Editing Equipartition theorem (section)

==History==
{{hatnote|This article uses the non-[[International System of Units|SI]] unit of [[calorie|cal]]/([[mole (unit)|mol]]·[[Kelvin|K]]) for heat capacity, because it offers greater accuracy for single digits. For an approximate conversion to the corresponding SI unit of J/(mol·K), such values should be multiplied by 4.2&nbsp;[[Joule|J]]/cal.}}

The equipartition of kinetic energy was proposed initially in 1843, and more correctly in 1845, by [[John James Waterston]].<ref>{{cite book | last = Brush | first = SG | year = 1976 | title = The Kind of Motion We Call Heat, Volume 1 | publisher = North Holland | location = Amsterdam | pages = 134–159 | isbn = 978-0-444-87009-4}}<br />{{cite book | last = Brush | first = SG | year = 1976 | title = The Kind of Motion We Call Heat, Volume 2 | publisher = North Holland | location = Amsterdam | pages = 336–339 | isbn = 978-0-444-87009-4}}<br />{{cite journal | last = Waterston | first = JJ | author-link = John James Waterston | year = 1846 | title = On the physics of media that are composed of free and elastic molecules in a state of motion | journal = Proc. R. Soc. Lond. | volume = 5 | pages = 604 | doi = 10.1098/rspl.1843.0077 | postscript=none| doi-access = free }} (abstract only). Published in full {{cite journal | title= On the Physics of Media that are Composed of Free and Perfectly Elastic Molecules in a State of Motion| journal = [[Philosophical Transactions of the Royal Society]] | year = 1893 | volume = A183 |pages = 1–79 | doi = 10.1098/rsta.1892.0001 |bibcode = 1892RSPTA.183....1W | last1 = Waterston | first1 = J. J. | last2 = Rayleigh | first2 = L. | doi-access = free }} Reprinted {{cite book | editor = J.S. Haldane | title = The collected scientific papers of John James Waterston | url = https://archive.org/details/b29487468 | year = 1928 | location = Edinburgh | publisher = Oliver & Boyd}}<br />{{cite book | last = Waterston | first = JJ | author-link = John James Waterston | year = 1843 | title = Thoughts on the Mental Functions }} (reprinted in his ''Papers'', '''3''', 167, 183.)<br />{{cite journal | last = Waterston | first = JJ | author-link = John James Waterston | year = 1851 | journal = British Association Reports | volume = 21 | pages = 6}}
Waterston's key paper was written and submitted in 1845 to the [[Royal Society]]. After refusing to publish his work, the Society also refused to return his manuscript and stored it among its files. The manuscript was discovered in 1891 by [[John Strutt, 3rd Baron Rayleigh|Lord Rayleigh]], who criticized the original reviewer for failing to recognize the significance of Waterston's work. Waterston managed to publish his ideas in 1851, and therefore has priority over Maxwell for enunciating the first version of the equipartition theorem.</ref> In 1859, [[James Clerk Maxwell]] argued that the kinetic heat energy of a gas is equally divided between linear and rotational energy.<ref>{{cite book | last = Maxwell | first = JC | author-link = James Clerk Maxwell | year = 2003 | chapter = Illustrations of the Dynamical Theory of Gases | title = The Scientific Papers of James Clerk Maxwell | editor = WD Niven | pages = Vol.1, pp. 377–409 | publisher = Dover | location = New York | isbn = 978-0-486-49560-6 | no-pp = true}} Read by Prof. Maxwell at a Meeting of the British Association at Aberdeen on 21 September 1859.</ref> In 1876, [[Ludwig Boltzmann]] expanded on this principle by showing that the average energy was divided equally among all the independent components of motion in a system.<ref>{{cite journal | last = Boltzmann | first = L | author-link = Ludwig Boltzmann | year = 1871 | title = Einige allgemeine Sätze über Wärmegleichgewicht (Some general statements on thermal equilibrium) | journal = Wiener Berichte | volume = 63 | pages = 679–711|language=de}} In this preliminary work, Boltzmann showed that the average total kinetic energy equals the average total potential energy when a system is acted upon by external harmonic forces.</ref><ref>{{cite journal | last = Boltzmann | first = L | author-link = Ludwig Boltzmann | year = 1876 | title = Über die Natur der Gasmoleküle (On the nature of gas molecules) | journal = Wiener Berichte | volume = 74 | pages = 553–560|language=de}}</ref> Boltzmann applied the equipartition theorem to provide a theoretical explanation of the [[Dulong–Petit law]] for the [[specific heat|specific heat capacities]] of solids.

