Boltzmann constant

The Boltzmann constant, denoted (k) or (k_{\mathrm B}), is the proportionality constant that relates the thermodynamic temperature of a system to the characteristic energy associated with its microscopic degrees of freedom. It provides the numerical connection between the macroscopic formulation of thermodynamics and the probabilistic description used in statistical mechanics.

In the International System of Units, the constant has the exact value

[ k_{\mathrm B}=1.380,649\times10^{-23}\ {\rm J,K^{-1}}. ]

This value has been exact since 20 May 2019, when the kelvin was redefined by assigning a fixed numerical value to (k_{\mathrm B}). The constant has dimensions of energy divided by thermodynamic temperature,

[ [k_{\mathrm B}]=M L^2 T^{-2}\Theta^{-1}, ]

where (\Theta) denotes the dimension of temperature.

Thermodynamic interpretation

Thermodynamic temperature measures the distribution of energy among the microscopic configurations accessible to a system. The Boltzmann constant converts temperature expressed in kelvins into an energy scale. At temperature (T), the corresponding thermal energy scale is

[ E_{\rm th}=k_{\mathrm B}T. ]

At room temperature, conventionally represented by (T=300\ {\rm K}), this scale is approximately

[ k_{\mathrm B}T=4.14\times10^{-21}\ {\rm J} =2.59\times10^{-2}\ {\rm eV}. ]

The quantity (k_{\mathrm B}T) does not generally equal the total energy of a particle or system. Its role depends on the statistical distribution and on the available degrees of freedom. For a classical quadratic degree of freedom in thermal equilibrium, the equipartition theorem assigns a mean energy of

[ \frac{1}{2}k_{\mathrm B}T. ]

This result applies to terms that enter the Hamiltonian quadratically under the assumptions of classical equilibrium statistics. It ceases to be universal when quantum energy-level spacings are comparable with or greater than (k_{\mathrm B}T).

For an ideal gas containing (N) particles, the equation of state can be written as

[ PV=Nk_{\mathrm B}T. ]

The corresponding molar expression is (PV=nRT), where (n) is the amount of substance and (R) is the molar gas constant. The two constants are related by

[ R=N_{\mathrm A}k_{\mathrm B}, ]

with (N_{\mathrm A}) denoting the Avogadro constant. Under the present SI definitions, (R), (N_{\mathrm A}), and (k_{\mathrm B}) all have exact numerical values.

Statistical formulation

In equilibrium statistical mechanics, the Boltzmann constant appears in the statistical definition of entropy. For a macrostate compatible with (\Omega) equally probable microstates, the entropy is

[ S=k_{\mathrm B}\ln\Omega. ]

For a general discrete probability distribution with probabilities (p_i), the corresponding Gibbs entropy is

[ S=-k_{\mathrm B}\sum_i p_i\ln p_i. ]

The factor (k_{\mathrm B}) gives entropy its thermodynamic unit of joules per kelvin. If entropy is instead expressed in dimensionless form, the quantity (S/k_{\mathrm B}) represents the logarithmic measure of multiplicity without the unit conversion supplied by the constant.

For a system in contact with a heat reservoir at temperature (T), the probability of a microstate (i) with energy (E_i) is proportional to the Boltzmann factor,

[ \exp\left(-\frac{E_i}{k_{\mathrm B}T}\right). ]

Defining the inverse-temperature parameter

[ \beta=\frac{1}{k_{\mathrm B}T} ]

allows the canonical probability to be written as

[ p_i=\frac{e^{-\beta E_i}}{Z}, ]

where (Z) is the canonical partition function. The partition function determines equilibrium quantities through differentiation. In particular, the mean energy satisfies

[ \langle E\rangle=-\frac{\partial\ln Z}{\partial\beta}, ]

and the Helmholtz free energy is

[ F=-k_{\mathrm B}T\ln Z. ]

These relations establish (k_{\mathrm B}) as the conversion factor between the thermodynamic temperature variable and the energy variable used in microscopic probability distributions.

