quantity of energy which is on the order of, though rarely exactly the same as, the energy of a given atom in a substance with a temperature T.

Macroscopically, the ideal gas law
Ideal gas law
The ideal gas law is the equation of state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions, although it has several limitations. It was first stated by Émile Clapeyron in 1834 as a combination of Boyle's law and Charles's law...

states that, for an ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. 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.At normal conditions such as...

, the product of pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

P and volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

V is proportional to the product of amount of substance
Amount of substance
Amount of substance is a standards-defined quantity that measures the size of an ensemble of elementary entities, such as atoms, molecules, electrons, and other particles. It is sometimes referred to as chemical amount. The International System of Units defines the amount of substance to be...

n (in moles
Mole (unit)
The mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities as there are atoms in 12 grams of pure carbon-12 , the isotope of carbon with atomic weight 12. This corresponds to a value...

) and absolute temperature T:

where R is the gas constant
Gas constant
The gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...

({{nowrap|8.314 4621(75) J K−1 mol−1}}). Introducing the Boltzmann constant transforms the ideal gas law into an equation about the microscopic properties of molecules,{{Citation needed|date=July 2011}}{{Dubious|date=July 2011}}
where N is the number of molecules of gas. (For n = 1 i.e. for 1 mole
Mole (unit)
The mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities as there are atoms in 12 grams of pure carbon-12 , the isotope of carbon with atomic weight 12. This corresponds to a value...

, N is equal to NA, the Avogadro constant.) Thus, the left hand side of the equation is a macroscopic amount of pressure-volume work represented by the state of the bulk gas. The right hand side divides this energy into N units, one for each gas molecule, each of which represents kT amount of energy.

## Role in the equipartition of energy

Given a thermodynamic
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

system at an absolute temperature
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....

T, the thermal energy carried by each microscopic "degree of freedom" in the system is on the order of magnitude
Order of magnitude
An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. In its most common usage, the amount being scaled is 10 and the scale is the exponent being applied to this amount...

of kT/2 (i. e., about 2.07{{e|−21}} J, or 0.013 eV, at room temperature).

### Application to simple gas thermodynamics

In classical
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

, this average is predicted to hold exactly for homogeneous ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. 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.At normal conditions such as...

es. Monatomic ideal gases possess three degrees of freedom
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

per atom, corresponding to the three spatial directions, which means a thermal energy of 1.5kT per atom (in the general case, DkT/2, where D is the number of spatial dimensions). As indicated in the article on heat capacity, this corresponds very well with experimental data. The thermal energy can be used to calculate the root mean square
Root mean square
In mathematics, the root mean square , also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. It is especially useful when variates are positive and negative, e.g., sinusoids...

speed of the atoms, which is inversely proportional to the square root of the atomic mass
Atomic mass
The atomic mass is the mass of a specific isotope, most often expressed in unified atomic mass units. The atomic mass is the total mass of protons, neutrons and electrons in a single atom....

. The root mean square speeds found at room temperature accurately reflect this, ranging from 1370 m/s for helium
Helium
Helium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas that heads the noble gas group in the periodic table...

, down to 240 m/s for xenon
Xenon
Xenon is a chemical element with the symbol Xe and atomic number 54. The element name is pronounced or . A colorless, heavy, odorless noble gas, xenon occurs in the Earth's atmosphere in trace amounts...

.

Kinetic theory gives the average pressure P for an ideal gas as
Substituting that the average translational kinetic energy is
gives
so the ideal gas equation is regained.

The ideal gas equation is also followed quite well for molecular gases; but the form for the heat capacity is more complicated, because the molecules possess new 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 seven degrees of simple freedom per molecule that are related to atomic motion (three translational, two rotational, and two 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 thermal energy available.

## Role in Boltzmann factors

More generally, systems in equilibrium at temperature T have probability p of occupying a state i with energy E weighted by the corresponding Boltzmann factor
Boltzmann factor
In physics, the Boltzmann factor is a weighting factor that determines the relative probability of a particle to be in a state i in a multi-state system in thermodynamic equilibrium at temperature T...

:
Where Z is the partition function
Partition function (statistical mechanics)
Partition functions describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...

.
Again, it is the energy-like quantity kT
KT (energy)
kT is the product of the Boltzmann constant, k, and the temperature, T. This product is used in physics as a scaling factor for energy values in molecular-scale systems , as the rates and frequencies of many processes and phenomena depend not on their energy alone, but on the ratio of that energy...

which takes central importance.