[[Image:DiatomicSpecHeat1.png|thumb|upright=1.5|Figure 4. Idealized plot of the [[specific heat capacity|molar specific heat]] of a diatomic gas against temperature. It agrees with the value (7/2)''R'' predicted by equipartition at high temperatures (where ''R'' is the [[gas constant]]), but decreases to (5/2)''R'' and then {{sfrac|3|2}}''R'' at lower temperatures, as the vibrational and [[rotational modes]] of motion are "frozen out". The failure of the equipartition theorem led to a paradox that was only resolved by [[quantum mechanics]]. For most molecules, the transitional temperature T<sub>rot</sub> is much less than room temperature, whereas ''T''<sub>vib</sub> can be ten times larger or more. A typical example is [[carbon monoxide]], CO, for which ''T''<sub>rot</sub> ≈ 2.8&nbsp;[[Kelvin|K]] and ''T''<sub>vib</sub>&nbsp;≈ 3103&nbsp;[[Kelvin|K]]. For molecules with very large or weakly bound atoms, ''T''<sub>vib</sub> can be close to room temperature (about 300&nbsp;K); for example, ''T''<sub>vib</sub>&nbsp;≈ 308&nbsp;K for [[iodine]] gas, I<sub>2</sub>.<ref name="mcquarrie_2000c" />]]

The history of the equipartition theorem is intertwined with that of [[specific heat capacity]], both of which were studied in the 19th century. In 1819, the French physicists [[Pierre Louis Dulong]] and [[Alexis Thérèse Petit]] discovered that the specific heat capacities of solid elements at room temperature were inversely proportional to the atomic weight of the element.<ref>{{cite journal | last = Petit | first = AT | author-link = Alexis Thérèse Petit |author2=Dulong PL |author-link2=Pierre Louis Dulong  | year = 1819 | title = Recherches sur quelques points importants de la théorie de la chaleur (Studies on key points in the theory of heat) | url = http://web.lemoyne.edu/~giunta/PETIT.html | journal = [[Annales de Chimie et de Physique]] | volume = 10 | pages = 395–413|language=fr}}</ref> Their law was used for many years as a technique for measuring atomic weights.<ref name="pais_1982">{{cite book | last = Pais | first = A | author-link = Abraham Pais | year = 1982 | title = Subtle is the Lord | publisher = Oxford University Press | isbn = 0-19-853907-X | url = https://archive.org/details/subtleislordscie00pais }}</ref> However, subsequent studies by [[James Dewar]] and [[Heinrich Friedrich Weber]] showed that this [[Dulong–Petit law]] holds only at high [[temperature]]s;<ref>{{cite journal | last = Dewar | first = J | author-link = James Dewar | year = 1872 | title = The Specific Heat of Carbon at High Temperatures | journal = [[Philosophical Magazine]] | volume = 44 | pages = 461}}<br />{{cite journal | last = Weber | first = HF | author-link = Heinrich Friedrich Weber | year = 1872 | title = Die specifische Wärme des Kohlenstoffs (The specific heat of carbon) | journal = [[Annalen der Physik]] | volume = 147 | issue = 10 | pages = 311–319 | url = http://gallica.bnf.fr/ark:/12148/bpt6k152316|language=de|doi=10.1002/andp.18722231007|bibcode = 1872AnP...223..311W }}<br />{{cite journal | last = Weber | first = HF | author-link = Heinrich Friedrich Weber | year = 1875 | title = Die specifische Wärmen der Elemente Kohlenstoff, Bor und Silicium (The specific heats of elemental carbon, boron, and silicon) | journal = [[Annalen der Physik]] | volume = 154 | issue = 3 | pages = 367–423, 553–582 |doi=10.1002/andp.18752300307 | url = http://gallica.bnf.fr/ark:/12148/bpt6k15238m|language=de|bibcode = 1875AnP...230..367W }}</ref> at lower temperatures, or for exceptionally hard solids such as [[diamond]], the specific heat capacity was lower.