The Boltzmann constant, named after Ludwig Boltzmann, is a physical constant relating
energy at the individual particle level with temperature. It is the gas constant R divided
by the Avogadro constant NA:
It has the same dimension as entropy. The accepted value in SI units is 1.3806488(13)×10−23 J/K.
Bridge from macroscopic to microscopic physics Boltzmann's constant, kB, 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 J K−1 mol−1). Introducing the Boltzmann constant transforms
the ideal gas law into an alternative form:
where N is the number of molecules of gas. For n = 1 mol, N is equal to the number of
particles in one mole. The left-hand side of the equation is a macroscopic
amount of pressure-volume energy representing the state of the bulk gas. The right-hand
side divides this energy into N units, one for each gas particle, each of which has an
average kinetic energy equal to kBT. Role in the equipartition of energy
Given a thermodynamic system at an absolute temperature T, the average thermal energy
carried by each microscopic "degree of freedom" in the system is on the order of magnitude
of kBT/2. 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 1.5kBT per atom. This
corresponds very well with experimental data. The thermal energy can be used to calculate
the root-mean-square speed of the atoms, which turns out to be 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
Combination with the ideal gas law
shows that the average translational kinetic energy is
Considering that the translational motion velocity vector v has three degrees of freedom
gives the average energy per degree of freedom equal to one third of that, i.e. kBT / 2.
The ideal gas equation is also obeyed closely by molecular gases; but the form for the heat
capacity is more complicated, because the molecules possess additional 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 six degrees of simple freedom per
molecule that are related to atomic motion. 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 relevant thermal energy per molecule.
Role in Boltzmann factors More generally, systems in equilibrium at
temperature T have probability Pi of occupying a state i with energy E weighted by the corresponding
Boltzmann factor:
where Z is the partition function. Again, it is the energy-like quantity kBT which takes
central importance. Consequences of this include the Arrhenius
equation in chemical kinetics. Role in the statistical definition of entropy
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:
This equation, which relates the microscopic details, or microstates, of the system to
its macroscopic state, is the central idea of statistical mechanics. Such is its importance
that it is inscribed on Boltzmann's tombstone. The constant of proportionality kB 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. The characteristic energy kBT 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 VT. The thermal voltage depends on absolute temperature T as
where q is the magnitude of the electrical charge on the electron with a value 1.602 176 565(35)×10−19 C
and k is the Boltzmann's constant, 1.3806488(13) × 10−23 J/K. In electronvolts, the Boltzmann
constant is 8.617 3324(78)×10−5 eV/K, making it easy to calculate that at room temperature,
the value of the thermal voltage is approximately 25.85 millivolts ≈ 26 mV. 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 first introduced kB, and
gave an accurate value for it, 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 the Boltzmann 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 = kB ln 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 lecture in 1920,
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" were the same as "physical molecules". To continue the quotation from Planck's 1920
lecture:
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.
In 2013 the UK National Physical Laboratory used microwave and acoustic resonance measurements
to determine the speed of sound of a monatomic gas in a triaxial ellipsoid chamber to determine
a more accurate value for the constant as a part of the revision of the International
System of Units. The new value was calculated as 1.380 651 56 × 10−23 J K−1 and is
expected to be accepted by the International System of Units following a review.
Value in different units
Since kB is a physical constant of proportionality between temperature and energy, its numerical
value depends on the choice of units for energy and temperature. The small numerical value
of the Boltzmann constant in SI units reflects the small energy in Joule required to increase
a particle's energy by raising the temperature by 1 K. 1 °C is defined to be the same
as 1 K. The characteristic energy kB 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/kB. In physics research another definition is often encountered in setting kB to unity,
resulting in the Planck units or natural units 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:
where Pi is the probability of each microstate. The value chosen for a unit of the Planck
temperature is that corresponding to the energy of the Planck mass or 1.416 833(85)×1032 K.
References
External links Draft Chapter 2 for SI Brochure, following
redefinitions of the base units Big Step Towards Redefining the Kelvin: Scientists
Find New Way to Determine Boltzmann Constant