# Boltzmann's constant

Physical constant
Surname Boltzmann's constant
Formula symbol ${\ displaystyle k}$ or ${\ displaystyle k _ {\ mathrm {B}}}$ value
SI 1.380 649e-23 ${\ displaystyle \ textstyle {\ frac {\ mathrm {J}} {\ mathrm {K}}}}$ Uncertainty  (rel.) (exactly)
Planck units 1
Sources and Notes
Source SI value: CODATA 2018 ( direct link )

The Boltzmann constant (symbol or ) is a natural constant that plays a central role in statistical mechanics . It was introduced by Max Planck and named after the Austrian physicist Ludwig Boltzmann , one of the founders of statistical mechanics. ${\ displaystyle k}$ ${\ displaystyle k _ {\ mathrm {B}}}$ ## value

The Boltzmann constant has the dimension energy / temperature .

Their value is:

${\ displaystyle k _ {\ mathrm {B}} = 1 {,} 380 \, 649 \ cdot 10 ^ {- 23} \; \ mathrm {J / K}}$ This value applies exactly because the unit of measurement " Kelvin " has been defined since 2019 by assigning this value to the Boltzmann constant. Previously, the Kelvin was defined differently and was a variable to be determined experimentally. ${\ displaystyle k _ {\ mathrm {B}}}$ If the value of the Boltzmann constant is specified in, the numerical value as the quotient of the two exact numbers 1.380649 · 10 −23 and 1.602176634 · 10 −19 does not have a finite representation of decimal places and must therefore be abbreviated as ...: 8${\ displaystyle \ mathrm {eV} / \ mathrm {K}}$ .617 333 262 ...e-5 ${\ displaystyle \ textstyle {\ frac {\ mathrm {eV}} {\ mathrm {K}}}}$ , but the value is still exact.

The universal gas constant is calculated from the Boltzmann constant :

${\ displaystyle R _ {\ mathrm {m}} = N _ {\ mathrm {A}} \ cdot k _ {\ mathrm {B}}}$ ,

where with the unit of measurement 1 / mol is the Avogadro constant . ${\ displaystyle N _ {\ mathrm {A}}}$ ## Definition and connection with entropy

Specifying Ludwig Boltzmann's ideas, the fundamental relationship found by Max Planck is:

${\ displaystyle S = k _ {\ mathrm {B}} \, \ ln \ Omega \ ,.}$ This means that the entropy of a macrostate of a closed system in thermal equilibrium is proportional to the natural logarithm of the number ( result space ) of the corresponding possible microstates (or, in other words, to the degree of “disorder” of the macrostate). The statistical weight is a measure of the probability of a certain microstate. ${\ displaystyle S}$ ${\ displaystyle \ Omega}$ ${\ displaystyle \ Omega}$ This equation links - using the Boltzmann constant as a proportionality factor - the micro-states of the closed system with the macroscopic size of the entropy and forms the central basis of statistical physics . It is engraved in a slightly different nomenclature on Ludwig Boltzmann's tombstone at Vienna's Central Cemetery .

The change in entropy is defined in classical thermodynamics as

${\ displaystyle \ Delta S = \ int {\ frac {\ mathrm {d} Q} {T}}}$ with the amount of heat . ${\ displaystyle Q}$ An increase in entropy corresponds to a transition to a new macrostate with a larger number of possible microstates. This is always the case in a closed (isolated) system ( Second Law of Thermodynamics ). ${\ displaystyle \ Delta S> 0}$ In relation to the microscopic partition function , entropy can also be defined as the quantity of the dimension number :

{\ displaystyle {\ begin {aligned} S ^ {\, '} & = {\ frac {S} {k _ {\ mathrm {B}}}} = \ ln \ Omega \\\ Rightarrow \ Delta S ^ {\ , '} & = \ int {\ frac {\ mathrm {d} Q} {k _ {\ mathrm {B}} T}}. \ end {aligned}}} In this “natural” form, the entropy corresponds to the definition of entropy in information theory and forms a central measure there. The term represents the energy needed to raise the entropy by one nit . ${\ displaystyle k _ {\ mathrm {B}} T}$ ${\ displaystyle S ^ {\, '}}$ ## Ideal gas law

The Boltzmann constant allows the calculation of the mean thermal energy of a monatomic free particle from the temperature according to

${\ displaystyle E _ {\ mathrm {therm}} = {\ frac {3} {2}} k _ {\ mathrm {B}} \, T,}$ and occurs, for example, in the gas law for ideal gases as one of the possible proportionality constants:

${\ displaystyle pV = N \, k _ {\ mathrm {B}} \, T}$ .

