Everything about Boltzmann Constant totally explained
The
Boltzmann constant (
k or
kB) is the
physical constant relating
energy at the particle level with
temperature observed at the bulk level. It is the
gas constant divided by the
Avogadro constant:
»
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
is a bridge between
macroscopic and microscopic physics. Macroscopically, the
ideal gas law states that, for an
ideal gas, the product of
pressure and
volume is proportional to the product of
amount of substance (in number of
moles) and
absolute temperature .
»
where
» is called the
gas constant [8.314 472 m
3·Pa·K
−1·mol
−1],
Introducing Boltzmann's constant transforms this into an equation about the
microscopic properties of molecules,
»
where
is the number of molecules of gas, and
is Boltzmann's constant.
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 (for example, about 2.07 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 3 degrees of freedom per atom, corresponding to the three spatial directions, which means a thermal energy of 1.5
kT 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
»
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
1.41679 K.
Historical note
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, in his derivation of the
law of black body radiation in December 1900. 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 1918
Nobel Prize lecture,
» "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. 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."
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Before 1900, equations involving Boltzmann factors were not written using the energies per molecule and Boltzmann's constant, but rather using the
gas constant R, and macroscopic energies for macroscopic quantities of the substance; as for convenience is still generally the case in chemistry to this day.
Value in different units
The digits in parentheses are the
standard measurement uncertainty in the last two digits of the measured value.
k can also be expressed with the unit
mol (such as 1.99 calories/mole-kelvin); for historical reasons it's then called the
gas constant.
The
numerical value of
k has no particular fundamental significance in itself: It merely reflects a preference for measuring temperature in units of familiar
kelvins, based on the macroscopic physical properties of water. What is physically fundamental is the characteristic energy
kT of a particular temperature. The numerical value of
k measures the conversion factor for mapping from this characteristic microscopic energy
E to the macroscopically-derived temperature scale
T = E/k . If, instead of talking of
room temperature as 300 K (27
°C or 80
°F), it were conventional to speak of the corresponding energy
kT of 4.14 J, or 0.0259
eV, then Boltzmann's constant wouldn't be needed.
Further Information
Get more info on 'Boltzmann Constant'.
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