Consequences of this include (in addition to the results for ideal gases above) the Arrhenius equation
Arrhenius equation
The Arrhenius equation is a simple, but remarkably accurate, formula for the temperature dependence of the reaction rate constant, and therefore, rate of a chemical reaction. The equation was first proposed by the Dutch chemist J. H. van 't Hoff in 1884; five years later in 1889, the Swedish...

in chemical kinetics
Chemical kinetics
Chemical kinetics, also known as reaction kinetics, is the study of rates of chemical processes. Chemical kinetics includes investigations of how different experimental conditions can influence the speed of a chemical reaction and yield information about the reaction's mechanism and transition...

.

## Role in the statistical definition of entropy

{{further|Entropy (statistical thermodynamics)}}

In statistical mechanics, the entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

S of an isolated system
Isolated system
In the natural sciences an isolated system, as contrasted with an open system, is a physical system without any external exchange. If it has any surroundings, it does not interact with them. It obeys in particular the first of the conservation laws: its total energy - mass stays constant...

at thermodynamic equilibrium
Thermodynamic equilibrium
In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...

is defined as the natural logarithm
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

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 rather more natural form; and this rescaled entropy exactly corresponds to Shannon's subsequent information entropy
Information entropy
In information theory, entropy is a measure of the uncertainty associated with a random variable. In this context, the term usually refers to the Shannon entropy, which quantifies the expected value of the information contained in a message, usually in units such as bits...

.

The characteristic energy kT is thus the heat required to increase the rescaled entropy by one nat
Nat (information)
A nat is a logarithmic unit of information or entropy, based on natural logarithms and powers of e, rather than the powers of 2 and base 2 logarithms which define the bit. The nat is the natural unit for information entropy...

.

## Role in semiconductor physics: the thermal voltage

In semiconductors, the relationship between the flow of electrical current and the electrostatic potential across a p-n junction
P-n junction
A p–n junction is formed at the boundary between a P-type and N-type semiconductor created in a single crystal of semiconductor by doping, for example by ion implantation, diffusion of dopants, or by epitaxy .If two separate pieces of material were used, this would...

depends on a characteristic voltage called the thermal voltage, denoted VT. The thermal voltage depends on absolute temperature T as
where q is the magnitude of the electrical charge on the electron
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...

with a value {{nowrap|1.602 176 565(35){{e|−19}} C}} and k is the Boltzmann constant in Joules/K. In electronvolt
Electronvolt
In physics, the electron volt is a unit of energy equal to approximately joule . By definition, it is equal to the amount of kinetic energy gained by a single unbound electron when it accelerates through an electric potential difference of one volt...

s, the Boltzmann constant is {{nowrap|8.617 3324(78){{e|−5}} eV/K}}, making it easy to calculate that at room temperature
Room temperature
-Comfort levels:The American Society of Heating, Refrigerating and Air-Conditioning Engineers has listings for suggested temperatures and air flow rates in different types of buildings and different environmental circumstances. For example, a single office in a building has an occupancy ratio per...

(≈ 300 K), the value of the thermal voltage is approximately 25.85 millivolts ≈ 26 mV http://www.google.com/search?hl=en&q=300+kelvin+*+k+%2F+elementary+charge+in+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
Max Planck
Max Karl Ernst Ludwig Planck, ForMemRS, was a German physicist who actualized the quantum physics, initiating a revolution in natural science and philosophy. He is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.-Life and career:Planck came...

first introduced k , and gave an accurate value for it (1.346{{e|−23}} J/K, 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 Boltzmann's constant, but rather using a form of the gas constant
Gas constant
The gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...

R, and macroscopic energies for macroscopic quantities of the substance. The iconic terse form of the equation S = k log W on Boltzmann's tombstone is in fact due to Planck, not Boltzmann. Planck actually introduced it in the same work as his h.

As Planck wrote in his Nobel Prize
Nobel Prize
The Nobel Prizes are annual international awards bestowed by Scandinavian committees in recognition of cultural and scientific advances. The will of the Swedish chemist Alfred Nobel, the inventor of dynamite, established the prizes in 1895...

lecture in 1920,
{{quotation|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" can be understood 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
Heuristic
Heuristic refers to experience-based techniques for problem solving, learning, and discovery. Heuristic methods are used to speed up the process of finding a satisfactory solution, where an exhaustive search is impractical...

