arXiv:1807.02270v1 [physics.hist-ph] 6 Jul 2018 ˆ m aai´ orl chaleur”. la mˆeme capacit´e pour appeared Dulong Pierre–Louis the 1819 before April 19 On Introduction 1 science. give modern framework of to the development in and the deserve per- of they scientists work place great the the back two of them attempt these view an by balanced is more formed work a This restore to raised. been and have fabrication’ ‘data fraud, criticisms of Petit bitter and however, Dulong was Recently, charging theory infancy. atomic its Dalton’s Lavoisier’s in and using caloric of explained in thermal concept still announced, bear was were we law phenomena if this when surprising that, more mind even con- impressive are quantum These sequences the solids. the of to way theory mechanical the accounted paved temperature, Einstein, low by at for failure its Even im- mechan- ics. an statistical gave Boltzmann’s and to table support portant periodic the devising thermo- powerful for in a tool with results Mendeleev general provided first has the dynamics, of one heats specific solids, the of for law limiting Dulong-Petit The Abstract l ipebde aeeatytesm capacity same the of exactly atoms heat” have for “the bodies that words, simple own all had their Petit in and in was, Dulong step found What fundamental a [1]. as physics stand statistical to going was Alexis-Th´er`esewhich with prepared jointly Petit, paper, a utfiaino hti o nw steDulong– the as the known fact, now of is was what matter heat) of a justification specific As molar constant. the approximately (hence their times mass elements chemical atomic 13 of heat specific the 1 Lsaoe etu e op ipe n xceetla exactement ont simples corps les tous de atomes “Les 1 r oeseicly httepoutof product the that specifically, more or, cde i e Sciences Acad´emie des h tag aeo r ei n r Dulong Mr. and Petit Dr. of case strange The oienc iMln,Paz enrod ic,3 03 M 20133 - 32 Vinci, da Leonardo Piazza Milano, di Politecnico iatmnod hmc,Mtraie nenraChimica Ingegneria ed Materiali Chimica, di Dipartimento nPrst read to Paris in oet Piazza Roberto 1 it,ee fbaigte ffadad“aafab- “data and fraud of of them stage blaming scien- the of two even to the tists, of progressed integrity and scientific recently life the 8] questioning their 7, of [6, studies work authoritative com- in up even line accounts and contemporary not in concerns does Elusive that some- pletely. something is there odd, finding thing revolutionary their support truly are 27), . just . yet. Petit And and impressive. 34, scien- was young two (Dulong these tists by the in discovery paved final see, and the shall to methods we way as experimental that, interpretation the advance- data the in Indeed, both ments inaccurate. rather Black, still Joseph by were chemist heat Scottish latent (French–born) of late discovery the bod- the the in from of developed century slowly heat” XVIII had for that concept “capacity quan- a the Besides, ies, of 5]. [4, values atomic before titative of years table six first only a weights published Ja- had and Berzelius earlier, cob decades two than less his theory theory introduced atomic had caloric Dalton its the Laplace, and on in Lavoisier France of still ob- in thermodynamics, was based time was it infancy, that which At in context histori- tained. geographical the and within framed cal when remarkable “second more a law quantum DP a the to [3]. granted wind” close has heat point anomalous specific the critical the of study of the behavior today, Even of [2]. capacity heat solids the of a theory develop vibrational to consistent Debye and physics matter concepts condensed quantum in introduce to giving thermodynamics, Einstein of besides motivated Law temperatures, Third Nernst’s low to at support failure law crushing DP the the of achievements. of evidence great hand, other Boltzmann’s the of On one was specific (vibrational) heat the for law limiting (DP) Petit e,i h aapeetdb ei n uogto Dulong and Petit by presented data the in Yet, even is Petit and Dulong of accomplishment The ilano rication” [9]. To be true, these harsh remarks seem to give you a couple of hints, you should find out to have been confined to a rather restricted audi- that: ence2. So, I regarded as useful to bring it to the 1. Even the most trivial check, namely that Col- attention of physicists, perhaps from the slightly umn 1 × Column 2 = Column 3 fails in one different perspective that I may have of the devel- case; opment of scientific ideas. 2. Much worse, even if the value of the atomic weight of at least two of the elements is largely wrong, the corresponding molar specific heats in Column 3 are quite close to modern values. Which means of course that there are com- pensating errors in the experimental values of Column 1. There are also some additional puzzling values that, with a deeper investigation, you may have discover and that we shall discuss later. In any case, if you have done the exercise, you may understand why it did not take me long before falling in a state of deep consternation, concluding that either a) there was something fishy, or b) my neurons have already sub- Figure 1: Data used by Petit and Dulong to derive limated more than I am aware of. Although the sec- their law (for convenience of the English readers, I ond possibility could not definitely be excluded, I plot the version published in the contemporary trans- went into a rather frustrating bibliographical search lation of Ref [1] that appeared in the Annals of Philos- that lead me to discover just a single paper written ophy [10]). in 2002 by Carmen Giunta, a professor in chem- istry at Le Moyne College in Syracuse, with the Before venturing into this enterprise, rather de- rather severe and accusatory title “Dulong and Pe- manding for a physicist on the job who does not tit: a case of data fabrication?” [9]. Although, as pretend at all to be a professional in historical is- you will see, I do not share Giunta’s bitter criti- sues, let me however tell you how I became aware cism, I think that this work has not attracted the of this strange story. While teaching statistical attention it deserves3. To avoid jumping to rushed physics in my university I have always found ex- conclusions, I had to embark upon a rather long in- tremely helpful to complement my technical pre- vestigation that arguably provided me with a more sentation of a subject with some historical remarks balanced perspective of this story, and which I hope that students generally