Boltzmann Constant

The Boltzmann constant, symbol $k$ or $k_{\mathrm B}$, is a physical constant connecting temperature with the characteristic energy scale of microscopic systems. It also relates entropy to the number of accessible microscopic states. Central to statistical mechanics and thermodynamics, it has the exact value $1.380649\times10^{-23}\ \mathrm{J\,K^{-1}}$ in the International System of Units (SI). Its fixed numerical value defines the kelvin, the SI unit of thermodynamic temperature. (nist.gov)

Value and physical meaning

The defining value is

Its dimensions are energy divided by temperature. Multiplication by absolute temperature gives $k_{\mathrm B}T$, an energy scale that can be compared with molecular excitation energies or other microscopic energy differences. This does not mean that every particle has energy $k_{\mathrm B}T$, or that a system’s total internal energy is necessarily proportional to temperature. Those relationships depend on its available states and physical model. (nist.gov)

For example, direct calculation from the defining value gives $k_{\mathrm B}T=4.141947\times10^{-21}\ \mathrm J$ at $300\ \mathrm K$. This small energy is significant for individual atoms and molecules, whereas macroscopic quantities involve very large numbers of particles. The constant supplies the conversion between temperatures expressed in kelvins and thermal energy scales expressed in joules. (nist.gov)

Ideal gases and equipartition

For an ideal gas, the microscopic form of its equation of state is

where $p$ is pressure, $V$ volume, and $N$ particle number. The equivalent molar expression is $pV=nRT$, with $n$ the amount of substance and $R$ the molar gas constant. Their relationship is $R=N_{\mathrm A}k_{\mathrm B}$, where $N_{\mathrm A}$ is the Avogadro constant. Thus, $R$ describes the molar scale, while $k_{\mathrm B}$ describes the particle scale. (cccbdb.nist.gov)

In classical equilibrium statistical mechanics, the equipartition theorem assigns mean energy $\tfrac12 k_{\mathrm B}T$ to each independent quadratic term in the energy. A monatomic ideal-gas particle has three translational degrees of freedom, giving

A vibrational coordinate can contribute both kinetic and potential quadratic terms. However, equipartition is not universally valid: in quantum mechanics, widely separated energy levels can make excitations negligible when their spacing greatly exceeds $k_{\mathrm B}T$. Such modes are described as thermally “frozen out.” (mit.edu)

Statistical entropy

For a system with $\Omega$ equally probable accessible microstates, the Boltzmann entropy formula is

A microstate specifies microscopic details, whereas a macroscopic state specifies bulk properties such as energy and volume. The logarithm converts multiplicative state counts for independent systems into additive entropy. The constant converts this dimensionless logarithmic quantity into entropy measured in joules per kelvin. (nist.gov)

For a discrete ensemble with microstate probabilities $p_i$, the Gibbs expression is

When every accessible state has probability $1/\Omega$, it reduces to the preceding formula. Its mathematical form resembles information entropy, but physical entropy includes the dimensional factor $k_{\mathrm B}$. If the corresponding information entropy is expressed in bits, the conversion factor is $k_{\mathrm B}\ln2$, rather than $k_{\mathrm B}$ alone. (ocw.mit.edu)

Boltzmann distribution

For a system weakly coupled to a large heat reservoir in thermodynamic equilibrium, with fixed volume and particle number, the probability of a microstate of energy $E_i$ is

This is the Boltzmann distribution; $Z$ is the partition function, which normalizes the probabilities. The dimensionless ratio $E_i/(k_{\mathrm B}T)$ controls the statistical weight. (ocw.mit.edu)

For two individual microstates separated by energy $\Delta E$, their probability ratio is $e^{-\Delta E/(k_{\mathrm B}T)}$. At positive temperatures, higher-energy microstates are therefore less probable individually. Populations of whole energy levels also depend on how many microstates share each energy. This framework explains the temperature-dependent occupation of molecular and quantum energy levels and is used in understanding blackbody radiation. (ocw.mit.edu)

Historical development

The constant is named after Ludwig Boltzmann, whose statistical treatment connected entropy with microscopic probabilities. Max Planck made the constant explicit in his radiation theory around 1900. In his Nobel lecture, Planck distinguished Boltzmann’s underlying statistical ideas from the introduction and numerical evaluation of the constant, noting that Boltzmann himself had not introduced it in that form. The development accompanied Planck’s introduction of the Planck constant and his account of thermal radiation. (nobelprize.org)

Kelvin redefinition and measurement

The revised SI took effect on 20 May 2019. Previously, the kelvin was defined using the triple point of water; the revised definition instead fixes $k_{\mathrm B}$. Consequently, its SI numerical value has no measurement uncertainty, although experimental realizations of temperature still do. The change reversed the metrological relationship: experiments formerly determined the constant using a defined temperature scale; they can now determine temperature using the fixed constant. (nist.gov)

Measurements supporting the redefinition included acoustic gas thermometry, which uses gas sound speeds; dielectric-constant gas thermometry, which measures electrical response; and Johnson-noise thermometry, which measures thermal electrical noise. Their agreement provided independent checks before the value was fixed. The water triple point remains useful for thermometer calibration, but it no longer defines the unit. (nist.gov)