, a useful tool for solving problems. Nor was there agreement as to whether "chemical molecules" (as measured by atomic weight
Atomic weight
Atomic weight is a dimensionless physical quantity, the ratio of the average mass of atoms of an element to 1/12 of the mass of an atom of carbon-12...

s) were the same as "physical molecules" (as measured by kinetic theory
Kinetic theory
The kinetic theory of gases describes a gas as a large number of small particles , all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container...

). To continue the quotation from Planck's 1920 lecture:
{{quotation|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.}}

## Value in different units

1.380 6488(13){{e|−23}} J
Joule
The joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...

/K
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...

SI
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...

units, 2010 CODATA value
8.617 3324(78){{e|−5}} eV
Electronvolt
In physics, the electron volt is a unit of energy equal to approximately joule . By definition, it is equal to the amount of kinetic energy gained by a single unbound electron when it accelerates through an electric potential difference of one volt...

/K
2010 CODATA value
electronvolt
Electronvolt
In physics, the electron volt is a unit of energy equal to approximately joule . By definition, it is equal to the amount of kinetic energy gained by a single unbound electron when it accelerates through an electric potential difference of one volt...

= 1.602 176 565(35)
Values of k Units
{{gaps|1.380|648|8(13)}}{{e|−23}} J
Joule
The joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...

K
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...

−1
{{gaps|8.617|332|4(78)}}{{e|−5}} eV K−1
{{gaps|1.380|648|8(13)}}{{e|−16}} erg
Erg
An erg is the unit of energy and mechanical work in the centimetre-gram-second system of units, symbol "erg". Its name is derived from the Greek ergon, meaning "work"....

K−1
For details, see Value in different units below.

The Boltzmann constant (k or kB) is the physical constant
Physical constant
A physical constant is a physical quantity that is generally believed to be both universal in nature and constant in time. It can be contrasted with a mathematical constant, which is a fixed numerical value but does not directly involve any physical measurement.There are many physical constants in...

relating energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

at the individual particle level with temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

, which must necessarily be observed at the collective or bulk level. It is the gas constant
Gas constant
The gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...

R divided by the Avogadro constant NA:

It has the same units as entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

. It is named after the Austria
Austria
Austria , officially the Republic of Austria , is a landlocked country of roughly 8.4 million people in Central Europe. It is bordered by the Czech Republic and Germany to the north, Slovakia and Hungary to the east, Slovenia and Italy to the south, and Switzerland and Liechtenstein to the...

n physicist Ludwig Boltzmann
Ludwig Boltzmann
Ludwig Eduard Boltzmann was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics...

.

## Bridge from macroscopic to microscopic physics

Boltzmann's constant, k, is a bridge between macroscopic and microscopic physics, since temperature (T) makes sense only in the macroscopic world, while the quantity kT gives a quantity of energy which is on the order of, though rarely exactly the same as, the energy of a given atom in a substance with a temperature T.

Macroscopically, the ideal gas law
Ideal gas law
The ideal gas law is the equation of state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions, although it has several limitations. It was first stated by Émile Clapeyron in 1834 as a combination of Boyle's law and Charles's law...

states that, for an ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. 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.At normal conditions such as...

, the product of pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

P and volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

V is proportional to the product of amount of substance
Amount of substance
Amount of substance is a standards-defined quantity that measures the size of an ensemble of elementary entities, such as atoms, molecules, electrons, and other particles. It is sometimes referred to as chemical amount. The International System of Units defines the amount of substance to be...

n (in moles
Mole (unit)
The mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities as there are atoms in 12 grams of pure carbon-12 , the isotope of carbon with atomic weight 12. This corresponds to a value...

) and absolute temperature T:

where R is the gas constant
Gas constant
The gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...

({{nowrap|8.314 4621(75) J K−1 mol−1}}). Introducing the Boltzmann constant transforms the ideal gas law into an equation about the microscopic properties of molecules,{{Citation needed|date=July 2011}}{{Dubious|date=July 2011}}
where N is the number of molecules of gas. (For n = 1 i.e. for 1 mole
Mole (unit)
The mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities as there are atoms in 12 grams of pure carbon-12 , the isotope of carbon with atomic weight 12. This corresponds to a value...

, N is equal to NA, the Avogadro constant.) Thus, the left hand side of the equation is a macroscopic amount of pressure-volume work represented by the state of the bulk gas. The right hand side divides this energy into N units, one for each gas molecule, each of which represents kT amount of energy.