appreciate. The very early you may wish to follow. Let us first introduce the stage at which the DP law was obtained has always two main characters we shall deal with. intrigued me, hence I decided, while writing the En- glish version of my notes [11], to investigate a bit more the matter by reading the original paper by 2 A hapless physician and a Petit and Dulong. As an experimentalist, the first thing I did was of course checking the data that boy wonder they presented in the table reproduced in Fig. (1), Life was not kind to Dulong, at least at the be- comparing at the same time their values for atomic 4 ginning. Born in Rouen on February 13, 1785 , or- weights and molar specific heats (at constant pres- phaned at the age of four, he was raised by his aunt sure) with modern ones. As an appetizer, my read- ers may want to to repeat the same exercise. Just 3Giunta’s paper was actually presented at the 221st Na- tional ACS Meeting with the (not much lighter) title “Du- 2So far, I have not found any colleagues aware of this long and Petit: a Case of Scientific Misconduct?”. diatribe, with the exception of Albert Philipse, a valuable 4Most biographical accounts, and Wikipedia too, state Dutch chemist at the van ‘t Hoff Laboratory in Utrecht, with that Dulong was born on February 13, but this is a long- whom I share the curiosity for the history of science. standing historical mistake. See the results of the detailed

2 and godmother Mme Faurax in Auxerre, where she Laplace included among its members great physi- took care of his education “with all the tenderness cists such as Biot, Gay–Lussac, Arago and Pois- of a mother” [13]. Mostly by dint of his own efforts, son, was at that time a stronghold of Lavoisier’s he prepared himself for the Ecole´ Polytechnique at caloric theory, where the “vibrational” (kinetic) Paris and matriculated at sixteen, the minimum theory of heat was strongly opposed and Dalton’s entrance age. Yet, his studies at the Ecole´ were atomic theory was still far from being generally ac- plagued by sick leaves that prevented his admis- cepted [8, 4]: The discovery by Dulong and Petit sion to the artillery corps, so that he was eventually was going to change this mood drastically. In Ar- forced to abandon the Ecole´ without completing cueil Dulong performed his first important chemi- the course. Dulong’s impaired physical condition cal study in which he extended Berthollet’s studies turned his attention to medicine, which in those of salt decomposition to show that also insoluble days did not require lengthy or deep studies5. Yet, salts are capable of exchanging constituents with this did not turn out to be a good choice, at least soluble electrolytes [15], a study that fostered the financially. He started practicing medicine in one of development of the law of mass action [6, 16]. Du- the poorest neighborhoods of Paris 12th arrondisse- long began to gain a reputation as a brilliant and ment, where, according to Arago [14], extremely careful experimentalist swiftly. However, life was not tired to ambush him yet. In October The clientele was increasing visibly, but 1811 the synthesis of NCl3, which turned out to his fortune diminished with the same ra- be one the most violently explosive substance ever pidity, for Dulong never saw an unfortu- discovered, costed him two fingers, so that he man- nate man without succouring him; because aged to resume his work only after several months7. he had felt obliged to have an account open Anyway, his academic career was settled: in 1820 at the pharmacist, at the benefit of the pa- he became professor of chemistry at Sorbonne, and tients who, without this, could not have ´ 6 then professor of physics at Ecole Polytechnique re- made use of his prescriptions. placing exactly Petit after his untimely death (see below). He concluded his life as Director of Studies Obviously Dulong could not endure this state of of the Ecole,´ respected if not exactly popular as a affairs for a long time and after some years he 8 professor , dying of cancer on July 19, 1838. was forced to give up with his career as a physi- Biographical accounts of the life of Petit are cian. Luckily, the passion for chemistry that he much poorer, and substantially based on the fu- had steadfastly cultivated during his medical stud- neral speech delivered by Biot [17]. For sure we ies draw the attention of Louis-Jacques Th´enard, can say that, compared to Dulong’s life, it was who took him as r´ep´etiteur of his course in Ecole´ almost all the way around. Born in Vesoul (not Polytechnique. The real turning point in Dulong’s far from Besan¸con) on October 2, 1791, he soon life took place when Th´enard introduced him to proved to be a child prodigy by completing all en- Claude–Louis Berthollet, who recognized at once trance requirements of the Ecole´ Polytechnique be- Dulong’s talent and invited him to become a mem- fore turning 11. Even for a boy wonder this was ber of the Societ´ede l’Arcueil, a kind of “country club” of French scientists that he had created to- 7 Nitrogen trichloride has been a bane for other scientists gether with Laplace. For our purposes, it is worth too. Intrigued by Dulong’s discovery, Humphrey Davy re- noticing that the Society of Arcueil, which besides peated the synthesis: this led to another explosion the that lodged a piece of glass in his cornea. In a pitiful way, this was investigation performed in 1843 by a commission of the beneficial to the development of science, since it is because Royal Academy of Rouen [12], where his original act de of his temporary blindness that Davy was forced to hire a bactˆeme is presented. young assistant: Michael Faraday. But Faraday was not ex- 5 Thus Dulong eventually became a medical doctor, but empt from the curse of NCl3 either: An explosion that took he never finished a doctorate. place while he was holding a test-tube containing grains of 6“La client`ele s’augmentait `avue d’oeil, mais la fortune nitrogen trichloride tore away his nails and burnt his fingers diminuait avec la mˆeme rapidit´e, car Dulong ne vit jamais un so much that he was unable to use them like before for a malheureux sans le secourir; car il s’´etait cru oblig´ed’avoir long time. un compte ouvert chez le pharmacien, au profit des malades 8His lectures do not seem to have been exactly exciting, qui, sans cela, n’auraient pas pu faire usage de ses prescrip- for he despised display and regarded any extra word as a tions”. word wasted [6].