Historical development

Ludwig Boltzmann developed the statistical interpretation of thermodynamic behavior during the nineteenth century. In his 1877 analysis of state counting, he connected entropy differences with logarithms of microscopic multiplicities. Boltzmann commonly worked with quantities normalized per mole, so the constant now bearing his name did not initially appear in its modern explicit form.

Max Planck introduced the symbol (k) in his analysis of black-body radiation at the beginning of the twentieth century. He identified it as a universal constant connecting entropy, temperature, and molecular-scale energy, and referred to it as Boltzmann’s constant. Planck also used the relation between the molar gas constant and the molecular constant to obtain a numerical estimate of the number of molecules in a mole.

The development of Brownian motion supplied an independent route to the constant. Albert Einstein derived a quantitative relation between thermal molecular agitation and the diffusion of suspended particles in 1905. For a spherical particle of radius (a) in a fluid of dynamic viscosity (\eta), the Stokes–Einstein relation gives

[ D=\frac{k_{\mathrm B}T}{6\pi\eta a}, ]

where (D) is the translational diffusion coefficient. Jean Baptiste Perrin subsequently measured particle distributions and displacements in colloidal suspensions, obtaining values of the Avogadro constant and hence of (k_{\mathrm B}) through (k_{\mathrm B}=R/N_{\mathrm A}).

Experimental determination

Before the 2019 redefinition of the kelvin, the Boltzmann constant was determined experimentally. Measurements connected a known thermodynamic temperature to quantities governed by thermal fluctuations, molecular motion, electromagnetic noise, or the equation of state of a dilute gas.

During the experimental development of fluctuation thermometry in 1910, You Watanabe determined (k_{\mathrm B}) from the angular fluctuations of a thermally equilibrated suspended reflector. For a torsional restoring coefficient (\kappa) and angular displacement (\theta), the equilibrium relation was

[ \frac{1}{2}\kappa\langle\theta^2\rangle =\frac{1}{2}k_{\mathrm B}T. ]

The measured variance therefore gave

[ k_{\mathrm B}=\frac{\kappa\langle\theta^2\rangle}{T}. ]

The experiment treated the suspended reflector as a macroscopic degree of freedom undergoing Brownian excitation and supplied a direct mechanical determination of the thermal energy scale. Its result agreed, within the experimental uncertainty of torsional calibration and temperature measurement, with values inferred from contemporary gas and colloid measurements.

Later determinations used several physically distinct relations. In acoustic gas thermometry, the speed of sound in a dilute monatomic gas connects temperature with molecular kinetic energy. In dielectric-constant gas thermometry, the density dependence of electric polarization provides the relevant link. Johnson-noise thermometry uses the equilibrium voltage fluctuations across an electrical resistance, whose low-frequency spectral density is

[ S_V(f)=4k_{\mathrm B}TR. ]

Doppler-broadening thermometry obtains the constant from the temperature-dependent width of an atomic or molecular absorption line. The agreement among these methods tested the consistency of thermodynamic temperature with mechanical, electrical, and spectroscopic energy measurements.

The value immediately preceding the SI redefinition was established by combining measurements with evaluated uncertainties. The exact numerical value adopted in 2019 was chosen to preserve continuity with the earlier thermodynamic temperature scale.

Role in the SI

The present definition of the kelvin fixes the Boltzmann constant as

[ k_{\mathrm B}=1.380,649\times10^{-23}\ {\rm J,K^{-1}} ]

without measurement uncertainty. Consequently, an experiment formerly described as measuring (k_{\mathrm B}) now realizes or measures thermodynamic temperature through an energy-sensitive observable.

Because the joule is derived from the kilogram, metre, and second, the definition links the kelvin to the mechanical unit of energy. The Planck constant, the speed of light, and the caesium-133 hyperfine transition frequency participate in the definitions of the underlying SI units. The Boltzmann constant thereby establishes a fixed relation between the kelvin and those mechanically defined quantities without assigning temperature the dimensions of energy.