## Role in the equipartition of energy

Given a thermodynamic
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

system at an absolute temperature
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....

T, the thermal energy carried by each microscopic "degree of freedom" in the system is on the order of magnitude
Order of magnitude
An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. In its most common usage, the amount being scaled is 10 and the scale is the exponent being applied to this amount...

of kT/2 (i. e., about 2.07{{e|−21}} J, or 0.013 eV, at room temperature).

### Application to simple gas thermodynamics

In classical
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

, this average is predicted to hold exactly for homogeneous ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. 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.At normal conditions such as...

es. Monatomic ideal gases possess three degrees of freedom
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

per atom, corresponding to the three spatial directions, which means a thermal energy of 1.5kT per atom (in the general case, DkT/2, where D is the number of spatial dimensions). As indicated in the article on heat capacity, this corresponds very well with experimental data. The thermal energy can be used to calculate the root mean square
Root mean square
In mathematics, the root mean square , also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. It is especially useful when variates are positive and negative, e.g., sinusoids...

speed of the atoms, which is inversely proportional to the square root of the atomic mass
Atomic mass
The atomic mass is the mass of a specific isotope, most often expressed in unified atomic mass units. The atomic mass is the total mass of protons, neutrons and electrons in a single atom....

. The root mean square speeds found at room temperature accurately reflect this, ranging from 1370 m/s for helium
Helium
Helium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas that heads the noble gas group in the periodic table...

, down to 240 m/s for xenon
Xenon
Xenon is a chemical element with the symbol Xe and atomic number 54. The element name is pronounced or . A colorless, heavy, odorless noble gas, xenon occurs in the Earth's atmosphere in trace amounts...

.

Kinetic theory gives the average pressure P for an ideal gas as
Substituting that the average translational kinetic energy is
gives
so the ideal gas equation is regained.

The ideal gas equation is also followed quite well for molecular gases; but the form for the heat capacity is more complicated, because the molecules possess new 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 seven degrees of simple freedom per molecule that are related to atomic motion (three translational, two rotational, and two 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 thermal energy available.

## Role in Boltzmann factors

More generally, systems in equilibrium at temperature T have probability p of occupying a state i with energy E weighted by the corresponding Boltzmann factor
Boltzmann factor
In physics, the Boltzmann factor is a weighting factor that determines the relative probability of a particle to be in a state i in a multi-state system in thermodynamic equilibrium at temperature T...

:
Where Z is the partition function
Partition function (statistical mechanics)
Partition functions describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...

.
Again, it is the energy-like quantity kT
KT (energy)
kT is the product of the Boltzmann constant, k, and the temperature, T. This product is used in physics as a scaling factor for energy values in molecular-scale systems , as the rates and frequencies of many processes and phenomena depend not on their energy alone, but on the ratio of that energy...

which takes central importance.

Consequences of this include (in addition to the results for ideal gases above) the Arrhenius equation
Arrhenius equation
The Arrhenius equation is a simple, but remarkably accurate, formula for the temperature dependence of the reaction rate constant, and therefore, rate of a chemical reaction. The equation was first proposed by the Dutch chemist J. H. van 't Hoff in 1884; five years later in 1889, the Swedish...

in chemical kinetics
Chemical kinetics
Chemical kinetics, also known as reaction kinetics, is the study of rates of chemical processes. Chemical kinetics includes investigations of how different experimental conditions can influence the speed of a chemical reaction and yield information about the reaction's mechanism and transition...

.

## Role in the statistical definition of entropy

{{further|Entropy (statistical thermodynamics)}}

In statistical mechanics, the entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

S of an isolated system
Isolated system
In the natural sciences an isolated system, as contrasted with an open system, is a physical system without any external exchange. If it has any surroundings, it does not interact with them. It obeys in particular the first of the conservation laws: its total energy - mass stays constant...

at thermodynamic equilibrium
Thermodynamic equilibrium
In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...

is defined as the natural logarithm
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

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 rather more natural form; and this rescaled entropy exactly corresponds to Shannon's subsequent information entropy
Information entropy
In information theory, entropy is a measure of the uncertainty associated with a random variable. In this context, the term usually refers to the Shannon entropy, which quantifies the expected value of the information contained in a message, usually in units such as bits...

.

The characteristic energy kT is thus the heat required to increase the rescaled entropy by one nat
Nat (information)
A nat is a logarithmic unit of information or entropy, based on natural logarithms and powers of e, rather than the powers of 2 and base 2 logarithms which define the bit. The nat is the natural unit for information entropy...