3 too early to be admitted to the most prestigious time, he was no longer able to teach. In June 1820, school in France, so he was placed in a prepara- when he was not yet thirty, Petit died from tu- tory school to fill in the time with math and lit- berculosis, the same disease that carried away his erature before reaching the minimal admission age beloved spouse [17]. It is at least comforting to ob- of 16. Petit finished his two–year course at the serve that the great result ha obtained with Dulong Ecole´ with unheard-of distinction, graduating hors brought him undying fame. de ligne (namely, he outranked completely all of his It may useful adding to this biographical facts classmates) in 1809, while it took him just another some ideas about the very different characters and two years to get his doctorate. Leafing through his way of working of Dulong and Petit, which I found thesis, which concerns and extends Laplace’s theory in a more informal account written in 1855 by Jules of capillarity [18], one cannot avoid being deeply Jamin [13], at that time professor of physics in the impressed by the mathematical skill, the clarity, the Ecole´ 12. According to Jamin: elegance of presentation displayed by a 20 years old lad9. Such a remarkable talent allowed Petit to be Petit had a lively intelligence, an elegant nominated adjoint professor at the Ecole´ Polytech- and easy speech, he seduced with an ami- nique when he was just 23 and and was advanced able look, got easily attached, and surren- to the full professorship of physics (professor titu- dered himself to his tendencies rather than laire) the following year (1815). A few months ear- governing them. He was credited with an lier he had become brother in law of Arago (their instinctive scientific intuition, a power of spouses were sisters), with whom he developed a premature invention, certain presages of close and friendly relationship10 and wrote his first an assured future that everyone foresaw important paper about the refractive power of sub- and even desired, so great was the benev- stances in different states of aggregation [20]. In a olence which he inspired. Dulong was few years, Petit was already regarded as one of the the opposite: His language was thought- most brilliant promises of French science, but un- ful, his attitude serious and his appear- happily his potential was never fully realized. The ance cold[. . . ] He worked slowly but with sudden death of his wife in 1817, just six months certainty, with a continuity and a power of after their marriage, drove Petit to a deep state of will that nothing stopped, I should say with depression in which he showed exhibited alarming a courage that no danger could push back. symptoms of premature senescence11. In a short In the absence of that vivacity of the mind which invents easily, but likes to rest, he 9 Among the other results, Petit derives an expression for had the sense of scientific exactness, the the contact angle of a liquid on the wall of a capillary in terms of the ratio between the adhesion force between the gusto for precision experiments, the talent wall and the liquid and the internal cohesion of the liquid of combining them, the patience of com- itself, providing also the condition for complete wetting. Cu- pleting them, and the art, unknown before riously, he was apparently not aware of the basic equation him, to carry them to the limits of ac- already derived by Young in 1805. 10Arago and Petit also shared a common view about the curacy[. . . ] Petit had more mathematical way mathematics should be taught to students in engineer- tendency, Dulong was more experimental; ing. In fact, they jointly filed a complaint about Cauchy, the first carried in the work more brilliant expressing concerns towards his insistence on teaching sub- easiness, the second more continuity; One jects “which mainly had to do with series and which the students would never have occasion to use in the services”, represented imagination, the other reason, 13 at the detriment of applied calculus. Because of this criti- which moderates and contains it. cism, the great mathematician eventually received a severe reprimand from the Director of the Ecole.´ [19]. gether with the fatigue from my own work, have profoundly 11About the state of health of Petit in his last days, Du- affected my health. Perhaps I am destined to follow him long wrote to Berzelius: “He never realized his condition. soon and by the same road” [6]. However, his last days were painful for me. His sickness 12To my knowledge this paper, which is also a beautiful had changed his character, he acquired an aversion for ev- example of popular science writing, went so far unnoticed eryone else around him; I was the only one who retained his 13“Petit avait l’intelligence vive, la parole ´el´egante et confidence, and he demanded that I stay near him to talk facile, il s´eduisait par des dehors aimables, il s’attachait with him about his health, during whatever spare time my ais´ement, et s’abandonnait `ases tendances plutˆot qu’il ne duties allowed me. The grief that this sight brought me, to- les gouvernait; on lui reconnaissait une facilit´ed’intuition

4 This lively portrait of the two scientists, and in par- out that determining the law of cooling required a ticular Jamin’s closing sentence, may give us some better definition of a rational temperature scale, clues about the curious story of their great discov- a problem that called for further studies15. Surely ery. they planned to carry on this investigation, but the political and military turmoil that followed16 forced them to give up the competition and to publish the 3 Prelude to the discovery following year a rather short paper that was noth- and a further DP law ing more than their original memory [21]. Luckily for them, no submission was judged to be worthy It is not clear how Dulong and Petit came into con- enough to award the prize, so that the same sub- 17 tact. Very likely, the trait d’union between them ject, with little but not irrelevant adjustments , has been Arago, member of Arcueil (to which Petit was proposed for the next year. has never been associated) and professor of analytic This time, Dulong and Petit produced a mas- 14 geometry at the Ecole´ (where Dulong was just a terpiece [22]. In a paper published in three parts modest examinateur). As we shall see, Arago may (adding up to a total of 113 pages), which was to have played a subtle and crucial role in the an- be acclaimed for a long time as a model of experi- nouncement of the DP law too. We know however mental method, they carefully and extensively ad- of a precise event that may have stirred up their dress all the questions presented above with un- common interest towards problems related to heat precedented rigour, remarkable clarity of presen- transfer, namely the announcement of the subjects tation, extremely careful data analysis. A detailed for prizes to be awarded by the First Class of the and authoritative account of this monumental work French Institute in 1816–1817 made on 9 January has been given by Robert Fox [8]. Here I shall only 1815 [8]. Applicants were supposed to find: i) the expatiate on two aspects that highlight the pivotal expansion of mercury in a thermometer between role played by the synergy of Dulong’s experimen- ◦ ◦ 0 C and 200 C; ii) the law of cooling of a body tal ingenuity and accuracy with Petit physical in- in a vacuum; iii) the laws of cooling in air, hydro- tuition and mathematical rigour. gen and “carbonic acid” (actually, CO2) for several While introducing their investigation of mercury values of temperature gas density. Five month af- expansion, Dulong and Petit immediately highlight ter, Dulong and Petit had already presented at the a delicate aspect. To determine the volume expan- Institut de France a joint memory that partly ad- sion rate of a fluid, just measuring the elongation dressed the first of these question, but they pointed of the latter enclosed in a glass tube with temper- scientifique en quelque sorte instinctive, une puissance ature like in a standard thermometer is far from d’invention pr´ematur´ee, pr´esages certains d’un avenir assur´e being an accurate method. Indeed, the tube ex- que chacun pr´evoyait et mˆeme d´esirait, tant ´etait grande pands too, hence what is measured is just a rela- la bienveillance qu’il avait su inspirer. Dulong ´etait tout tive thermal expansion, which also depends on the l’oppos´e; son langage ´etait r´efl´echi, son attitude grave et son apparence froide[. . . ] Il travaillait lentement, mais avec material the tube is made of. How can one get sˆuret´e, avec une continuit´eet une puissance de volont´e que rid of this annoying problem and get an absolute rien n’arrtait, je devrais dire avec un courage qu’aucun expansion rate? The brilliant method devised by danger ne faisait reculer. A` d´efaut de cette vivacit´e de l’esprit qui invente ais´ement, mais qui aime `ase reposer, il avait le sentiment de l’exactitude scientifique, le goˆut des 15It is interesting to note that, although the subject of the exp´eriences de pr´ecision, le talent de les combiner, la pa- competition was not related to the problem of heat capacity, tience de les achever, et l’art, inconnu jusqu’`alui, de les Dulong and Petit already pointed out the crucial importance porter jusqu’`ala limite possible de l’exactitude[. . . ] Tels of checking for any temperature variation of the specific heat sont les traits principaux de ces deux hommes c´el`ebres. Pe- when developing thermometers based on the thermal expan- tit avait plus de tendance math´ematique, Dulong se mon- sion of a substance. trait plus exp´erimentateur; le premier portait dans le travail 16Recall that was the year of Les Cent-Jours of Napoleon plus de facilit´ebrillante, le second plus de continuit´e; celui-l`a and of the following rebellion against restoration that ended repr´esentait l’imagination, celui-ci la raison, qui la mod`ere with the Treaty of Paris. et la contient.” 17Determination of mercury expansion should have been 14Arago succeeded Gaspar Monge, the father of differen- performed down to −20◦C and compared with that of an air tial geometry who was very influential for the mathematical thermometer. For the announcement, see J. Chim. Phys., education of Petit. 4, 302-303.