<ref>{{cite journal | last = de la Rive | first = A |author2=Marcet F | year = 1840 | title = Quelques recherches sur la chaleur spécifique (Some research on specific heat) | journal = Annales de Chimie et de Physique | volume = 75 | pages = 113–144 | url=https://books.google.com/books?id=vBwAAAAAMAAJ&pg=RA1-PA3 | publisher = Masson.|language=fr}}<br />{{cite journal | last = Regnault | first = HV | author-link = Henri Victor Regnault | year = 1841 | title = Recherches sur la chaleur spécifique des corps simples et des corps composés (deuxième Mémoire) (Studies of the specific heats of simple and composite bodies) | journal = [[Annales de Chimie et de Physique]] | volume = 1 | series = (3me Série)| pages = 129–207 | url = http://gallica.bnf.fr/ark:/12148/bpt6k34742d|language=fr}} Read at l'Académie des Sciences on 11 January 1841.<br />{{cite journal | last = Wigand | first = A | year = 1907 | title = Über Temperaturabhängigkeit der spezifischen Wärme fester Elemente (On the temperature dependence of the specific heats of solids) | journal = [[Annalen der Physik]] | volume = 22 | issue = 1 | pages = 99–106|language=de|doi=10.1002/andp.19063270105 |bibcode = 1906AnP...327...99W | url = https://zenodo.org/record/1424079 }}</ref>

Experimental observations of the specific heat capacities of gases also raised concerns about the validity of the equipartition theorem. The theorem predicts that the molar heat capacity of simple monatomic gases should be roughly 3&nbsp;cal/(mol·K), whereas that of diatomic gases should be roughly 7&nbsp;cal/(mol·K). Experiments confirmed the former prediction,<ref name="kundt_1876">{{cite journal | last = Kundt | first = A | author-link = August Kundt |author2=Warburg E |author-link2=Emil Warburg  | year = 1876 | title = Über die specifische Wärme des Quecksilbergases (On the specific heat of mercury gases) | journal = [[Annalen der Physik]] | volume = 157 | issue = 3 | pages = 353–369 | url = http://gallica.bnf.fr/ark:/12148/bpt6k15241h|language=de|doi=10.1002/andp.18762330302 |bibcode = 1876AnP...233..353K }}</ref> but found that molar heat capacities of diatomic gases were typically about 5&nbsp;cal/(mol·K),<ref name="Wueller_1896">{{cite book | last = Wüller | first = A | year = 1896 | title = Lehrbuch der Experimentalphysik (Textbook of Experimental Physics) | publisher = Teubner | location = Leipzig | pages = Vol. 2, 507ff | no-pp = true|language=de}}</ref> and fell to about 3&nbsp;cal/(mol·K) at very low temperatures.<ref name="Eucken_1912">{{cite journal | last = Eucken | first = A | year = 1912 | title = Die Molekularwärme des Wasserstoffs bei tiefen Temperaturen (The molecular specific heat of hydrogen at low temperatures) | journal = Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften | volume = 1912 | pages = 141–151|language=de}}</ref> [[James Clerk Maxwell|Maxwell]] noted in 1875 that the disagreement between experiment and the equipartition theorem was much worse than even these numbers suggest;<ref name="maxwell_1875">{{cite book | last = Maxwell | first = JC | author-link = James Clerk Maxwell | year = 1890 | chapter = On the Dynamical Evidence of the Molecular Constitution of Bodies | title = The Scientific Papers of James Clerk Maxwell | url = https://archive.org/details/scientificpapers02maxwuoft | editor = WD Niven | pages = Vol.2, pp.418–438 | publisher = At the University Press | location = Cambridge | id = ASIN B000GW7DXY | no-pp = true | isbn = 0-486-61534-0}} A lecture delivered by Prof. Maxwell at the Chemical Society on 18 February 1875.</ref> since atoms have internal parts, heat energy should go into the motion of these internal parts, making the predicted specific heats of monatomic and diatomic gases much higher than 3&nbsp;cal/(mol·K) and 7&nbsp;cal/(mol·K), respectively.

