Boltzmann constant

For the constant pertaining to energy of black body radiation see Stefan-Boltzmann constant

The Boltzmann constant (k or k_B) is the physical constant relating energy at the particle level with temperature observed at the bulk level. It is the gas constant R divided by the Avogadro constant N_A:

It has the same units as entropy. It is named after the Austrian physicist Ludwig Boltzmann.

Bridge from macroscopic to microscopic physics

Boltzmann's constant k is a bridge between macroscopic and microscopic physics. Macroscopically, the ideal gas law states that, for an ideal gas, the product of pressure p and volume V is proportional to the product of amount of substance n and absolute temperature T:

where R is the gas constant (8.314 472(15) J K⁻¹ mol⁻¹). 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.

Role in the equipartition of energy

Given a thermodynamic system at an absolute temperature T, the thermal energy carried by each microscopic "degree of freedom" in the system is on the order of magnitude of kT/2 (i.e., about 2.07 × 10⁻²¹ J, or 0.013 eV at room temperature).

Application to simple gas thermodynamics

In classical statistical mechanics, this average is predicted to hold exactly for homogeneous ideal gases. Monatomic ideal gases possess three degrees of freedom per atom, corresponding to the three spatial directions, which means a thermal energy of 1.5kT per atom. 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 speed of the atoms, which is inversely proportional to the square root of the atomic mass. The root mean square speeds found at room temperature accurately reflect this, ranging from 1370 m/s for helium, down to 240 m/s for xenon.

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 (approximately) five degrees of freedom per molecule.

Role in Boltzmann factors

More generally, systems in equilibrium with a reservoir of heat at temperature T have probabilities p of occupying states with energy E weighted by the corresponding Boltzmann factor:

Again, it is the energy-like quantity kT which takes central importance.

Consequences of this include (in addition to the results for ideal gases above) the Arrhenius equation in chemical kinetics.

Role in the statistical definition of entropy

Boltzmann's grave in the Zentralfriedhof, Vienna, with bust and entropy formula.

In statistical mechanics, the entropy S of an isolated system at thermodynamic equilibrium is defined as the natural logarithm of W, the number of distinct microscopic states available to the system given the macroscopic constraints (such as a fixed total energy E):

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 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.

The characteristic energy kT is thus the heat required to increase the rescaled entropy by one nat.

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 depends on a characteristic voltage called the thermal voltage, denoted V_T. The thermal voltage depends on absolute temperature T as

where q is the magnitude of the electrical charge on the electron with a value 1.602 176 487(40) × 10⁻¹⁹ C. In electronvolts, the Boltzmann constant is 8.617 343(15) × 10⁻⁵ eV/K, making it easy to calculate that at room temperature (≈ 300 K), the value of the thermal voltage is approximately 25.85 millivolts ≈ 26 mV.

Boltzmann's constant in Planck units

Planck's system of natural units is one system constructed such that the Boltzmann constant is 1. This gives

as the average kinetic energy of a gas molecule per degree of freedom; and makes the definition of thermodynamic entropy coincide with that of information entropy:

The value chosen for the Planck unit of temperature is that corresponding to the energy of the Planck mass—a staggering 1.416 785(71) × 10³² K.

History

Although Boltzmann first linked entropy and probability in 1877, it seems the relation was never expressed with a specific constant until Max Planck first introduced k , and gave an accurate value for it (1.346 × 10⁻²³ J/K, about 2.5% lower than today's figure), in his derivation of the law of black body radiation in 1900–1901.^[2] 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 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.

As Planck wrote in his Nobel Prize lecture in 1920,^[3]

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, a useful tool for solving problems. Nor was there agreement as to whether "chemical molecules" (as measured by atomic weights) were the same as "physical molecules" (as measured by kinetic theory). To continue the quotation from Planck's 1920 lecture:^[3]

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

Since k is a constant of proportionality of temperature and energy, the numerical value of k depends on the choice of units for energy and temperature. The Kelvin temperature scale was chosen to conveniently divide up the liquid range of water into one hundred intervals. The very small numerical value of k merely reflects the small energy in joules required to increase a particle's energy through 1 K. The physically fundamental idea is the characteristic energy kT of a particular temperature.

The numerical value of k provides a mapping from this characteristic microscopic energy E to the macroscopically-derived temperature scale T = E/k. On the other hand, the Planck units of temperature and energy are defined in such a way that k = 1. If we choose to measure temperature in units of energy then Boltzmann's constant would not be needed at all.^[4]

References