5 Dulong and Petit was absolutely new18. In their that would be made by assuming a T -independent words (Ref. [22], pag. 125): expansivity: for instance, a mercury thermometer that agrees with an air thermometer at 0◦C would [The method] is based on this incon- be in differ from the latter 14◦C by more than at testable principle of hydrostatic, that when 300◦C. But this is not the whole story. Dulong and two liquid columns communicate between Petit went on by showing that they could simply them by a lateral tube, the vertical heights obtain the value of αV for a solid material by con- of these columns are precisely in inverse trasting the absolute expansivity of mercury with ratio to their densities. Therefore, if its apparent one in a tube made of that material. one could exactly measure the heights of With this brilliant trick, Dulong and Petit obtained two columns of mercury contained in the the thermal expansivity of glass, iron, copper, and branches of an inverted glass siphon, by platinum. keeping one in melting ice, for example, While one can almost hear the voice of Dulong while the other is brought to a known describing the investigation of thermal expansivity, temperature, one would easily deduce the the deep physical intuition and the great mathe- sought dilation. In fact, if h and h′ are matical skill of Petit stand out when we consider the vertical heights of two columns giving the second part of the paper, dedicated to the deter- the same pressure at temperatures t and mination of the law of cooling. This subject was ar- t′, we must have, by calling d and d′ the guably the main goal of the competition, motivated corresponding densities, hd = h′d′. by the controversies concerning Fourier’s model of heat conduction in solids for which he had been In simple words, they devised an instrument work- 20 ing on the principle of the barometer that is totally awarded the 1811 prize of the Academy . What independent from the material, size or uniformity was actually asked to the competitors was to in- of the columns and moreover yields a differential vestigate how heat is transferred in vacuum and in measurement of the density: I am sure that any fluids, with the main aim of proving or disproving experimentalists will immediately appreciate this the Law of Cooling enunciated by Newton, which last major advantage. The following 8-page long states that the time it takes for a sample to cool description of the set up they built and of experi- is proportional to the temperature difference with mental protocol they applied is by itself a piece of the surrounding environment. bravura. Of particular interest is the accuracy with Actually, Newton’s meditations and experiments which they measured the column heights using a about cooling cannot be exactly regarded as “crys- cathetometer equipped with a vernier that allowed tal clear”. In his original sparse writings about the appreciating displacements as small as 0.02 mm, subject, the great scientist does not distinguish be- probably built for them by the great instrument– tween the different modes of heat transfer. In fact, maker Henri–Prudence Gambey (see Ref. [14], pag. if a sample is left to cool freely, generating sponta- 604). With this instrument they measured the ab- neous (“natural” or “free”) convective currents in solute volume expansivity αV of mercury in a wide the surrounding air, Newton’s law does not hold as temperature range (0 − 300◦C) with a much higher a rule, while it does in the presence of forced con- accuracy than in previous studies (their data are vection, for instance if we blow fresh air around the 21 actually within a fraction of a percent with mod- sample with an hairdryer Newton was probably ern ones19 ). This also allow them to show the error later), Dulong and Petit provide not only “la moyenne d’un 18Actually, Dulong and Petit, who are very careful in cit- grand nombre de mesures”, but also “les valeurs extrˆemes”, ing previous works, state that this method was originally namely the full range of values they obtained (which was of suggested by Boyle, and that several other scientists had the order of 10−3 of the averages). thought of using it. Yet, they point out that its practical 20It is probably because of the severe criticisms raised application is far from being easy, in particular for large by the committee that judged this work, which also in- temperature differences. As a matter of fact, no one used it cluded Laplace, Lagrange and Legendre, that Fourier’s law before them. appeared in print only several years later [8]. 19It is worth noticing that, at a time when statistics was 21Technically, this is because in forced convection the Nus- still in its infancy (Gauss introduced the concept of “mean selt number, which is proportional to the ratio Q/˙ ∆T be- error”, the forerunner of standard deviation, only a few years tween the heat transfer rate and the temperature difference,

6 aware of the problem when, in his Scala graduum caloris [24], he wrote that the piece of iron he was studying was laid not in calm air but in a wind that blew uniformly on it22. Yet, this sentence was probably too cryptic to be adequately appreciated. In fact, most of the claims of failure of the Newton’s law made in the 18th century can hardly be trusted, because of the poor definition of the experimental conditions, adding up with a substantial degree of confusion about the concept of convection as op- posed to conduction. Newton himself, however, had already noticed in his Opticks (Query 18) that a cooled thermometer heats up even if enclosed in a transparent vessel Figure 2: Apparatus used by Dulong and Petit to in- wherefrom air has been pumped out. In his own vestigate cooling in vacuum and in air. For a detailed words: (almost an understatement) description of the setup, see Ref. [22] (source: Hathi Trust Digital Library, from Is not the Heat of the warm Room con- the original conserved in the library of the University vey’d through the Vacuum by the Vibra- of Virginia.) tions of a much subtiler medium than Air, which after the Air was drawn out re- mained in the Vacuum? And is not this during cooling the sample must be thermally ho- Medium the same with that Medium by mogeneous, a condition that is more easily satisfied which Light is refracted and reflected, and for small samples of liquids, so to take advantage of by whose Vibrations Light communicates convection for leveling out temperature differences. Heat to Bodies, and is put into Fits of easy The simplest choice was studying the cooling rate Reflexion and easy Transmission? in vacuum of mercury contained in the glass bulb of a thermometer. In order to obtain a general In other words, with a strike of genius Newton had picture, however, they subsequently scrutinize the fully realized that heat can also be transported by effects of a) the sample volume, b) the nature of the something akin to light, namely, he had discovered investigated liquid (by comparing mercury to wa- what later was called “radiant heat”. ter, ethanol, and sulfuric acid, and c) the container Dulong and Petit begin their discussion