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Concept# Boltzmann constant

Summary

The Boltzmann constant (kB or k) is the proportionality factor that relates the average relative thermal energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, and in Planck's law of black-body radiation and Boltzmann's entropy formula, and is used in calculating thermal noise in resistors. The Boltzmann constant has dimensions of energy divided by temperature, the same as entropy. It is named after the Austrian scientist Ludwig Boltzmann.
As part of the 2019 redefinition of SI base units, the Boltzmann constant is one of the seven "defining constants" that have been given exact definitions. They are used in various combinations to define the seven SI base units. The Boltzmann constant is defined to be exactly 1.380649e-23J.K-1.
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 molar gas constant (8.31446261815324J⋅K−1⋅mol−1). Introducing the Boltzmann constant as the gas constant per molecule k = R/NA transforms the ideal gas law into an alternative form:
where N is the number of molecules of gas.
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 1/2kT (i.e., about 2.07e−21J, or 0.013eV, at room temperature). This is generally true only for classical systems with a large number of particles, and in which quantum effects are negligible.
In classical statistical mechanics, this average is predicted to hold exactly for homogeneous ideal gases. Monatomic ideal gases (the six noble gases) possess three degrees of freedom per atom, corresponding to the three spatial directions. According to the equipartition of energy this means that there is a thermal energy of 3/2kT per atom. This corresponds very well with experimental data.

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The Rankine scale (ˈræŋkɪn) is an absolute scale of thermodynamic temperature named after the University of Glasgow engineer and physicist Macquorn Rankine, who proposed it in 1859. Similar to the Kelvin scale, which was first proposed in 1848, zero on the Rankine scale is absolute zero, but a temperature difference of one Rankine degree (°R or °Ra) is defined as equal to one Fahrenheit degree, rather than the Celsius degree used on the Kelvin scale. In converting from kelvin to degrees Rankine, 1 K = 9/5 °R or 1 K = 1.

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