.

## Role in semiconductor physics: the thermal voltage

In semiconductors, the relationship between the flow of electrical current and the electrostatic potential across a p-n junction
P-n junction
A p–n junction is formed at the boundary between a P-type and N-type semiconductor created in a single crystal of semiconductor by doping, for example by ion implantation, diffusion of dopants, or by epitaxy .If two separate pieces of material were used, this would...

depends on a characteristic voltage called the thermal voltage, denoted VT. The thermal voltage depends on absolute temperature T as
where q is the magnitude of the electrical charge on the electron
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...

with a value {{nowrap|1.602 176 565(35){{e|−19}} C}} and k is the Boltzmann constant in Joules/K. In electronvolt
Electronvolt
In physics, the electron volt is a unit of energy equal to approximately joule . By definition, it is equal to the amount of kinetic energy gained by a single unbound electron when it accelerates through an electric potential difference of one volt...

s, the Boltzmann constant is {{nowrap|8.617 3324(78){{e|−5}} eV/K}}, making it easy to calculate that at room temperature
Room temperature
-Comfort levels:The American Society of Heating, Refrigerating and Air-Conditioning Engineers has listings for suggested temperatures and air flow rates in different types of buildings and different environmental circumstances. For example, a single office in a building has an occupancy ratio per...

(≈ 300 K), the value of the thermal voltage is approximately 25.85 millivolts ≈ 26 mV http://www.google.com/search?hl=en&q=300+kelvin+*+k+%2F+elementary+charge+in+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
Max Planck
Max Karl Ernst Ludwig Planck, ForMemRS, was a German physicist who actualized the quantum physics, initiating a revolution in natural science and philosophy. He is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.-Life and career:Planck came...

first introduced k , and gave an accurate value for it (1.346{{e|−23}} J/K, 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 Boltzmann's constant, but rather using a form of the gas constant
Gas constant
The gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...

R, and macroscopic energies for macroscopic quantities of the substance. The iconic terse form of the equation S = k log W on Boltzmann's tombstone is in fact due to Planck, not Boltzmann. Planck actually introduced it in the same work as his h.

As Planck wrote in his Nobel Prize
Nobel Prize
The Nobel Prizes are annual international awards bestowed by Scandinavian committees in recognition of cultural and scientific advances. The will of the Swedish chemist Alfred Nobel, the inventor of dynamite, established the prizes in 1895...

lecture in 1920,
{{quotation|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" can be understood 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
Heuristic
Heuristic refers to experience-based techniques for problem solving, learning, and discovery. Heuristic methods are used to speed up the process of finding a satisfactory solution, where an exhaustive search is impractical...

, a useful tool for solving problems. Nor was there agreement as to whether "chemical molecules" (as measured by atomic weight
Atomic weight
Atomic weight is a dimensionless physical quantity, the ratio of the average mass of atoms of an element to 1/12 of the mass of an atom of carbon-12...

s) were the same as "physical molecules" (as measured by kinetic theory
Kinetic theory
The kinetic theory of gases describes a gas as a large number of small particles , all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container...

). To continue the quotation from Planck's 1920 lecture:
{{quotation|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.}}

## Value in different units

1.380 6488(13){{e|−23}} J
Joule
The joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...

/K
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...

SI
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...

units, 2010 CODATA value
8.617 3324(78){{e|−5}} eV
Electronvolt
In physics, the electron volt is a unit of energy equal to approximately joule . By definition, it is equal to the amount of kinetic energy gained by a single unbound electron when it accelerates through an electric potential difference of one volt...

/K
2010 CODATA value
electronvolt
Electronvolt
In physics, the electron volt is a unit of energy equal to approximately joule . By definition, it is equal to the amount of kinetic energy gained by a single unbound electron when it accelerates through an electric potential difference of one volt...

= 1.602 176 565(35){{e J
1/kB = 11 604.519(11) K/eV
2.083 6618(19){{e|10}} Hz
Hertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....

/K
2010 CODATA value
1 Hz·h = 6.626 069 57(29){{e|−34}} J
3.166 8114(29){{e|−6}} EH/K EH = 2R
Rydberg constant
The Rydberg constant, symbol R∞, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to atomic spectra in the science of spectroscopy. Rydberg initially determined its value empirically from spectroscopy, but Niels Bohr later showed that its value could be calculated...

hc = 4.359 744 34(19){{e|−18}} J
= 6.579 683 920 729(33) Hz·h
1.380 6488(13){{e|−16}} erg
Erg
An erg is the unit of energy and mechanical work in the centimetre-gram-second system of units, symbol "erg". Its name is derived from the Greek ergon, meaning "work"....