by shape (spherical vs. cylindrical) and material (a clearly stating that the process of cooling in air glass vs. an iron sphere). Thanks to this thorough takes place via two distinct mechanisms, radiation investigation, they manage to reach a key conclu- and heat transfer by the fluid, which may obey dif- sion, namely that “the law of the cooling of a liquid ferent laws and must therefore be separately inves- mass, variable with the state of the surface which tigated. They also attentively point out some gen- serves as its envelope, is nevertheless independent eral experimental requirements, in particular that the nature of this liquid, the shape and size of the must be a function of the Reynolds and Pecl´et numbers vase that contains it”. The original drawing of the alone, which are both T -independent. Conversely in nat- setup they used is shown in Fig. (2). ural convection, where no characteristic velocity scale (and What they still had to do was finding that law, therefore a meaningful Reynolds number) can be defined, and here is the genius of Petit that stands out. the heat transfer coefficient h = Q/A˙ ∆T , where A is the heat transfer surface, is in general a function of ∆T , unless There was already evidence, in particular thanks the latter is small. For instance, in the simple case of free to careful experiments performed by John Leslie, convection from a vertical plate, Q˙ = (∆T )α, where α in- that Newton’s law of cooling does not hold when creases from 5/4 for laminar flow to 4/3 for fully turbulent the temperature difference ∆T between the sam- flow [23]. 23 22“Locavi autem ferrum, non in aere tranquillo sed in ple and the environment is large , and their ex- vento uniformiter spirante ut aer a ferro calefactus semper abriperetur a vento & aer frigidus in locum ejus uniformi 23Leslie gave a key contribution to the study of radiant cum motu succederet”. heat by observing and quantifying the different emissivity

7 periments fully confirmed that the cooling rate in- progression” and that “the ratio of this geomet- creases with ∆T more than linearly. They clearly ric progression is the same whatever the excess of realize that they could have easily fitted the data temperature considered”. With a simple but not by adding terms of higher order in ∆T , but they ar- trivial calculation, Petit shows (it must have been gue that formulas of this kind, useful with no doubt him, necessarily!) that this implies F (T ) ∝ aT + c, when one needs to calculate intermediate values where a is a universal constant that their data in- within the interpolated range, almost always be- dicated to be a = 1.0077, and c another constant come inaccurate outside the limits between which that can be neglected by appropriately choosing they have been determined, and are never able to the zero of the temperature scale 26. Consequently, make known the laws of the phenomenon under v ∝ aT (a∆T −1). In the last part of their paper Du- study. In fact, as they state, their study had led long and Petit comply with the final requirement them to develop a comprehensive “theory of irradi- of the competition by performing extensive cooling ation”, something that, they believe, no physicist measurements in the presence of several gases (air, had done before. This is truly a qualitative leap, hydrogen, carbon dioxide, and ethylene) conclud- which to me amounts to recognize the difference ing that in the pressure range p ∈ [45 − 720] Torr between “taking measurements” and performing an the gas yield an additional contribution to the cool- experiment. ing rate proportional to ∆T bp c, where c depended To reach their goal, Dulong and Petit first use on the investigated gas (but, with the exception of the notion, qualitatively introduced by Prevost in hydrogen, was of the order of c ≃ 0.5), while b had 1791 [26], that equilibrium in radiation exchange a universal value b ≃ 1.23 (hence very close to the comes from a balance between emission and absorp- value expected for free convection!). 24 tion . Hence, accounting both for the radiation Curiously, till the end of the XIX century it was emitted by the sample at temperature T + ∆T and this one, and not the one about specific heats, that that received from the environment at temperature was considered among physicists as “the” Dulong– T , they write the cooling rate as v = dT/dt = Petit law. As we presently know, however, this F (T + ∆T ) − F (T ), where the unknown function “further” DP law is wrong, for the true expres- F (T ) is in fact the “radiation law” they were look- sion that governs the temperature dependence of 25 ing for. Of course, one recovers Newton’s law emission and absorption of radiation is given by only if F (T ) is linear in T . On the contrary, how- the Stefan–Boltzmann law, which in modern terms ever, according to Dulong and Petit their experi- states that the power radiated per unit surface area mental results could be summarized by stating that from a body is given by P = ǫσT 4, where σ ≃ “the cooling rate of a thermometer in the vacuum 5.67 × 10−8 W m−2K−4 is the Stefan–Boltzmann increases in geometric progression when the tem- constant and ǫ is the surface emissivity coefficient perature of the enclosure increases in arithmetic (ǫ = 1 for a black body). This implies that F (T ) does not increase exponentially with temperature, of surfaces (for instance black vs. metallic) using the “pho- but as a power–law. It took however a long time tometer” he invented (later called the “Leslie cube”). Curi- ously, although he held that light and heat were only differ- before discrepancies from (this) DP law could be ent states of the same substance, he claimed that heat could found (see for instance Ref. [5, 28].). Here I wish never be transmitted through a completely empty space, only to recall that Stefan started the investigation stating that “Were it possible to procure an absolute vac- 4 uum, a body thus insulated would indisputably retain for that led him to propose in 1879 the T −law by ever the same temperature” (see Ref. [25], pag. 142). Ra- arguing that the vacuum level that Dulong and Pe- diant heat also played an important role in the controversy tit could reach (about 2 − 3 mm of Hg) was not between the caloric and the vibrational theories of heat (for sufficient to ensure that the residual gas did not extensive accounts, see [4, 5]). 24“Dans la th´eorie adopt´ee des ´echanges de chaleur, le refroidissement d’un corps dans le vide n’est que l’exc`es de 26Dulong and Petit were aware that this exponential form son rayonnement sur celui des corps environnans”, see [22], for F (t) implies that the “absolute zero” temperature (still pag. 148. hypotetical at that time) must be −∞, but observe that this 25Which Dulong and Petit actually call “la loi de Rich- does not necessarily means that a body contains an infinite mann,”, from the extensive experiments by Georg Wilhelm amount of heat. To avoid this, indeed, is sufficient that the Richmann for water cooling in air, which fully supported the integral over T of the specific heat (which they already know law [27]. to decrease with T ) is finite.

