Boltzmann constant 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 (energy divided by temperature) 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 (in moles) and absolute temperature
T: where R is the gas constant (8.314 4621(75) 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 (Avogadro's number). 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 thermal energy carried
by each microscopic "degree of freedom" in the system is on the order of magnitude of
kBT/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 1.5kBT per atom (in the general case, DkBT/2, where D is the number
of spatial dimensions). 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 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 (three translational, two rotational, and
one vibrational). 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 thermal energy available. Role in Boltzmann factors
More generally, systems in equilibrium at temperature T have probability p 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 kB 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 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 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
(≈ 300 K), 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 (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. 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 log 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 "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: 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 (98) × 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.