Boltzmann constant
The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. It is named after the Austrian physicist Ludwig Boltzmann, who made important contributions to the theory of statistical mechanics, in which this constant plays a crucial role. Its experimentally determined value is (in SI units)
The digits in parentheses are the uncertainty (standard deviation) in the last two digits of the measured value. In principle, the Boltzmann constant could be a derived physical constant, as its value is determined by other physical constants and in the definition of unit of absolute temperature. However, calculating the Boltzmann constant from first principles is far too complex to be done with current knowledge. In a system of natural units, the natural unit of temperature would be such a temperature that would normalize the Boltzmann constant to unity.
The universal gas constant R is simply the Boltzmann constant multiplied by Avogadro's number. The gas constant is more useful when calculating numbers of particles in moles.
Given a thermodynamic system at an absolute temperature T, the Boltzmann constant defines an energy E = 0.5 kT that is the mean amount of kinetic energy carried by each microscopic particle in the system per degree of freedom of motion.
For homogeneous ideal gases, monatomic gases have 3 degrees of freedom per particle. An atom in a classical ideal gas has a mean kinetic energy of 1.5 kT (20.7 yJ/K per particle, 12.47 J/K per mole). Hence, comparing different ideal gases, at a given temperature the root mean square speed of the atoms is inversely proportial to the square root of the atomic mass (Larger atoms are slower.); at a given root mean square speed the temperature is proportional to the atomic mass (Larger atoms with the same speed form a gas of higher temperature.).
The atomic masses of ideal gases range from 4 to 131, hence the kinetic energy per kg ranges from 3120 down to 95.2 J/K. At 300K this is from 936 down to 28.6 kJ/kg, so at room temperature the root mean square speed of the atoms ranges from 1370 m/s for helium down to 240 m/s for xenon.
The article gas phase heat capacities shows that the 1.5 kT and 12.47 J/K per mole correspond very well with experimental data. For diatomic gases the case is more complex, but data correspond approximately to 5 degrees of freedom, for large molecules more.
The energy kT per particle per degree of freedom associated with room temperature, 300 K (27 °C, or 80 °F), is 2.07 × 10−21 J (13 meV).
Role in definition of entropy
In statistical mechanics, the entropy S of a system is defined as the natural logarithm of Ω, the number of distinct microscopic states available to the system given the macroscopic constraints (such as a fixed total energy E):
The constant of proportionality k is the Boltzmann constant. This equation, which relates the microscopic details of the system (via Ω) to its macroscopic state (via the entropy S), is the central idea of statistical mechanics.