8 did measure specific heats using the (now) classical method of mixtures. Right at the end of the first part of their paper they first obtain the specific 27 heat cp of iron within four different temperature ranges between 0◦C and 350◦C and then the spe- cific heat of mercury, zinc, antimony, silver, copper, platinum, and glass for two different T -ranges. Du- long and Petit were indeed clearly aware that c, like the thermal expansivity, does depend on T 28. But there is no attempt of finding any relation among these values, no sign of the simple but great insight that came to their mind roughtly a year later, al- though some of the values for the specific they had obtained may have hinted at that conclusion. Why it did not happen is the first of several little puzzles we shall encounter in the next sections.

Figure 3: Data obtained by Dulong and Petit for the 4 Everlasting fame: the great cooling rate v(∆T ) of a glass bulb filled with mercury ◦ achievement of 1819 (environment at T0 = 0 C), with fits to the Dulong– Petits and the Stefan–Boltzmann laws. The first odd thing about the paper that gave our two scientists everlasting fame is the order of names. In all previous papers, Dulong had been contribute to heat transport. Stefan indeed agreed the first author, and it was Dulong who read the that convection becomes negligible at these pres- paper at the Acad´emie des Sciences. After all Du- sures, but heat conduction, according to Maxwell long was much more experienced and recognized kinetic theory, does not depend on p over a large them Petit, not to say that he was also six years range, and should be taken into account. Never- older. But the DP law should properly be called theless, Stefan himself had to admit that the law the law of Petit and Dulong, for the first author he was proposing did not give a much better fit to 29 of Ref. [1] is Petit . This already suggests that the DP data (which evidently were still regarded it was the young rising star that had the brilliant as a reference 60 years later), and only an order of intuition of checking the products c × m of the magnitude comparison with the data obtained by p a ◦ specific heats times the atomic weights. But a hy- Tyndall for a platinum wire heated up to 1200 C pothetical backstory, advocated by Jean–Baptiste allowed him to state that, on a wider T -range, his Dumas, may also suggest that the two scientists law was much better obeyed [5]. Just to highlight had a dissimilar opinion about the experimental how hard was to disprove the DP law of cooling, I 27 have contrasted in Fig. (3) one of their original fits I use the symbol cp for the specific heat at constant pressure, which is of course what Dulong Petit were really with a fit to the same data made using the correct 2 Stefan–Boltzmann law. measuring, that is related to cv by cp = cv + α T/(ρβT ), where α is the thermal expansivity, βT the isothermal com- I have expatiated a lot on the 1817 paper, first pressibility, and ρ the mass density of the material. It is because it conveys a clear, undeniable message: worth noticing however that they were already aware of that Dulong was an exquisite experimentalist, and Pe- cp =6 cv, since this has already pointed out by Dalton. How- ever, they correctly estimate that the amount of heat spent tit a first-rate theorist. But there is a second rea- for volume expansion of an almost incompressible solid or son, which is no lesser importance for our purposes. liquid can be safely neglected. 28 Even if the competition announced by the French “Il en est donc des capacit´es des corps solides comme Institute had little to do with the heat capacity, Du- de leurs dilatabilit´es; elles croissent avec les temp´eratures mesur´ees sur le thermom`etre `aair”, Ref [22], pag. 147. long and Petit, motivated by the crucial role of the 29I made a little effort to patch over this historical injustice latter in establishing a proper thermometric scale, with the short title of this paper (see the top of this page)

9 evidence supporting their finding [8]. According to the “elements on which they are immediately de- the eulogy of Henri–Victor Regnault he presented pendent”, they indeed claim (clearly referring to in 1881[29], on 5 April 1918 (“a memorable date”, the content of the paper) that according to Dumas) Petit confidentially (and ea- The success that we have already attained gerly) showed his brother-in-law Arago a piece of makes us hope that this kind of reason- paper where he had discovered the unique similar- ing will not only contribute to the ulti- ity of the values of cp × ma. Arago immediately mate progress of physics, but that also the grasped the importance of the discovery but, appar- atomic theory will in its turn receive from ently, he had good reasons to suspect that Dulong it a new degree of probability, and that it might have objected to publication. Hence, to con- will there find sure criteria for the distinc- vince him, he leaks the news to his fellow members tion of the truth among hypotheses that of the Acad´emie. His stratagem evidently worked appear to be equally probable.32 out, since it did not take too for Dulong to present the joint paper to the Academy30. Such a bold statement implies that they were fully A story told more than 60 years later by some- aware of the great importance of their discovery. one who in 1819 was just a student in Geneva may Then, they specify that the attributes of matter sound apocryphal, even more because no other ac- they will focus on are those that “depend on the count of this meeting between Petit and Arago sur- action of heat”, and in particular the specific heat. vives. Yet, Dumas had close ties with Arago, who However, a terse review of previous investigations may have been the original source, and his inter- of this subject lead them to conclude that “The at- est for the work of Petit and Dulong dates back tempts hitherto made to discover some laws in the to 1826, al least31. Whatever the truth, this curi- specific heats of bodies have then been entirely un- ous anecdote would not be out of keeping with the successful”. Dulong and Petit identify the origin observations of Jamin on our two scientists. of this failure in the difficulty of finding accurate What is surely true is that the 1819 paper dras- methods of measurement. They admit that, among tically differs from the previous works of Dulong the proposed approaches, the method of mixtures and Petit in content, style, and length too, being “may doubtless, when properly conducted, lead to only 19 pages long. Petit and Dulong begin with very exact results”. Yet, it suffers from a major a very general incipit. After stating that they are drawback: it requires a sizeable amount of the in- persuaded that certain properties of matter would vestigate material, which