A third discrepancy concerned the specific heat of metals.<ref name="kittel_1996">{{cite book | last = Kittel | first = C | year = 1996 | title = [[Introduction to Solid State Physics]] | publisher = John Wiley and Sons | location = New York | isbn = 978-0-471-11181-8 | pages = 151–156}}</ref> According to the classical [[Drude model]], metallic electrons act as a nearly ideal gas, and so they should contribute {{math|{{sfrac|3|2}} ''N''<sub>e</sub>''k''<sub>B</sub>}} to the heat capacity by the equipartition theorem, where ''N''<sub>e</sub> is the number of electrons. Experimentally, however, electrons contribute little to the heat capacity: the molar heat capacities of many conductors and insulators are nearly the same.<ref name="kittel_1996" />

Several explanations of equipartition's failure to account for molar heat capacities were proposed. [[Ludwig Boltzmann|Boltzmann]] defended the derivation of his equipartition theorem as correct, but suggested that gases might not be in [[thermal equilibrium]] because of their interactions with the [[luminiferous aether|aether]].<ref>{{cite journal | last = Boltzmann | first = L | author-link = Ludwig Boltzmann | year = 1895 | title = On certain Questions of the Theory of Gases | journal = Nature | volume = 51 | issue = 1322 | pages = 413–415 | doi = 10.1038/051413b0|bibcode = 1895Natur..51..413B | s2cid = 4037658 | url = https://zenodo.org/record/1429366 }}</ref> [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] suggested that the derivation of the equipartition theorem must be incorrect, since it disagreed with experiment, but was unable to show how.<ref>{{cite book | last = Thomson | first = W | author-link = William Thomson, 1st Baron Kelvin | year = 1904 | title = Baltimore Lectures | publisher = Johns Hopkins University Press | location = Baltimore | pages = Sec. 27 | no-pp = true | isbn = 0-8391-1022-7 | url = https://archive.org/details/macromoleculesbe0000unse }} Re-issued in 1987 by MIT Press as ''Kelvin's Baltimore Lectures and Modern Theoretical Physics: Historical and Philosophical Perspectives'' (Robert Kargon and Peter Achinstein, editors). {{ISBN|978-0-262-11117-1}}</ref> In 1900 [[John Strutt, 3rd Baron Rayleigh|Lord Rayleigh]] instead put forward a more radical view that the equipartition theorem and the experimental assumption of thermal equilibrium were ''both'' correct; to reconcile them, he noted the need for a new principle that would provide an "escape from the destructive simplicity" of the equipartition theorem.<ref>{{cite journal | last = Rayleigh | first = JWS | author-link = John Strutt, 3rd Baron Rayleigh | year = 1900 | title = The Law of Partition of Kinetic Energy | journal = [[Philosophical Magazine]] | volume = 49 | issue = 296 | pages = 98–118 | doi=10.1080/14786440009463826| url = https://zenodo.org/record/1430610 }}</ref> [[Albert Einstein]] provided that escape, by showing in 1906 that these anomalies in the specific heat were due to quantum effects, specifically the quantization of energy in the elastic modes of the solid.