/K
erg
Erg
An erg is the unit of energy and mechanical work in the centimetre-gram-second system of units, symbol "erg". Its name is derived from the Greek ergon, meaning "work"....

= 1{{e|−7}} J
3.297 6230(30){{e|−24}} cal
Calorie
The calorie is a pre-SI metric unit of energy. It was first defined by Nicolas Clément in 1824 as a unit of heat, entering French and English dictionaries between 1841 and 1867. In most fields its use is archaic, having been replaced by the SI unit of energy, the joule...

/K
1 Steam Table calorie
Calorie
The calorie is a pre-SI metric unit of energy. It was first defined by Nicolas Clément in 1824 as a unit of heat, entering French and English dictionaries between 1841 and 1867. In most fields its use is archaic, having been replaced by the SI unit of energy, the joule...

= 4.1868 J
1.832 0128(17){{e|−24}} cal/°R  1 degree Rankine = 5/9 K
5.657 3016(51){{e|−24}} ft lb
Foot-pound force
The foot-pound force, or simply foot-pound is a unit of work or energy in the Engineering and Gravitational Systems in United States customary and Imperial units of measure. It is the energy transferred on applying a force of 1 pound-force through a displacement of 1 foot...

/°R
foot-pound force
Foot-pound force
The foot-pound force, or simply foot-pound is a unit of work or energy in the Engineering and Gravitational Systems in United States customary and Imperial units of measure. It is the energy transferred on applying a force of 1 pound-force through a displacement of 1 foot...

= 1.355 817 948 331 4004 J
0.695 034 76(63) cm−1
Wavenumber
In the physical sciences, the wavenumber is a property of a wave, its spatial frequency, that is proportional to the reciprocal of the wavelength. It is also the magnitude of the wave vector...

/K
2010 CODATA value
1 cm−1 ·hc = 1.986 445 683(87){{e|−23}} J
0.001 987 2041(18) kcal/mol
Mole (unit)
The mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities as there are atoms in 12 grams of pure carbon-12 , the isotope of carbon with atomic weight 12. This corresponds to a value...

/K
per mole form often used in statistical mechanics—using thermochemical cal=4.184 Joules
0.008 314 4621(75) kJ/mol/K per mole form often used in statistical mechanics
4.10 pN·nm kT in piconewton nanometer at 24°C, used in biophysics
−228.5991678(40) dBW/K/Hz in decibel watts, used in telecommunications

Since k is a physical constant
Physical constant
A physical constant is a physical quantity that is generally believed to be both universal in nature and constant in time. It can be contrasted with a mathematical constant, which is a fixed numerical value but does not directly involve any physical measurement.There are many physical constants in...

of proportionality between temperature and energy, its numerical value depends on the choice of units for energy and temperature. The Kelvin temperature scale is based on the Celsius scale which divides the temperature range of liquid water into one hundred increments. The small numerical value of the constant in the metric system reflects the small energy in joules required to increase a particle's energy by raising the temperature by 1 K. The characteristic energy kT is a term encountered in many physical relationships.

### Planck units

The Boltzmann constant provides a mapping from this characteristic microscopic energy E to the macroscopic temperature scale T = E/k. In physics research another definition is often encountered in setting k to unity, resulting in the Planck units or natural units
Natural units
In physics, natural units are physical units of measurement based only on universal physical constants. For example the elementary charge e is a natural unit of electric charge, or the speed of light c is a natural unit of speed...

for temperature and energy. In this context temperature is measured effectively in units of energy and the Boltzmann constant is not explicitly needed.:
This simplifies many physical relationships and makes the definition of thermodynamic entropy coincide with that of information entropy
Information entropy
In information theory, entropy is a measure of the uncertainty associated with a random variable. In this context, the term usually refers to the Shannon entropy, which quantifies the expected value of the information contained in a message, usually in units such as bits...

:

The value chosen for the Planck unit of temperature
Planck temperature
Planck temperature is the greatest physically-possible temperature, according the set of theories proposed by the German physicist Max Planck. It's part of a system of five natural units known as Planck units, based on universal physical constants....

is that corresponding to the energy of the Planck mass or {{nowrap|1.416 833(85){{e}}.