prevents its application appear in simpler form and would be expressed by to rare substances (or to expensive ones, like gold less complicated laws if one could relate them to or platinum!). Nevertheless, they claim, the ex- perience they made allowed them to single out a 30“Une heure apr`es, l’illustre secr´etaire perp´etuel conva- method that satisfies all critical requirements: the incu que Dulong, toujours h´eesitant, pourrait s’opposera ` la method of cooling. This statement may sound a bit divulgation de cette belle loi, en entretenait ses confr`eres, singular, since it seems to be based on Newton’s par une indiscr´etion calculee. Huit jours plus tard, les assumption that in their previous work they had deux collaborateurs l’´enon¸ccaient devant l’Acad´emie elle– mˆeme. . . ”[29] (Note that Dumas speaks of 8 days, while shown not to be an exact law. Yet, they are fully they actually seem to have been 14). aware that it does apply for sufficiently small tem- 31 Actually, in Ref. [29] Dumas claims to have been the perature differences. Accordingly, they write, “all one who spurred Regnault to carry on the work of Dulong our experiments were made in an interval of tem- and Petit. Dumas also developed an interesting method for ◦ ◦ measuring the molecular weight of volatile substances, which perature included between 10 and 5 centigrade basically consists in placing a small quantity of the that of excess above the ambient medium”. Operating substance into a flask of known volume, which is heated around the same temperature also allows them to until the substance turns into a vapour that replaces the air get rid of errors resulting from the graduation of in the flask: When the the substance has fully evaporated, the vessel is sealed, dried, and weighed. Unfortunately, as the thermometer. the shall briefly see, the results obtained by Dumas with this Although concise, the following discussion of the method, which has been a standard in organic chemistry for specific choices they made and of the apparatus a long time, were one of the main reasons why Avogadro’s fundamental insight was rejected by most chemists, first of 32This quotation and all those that follow are from the all Berzelius (a full account of this story can be read in [30]). contemporary English translation, Ref. [10].

10 they developed shows how much Petit and Dulong had learnt from their previous investigations. First, they observe that any precaution to improve tem- perature measurements would be “delusive” if the ambient temperature were not rigorously constant during the total duration of each experiment. To ensure this they plunged the samples into a vessel, blackened in the inside and surrounded by a thick coating of melting ice. Blackening the inside of the vessel had also the advantage of slowing down the sample cooling, an important problem when work- ing with small samples. To further reduce the cool- ing rate they exploit another evidence they found in their previous study, namely that “the velocity of cooling of a body may, ceteris paribus, be consid- erably diminished when its surface possesses but a very weak radiating power, and is plunged in an air very much dilated.” To this aim, they finely ground Figure 4: Sketch of the novel apparatus discussed by a tiny amount of each investigated substance, press- Petit and Dulong in Ref. [1], as shown in Ref. [31], ing then the powder into a small and thin cylindri- Deuxi`eme Fascicule, pag. 29 (freely downloadable from cal silver vessel with high surface reflectivity, the The Internet Archive.) axis of which was occupied by the thermometer that served to evaluate the rate cooling rate. By So, Petit and Dulong maintained that a forthcom- these precautions they managed to work with sam- ing extended paper will have clarified any ques- ples weighting less than 30 g even for very dense tions or misunderstandings about their findings. metals like platinum, still retaining cooling times We know however that things went differently, due of tens of minutes. Unfortunately, at variance with to the tragic end of Petit. Yet, we may wonder their previous paper (see Fig. (2)), they do not in- why Dulong did not keep this promise, at least to clude any picture of their setup. However, a draw- keep alive the memory of his good friend. Next ing reproducing the latter that may have been kept comes the climax of the paper, with the presenta- at the Ecole´ can be found in Ref. [31] and is shown tion and an insightful discussion of the table shown in Fig. (4). in Fig. (2), in which Petit and Dulong show that The following part of Ref. [1] shows the largest they are fully aware of the terrible blow their dis- departure from the style of presentation of their covery gives to the hypothesis of the caloric, and at previous works. Indeed, there is no trace of the the same time of the remarkable support it provides minute data presentation and analysis that charac- to the atomic theory33. terize their prize–winning publication. They actu- It is however worth pointing out what the law ally ‘apologize’ by stating that of Petit of Dulong does not say. First, the value of the product c × m had no explanation until It would now be requisite to give the p a Boltzmann gave it a precise physical meaning34, formula which served for the calculation of the observations; but the details into 33On 15 January 1820, Dulong wrote to Berzelius “We had which we should be obliged to enter re- already given a fatal blow to the chemical theory of warmth specting the manner of making the differ- in the memory we read at the Institute during your stay in Paris” and that “Despite the objections of M.Laplace and ent corrections depending on the method some others, I am convinced that this [atomic] theory is of proceeding would lead us into a discus- the most important concept of the century and in the next sion which we reserve for the publication twenty years it will bring about an incalculable extension to of the definitive results of all the direct ex- all parts of the physical sciences” [32] 34One must say that Boltzmann did not made his best periments which we have made on the sub- effort to ‘advertise’ his result, which is rather buried inside ject. a paper with the slightly misleading title “Analytical proof