Meaning of the symbols:

• ${\ displaystyle p}$ - pressure
• ${\ displaystyle V}$ - volume
• ${\ displaystyle N}$ - number of particles
• ${\ displaystyle T}$ - Absolute temperature

Based on normal conditions (temperature and pressure ) and using the Loschmidt constant , the gas equation can be reformulated to: ${\ displaystyle T_ {0}}$ ${\ displaystyle p_ {0}}$ ${\ displaystyle N _ {\ mathrm {L}} = {\ tfrac {N} {V_ {0}}}}$ {\ displaystyle {\ begin {aligned} \ Leftrightarrow {\ frac {N} {V}} & = {\ frac {1} {k _ {\ mathrm {B}}}} {\ frac {p} {T}} \\ & = \ left (N _ {\ mathrm {L}} {\ frac {T_ {0}} {p_ {0}}} \ right) {\ frac {p} {T}} \\ & = N_ { \ mathrm {L}} {\ frac {p} {p_ {0}}} {\ frac {T_ {0}} {T}}. \ end {aligned}}} ## Relationship with the kinetic energy

In general, for the mean kinetic energy of a classical point-shaped particle in thermal equilibrium with degrees of freedom , which are included in the Hamilton function as a square ( equipartition theorem ): ${\ displaystyle f}$ ${\ displaystyle \ langle E _ {\ mathrm {kin}} \ rangle = {\ frac {f} {2}} k _ {\ mathrm {B}} \, T.}$ For example, a point particle has three degrees of translational freedom:

${\ displaystyle \ langle E _ {\ mathrm {kin}} \ rangle = {\ frac {3} {2}} k _ {\ mathrm {B}} \, T.}$ Has a diatomic molecule

• without symmetry, three additional degrees of freedom of rotation, for a total of six
• with an axis of symmetry, two additional degrees of freedom of rotation , i.e. a total of five ( no energy can be stored by rotating along the axis of symmetry , since the moment of inertia is comparatively small here).

In addition, at sufficiently high temperatures, the atoms vibrate against each other along the bonds . In the case of individual substances, chemistry also contributes to the heat capacity: For example, water has an extremely high heat capacity because hydrogen bonds are broken when the temperature rises and energy is expended, and when the temperature falls, they are formed with the release of energy.

## Role in statistical physics

More generally, the Boltzmann constant occurs in the thermal probability density of any system of statistical mechanics in thermal equilibrium. This is: ${\ displaystyle \ rho _ {\ mathrm {th}}}$ ${\ displaystyle \ rho _ {\ mathrm {th}} = {\ frac {e ^ {- {\ frac {E} {k _ {\ mathrm {B}} T}}}} {Z}}}$ With

• the Boltzmann factor ${\ displaystyle e ^ {- {\ frac {E} {k _ {\ mathrm {B}} T}}}}$ • the canonical partition function as normalization constant.${\ displaystyle Z}$ ## Example from solid state physics

In semiconductors there is a dependency of the voltage across a pn junction on the temperature, which can be described with the help of the temperature voltage or : ${\ displaystyle \ phi _ {T}}$ ${\ displaystyle U_ {T}}$ ${\ displaystyle \ phi _ {T} = U_ {T} = {\ frac {k _ {\ mathrm {B}} \ cdot T} {e}}}$ It is

• ${\ displaystyle T}$ the absolute temperature in Kelvin
• ${\ displaystyle e}$ the elementary charge .

At room temperature ( T  = 293 K) the value of the temperature voltage is approximately 25 mV.