In 1920, Planck wrote in his Nobel Prize lecture: Planck actually introduced it in the same work as his eponymous h. The iconic terse form of the equation S = k ln W on Boltzmann's tombstone is in fact due to Planck, not Boltzmann. 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. Although Boltzmann first linked entropy and probability in 1877, the relation was never expressed with a specific constant until Max Planck first introduced k, and gave a more precise 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. The Boltzmann constant is named after its 19th century Austrian discoverer, Ludwig Boltzmann. the Nernst equation) 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. The thermal voltage is also important in plasmas and electrolyte solutions (e.g. Kinetic theory gives the average pressure p for an ideal gas as 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. 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. This corresponds very well with experimental data. According to the equipartition of energy this means that there is a thermal energy of 3 / 2 kT per atom. Monatomic ideal gases (the six noble gases) possess three degrees of freedom per atom, corresponding to the three spatial directions. In classical statistical mechanics, this average is predicted to hold exactly for homogeneous ideal gases. This is generally true only for classical systems with a large number of particles, and in which quantum effects are negligible. 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 / 2 kT (i.e., about 2.07 ×10 −21 J, or 0.013 eV, at room temperature).
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