<ref>{{cite journal | last = Einstein | first = A | author-link = Albert Einstein | year = 1906 | title = Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme (The Planck theory of radiation and the theory of specific heat) | journal = Annalen der Physik | volume = 22 | issue = 1 | pages = 180–190|bibcode = 1906AnP...327..180E|doi = 10.1002/andp.19063270110 | url = https://zenodo.org/record/1424081 |language=de}}<br />{{cite journal | last = Einstein | first = A | author-link = Albert Einstein | year = 1907 | title = Berichtigung zu meiner Arbeit: 'Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme' (Correction to previous article) | journal = Annalen der Physik | volume = 22 | issue = 4 | pages = 800 | doi = 10.1002/andp.19073270415|bibcode = 1907AnP...327..800E | s2cid = 122548821 |language=de| url = https://zenodo.org/record/1424089 }}<br />{{cite journal | last = Einstein | first = A | author-link = Albert Einstein | year = 1911 | title = Eine Beziehung zwischen dem elastischen Verhalten and der spezifischen Wärme bei festen Körpern mit einatomigem Molekül (A connection between the elastic behavior and the specific heat of solids with single-atom molecules) | journal = Annalen der Physik | volume = 34 | issue = 1 | pages = 170–174 | url = http://gallica.bnf.fr/ark:/12148/bpt6k15337j | doi = 10.1002/andp.19113390110|bibcode = 1911AnP...339..170E | s2cid = 122512507 |language=de}}<br />{{cite journal | last = Einstein | first = A | author-link = Albert Einstein | year = 1911 | title = Bemerkung zu meiner Arbeit: 'Eine Beziehung zwischen dem elastischen Verhalten and der spezifischen Wärme bei festen Körpern mit einatomigem Molekül' (Comment on previous article) | journal = Annalen der Physik | volume = 34 | issue = 3 | pages = 590 | url = http://gallica.bnf.fr/ark:/12148/bpt6k15337j | doi = 10.1002/andp.19113390312|bibcode = 1911AnP...339..590E |language=de}}<br />{{cite journal | last = Einstein | first = A | author-link = Albert Einstein | year = 1911 | title = Elementare Betrachtungen über die thermische Molekularbewegung in festen Körpern (Elementary observations on the thermal movements of molecules in solids) | journal = Annalen der Physik | volume = 35 | pages = 679–694 | url = http://gallica.bnf.fr/ark:/12148/bpt6k15338w|bibcode = 1911AnP...340..679E|doi = 10.1002/andp.19113400903 | issue = 9 |language=de}}</ref> Einstein used the failure of equipartition to argue for the need of a new quantum theory of matter.<ref name="pais_1982" /> [[Walther Nernst|Nernst's]] 1910 measurements of specific heats at low temperatures<ref>{{cite journal | last = Nernst | first = W | author-link = Walther Nernst | year = 1910 | title = Untersuchungen über die spezifische Wärme bei tiefen Temperaturen. II. (Investigations into the specific heat at low temperatures) | journal = Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften | volume = 1910 | pages = 262–282|language=de}}</ref> supported Einstein's theory, and led to the widespread acceptance of [[Quantum mechanics|quantum theory]] among physicists.<ref>{{cite book | last = Hermann | first = Armin | year = 1971 | title = The Genesis of Quantum Theory (1899–1913) | edition = original title: ''Frühgeschichte der Quantentheorie (1899–1913)'', translated by Claude W. Nash | publisher = The MIT Press | location = Cambridge, MA | isbn = 0-262-08047-8 | pages = [https://archive.org/details/genesisofquantum00herm/page/124 124–145] | lccn = 73151106 | url = https://archive.org/details/genesisofquantum00herm/page/124 }}</ref>