11 Second, since they knew very well from their pre- Table 1: Atomic weights, normalized to the atomic vious work that c decreases with T , they could not weight of oxygen and rounded to the second decimal regard that value as temperature independent. Fi- obtained by Berzelius in 1818 (B1818) and in 1826 nally, they did not think that they results applied (B1826), compared to the values used by Petit and only to solids but rather to any substance, simple Dulong (PD1819) and to the current values (fourth col- or compound, in any state of aggregation (as they umn). The last column is discussed in the next Section. explicitly state in the paper). In fact, after Petit’s departure Dulong not only tried to extend the law B1818 PD1819 B1826 Modern Ratio to gases and compounds, but also to other physical Bi 17.74 13.30 13.30 13.06 3/4 quantities besides the specific heat like the refrac- Pb 25.89 12.95 25.89 12.95 1/2 tive index (with Arago), with little success in both Au 24.86 12.43 12.43 12.31 1/2 Pt 12.15 11.16 12.15 12.19 9/10 cases. Sn 14.71 7.35 14.71 7.42 1/2 Actually, Petit and Dulong may have not even be Ag 27.03 6.75 13.52 6.74 1/4 fully convinced that what they had found was an Zn 8.06 4.03 4.03 4.09 1/2 exact Te 8.06 4.03 8.02 7.98 1/2 ‘law’. Indeed, just after the grand statement Cu 7.91 3.96 3.96 3.97 1/2 stated in the introduction, they add a caveat: Ni 7.40 3.69 3.70 3.67 1/2 Fe 6.78 3.39 3.39 3.49 1/2 If we recollect what has been said above Co 7.38 2.46 3.69 3.68 1/3 respecting the kind of uncertainty which S 2.01 2.01 2.01 2.00 1 exists in fixing the specific weight of the atoms, it will be easy to conceive that the law which we have just established will atomic or molecular weights he managed to esti- change if we adopt for the density of the mate were often wrong, also because he was using particles, a supposition different from that arbitrary assumptions like the ‘rule of greatest sim- which we have chosen [. . . ] But what- plicity’35. The true prince of atomic weight deter- ever opinion be adopted respecting this re- mination was Jacob Berzelius, a giant of XIX chem- lation, it will enable us hereafter to con- istry. But even Berzelius took a long time before trol the results of chemical analysis; and reaching consistent and accurate values. The first in certain cases will give us the most ex- and third columns of Table 1 compare the values act method of arriving at the knowledge of of ma for the elements investigated by Petit and the proportions of certain combinations. Dulong (normalized to the atomic weight of oxy- gen) obtained by Berzelius in 1818 and 1828, with What they suggest with these words is that their the atomic weights used in Ref. [1] and with the ‘law’, even if approximate, might provide a power- currently accepted values. It is worth noticing that ful tool to solve some crucial problems in the de- Dulong and Petit were surely aware of the first set termination of atomic weights that, at that time, of data, although Berzelius’ work was translated in was indeed far from being satisfactory. By mak- French only one year later, since Berzelius was in ing atomic weights the keystone of his theory and Paris from August 1818 to June 1819 and exten- using them as the chief criterion to set apart dif- sively worked with Dulong in Arcueil36. Table 1 ferent atomic species, Dalton went far beyond all will be useful to understand the reception and to previous abstract and generic models, thus attract- discuss the recent criticisms of Ref. [1]. Before that, ing on atoms the interest of chemists. But the few however, it is useful to take a ‘fresh look’ at the law of the second law of mechanical heat theory from sentences on the balance of living force” [33] that does not lead itself 35For instance, since water was known to be made of oxy- to be easily understood (to use a euphemism). A much gen and hydrogen, Dalton assumed that its formula was OH. more readable account was written only several years later For a very accurate account of the development of atomic by Franz Richarz [34]. This paper, which contains a detailed theory, see [30]. model for the transfer of heat to atom vibrations and also an 36Dulong and Berzelius developed a deep friendship, wit- interesting discussion of anharmonicity effects, was actually nessed by the copious and warm letters they exchanged regarded at the end of the XIX century as the most useful till Dulong death (see for instance footnote 33). In fact, source for the theoretical interpretation of the DP law (see Berzelius was really fond of Dulong, and described him as Ref. [35], pag. 37). “having the most brilliant mind in the world of chemistry”.

12 of Petit and Dulong by plotting the currently ac- 2. Cobalt and tellurium, whose molar heat capac- cepted values of the specific heat at constant pres- ities are quite close to the currently accepted −1 −1 sure cp versus ma for all the solid elements in the values (24.68 Jmol K to be compared with periodic table. Fig. (5) shows that, apart from the 24.81 Jmol−1K−1 for Co, 24.61 Jmol−1K−1 a few very anomalous cases, it would be quite unfair to be compared with 25.73 Jmol−1K−1 for −1 to deny that an inverse relation cp ∝ ma holds, at Te [36]), but whose atomic weight are underes- least approximately. Keeping this picture in mind, timated by a factor of 2/3 for Co and 2 for Te we are ready to put Dulong and Petit on trial. (which means, of course that the experimental value are overestimated by the same factors).

Glancing through Table 1, however, you will imme- diately notice that all values of ma used by Petit and Dulong, except the one for sulfur, differ (in most cases quite consistently) from those just pub- lished by the great Berzelius (who, remember, may well have been hanging about their lab on these days!). Yet, look at the last column of the table: all the ratios between the values they used and those by Berzelius are almost exactly simple fractions38. This gives us an important clue to understand why it took so long for them to realize the evidence. In- deed, it would be silly to think that a gifted mathe- matician like Petit should have taken a whole year to make a × b = c (in fact, this did not work). What Petit brilliantly guessed is that the products c Figure 5: Specific heat per unit mass p of those ele- of the experimental specific heats times Berzelius’ T ◦ ments that are solids at = 25 C. Full and open circles atomic weights, rescaled by suitable but simple fac- respectively indicate metals and nonmetals, while the tors, were very close. For Petit this was too nice semiconductors silicon and germanium are shown by half-full dots. In this double-log plot, the law of Petit to be fortuitous. Hence, he arguably concluded, and Dulong is given by the straight line. Those el- the true values of ma must have been the rescaled ements whose heat capacity deviates appreciably from ones, which would have turned their relation into the DP law are explicitly indicated, with two allotropes “the most exact method of arriving at the knowl- of carbon, diamond and graphite, marked by C(dia) and edge of the proportions of certain combinations”. C(gra). This view of Petit and Dulong’s results seems to have been appreciated by their contemporaries, and in particular by Berzelius himself who, seven years 5 Petit & Dulong on trial later, had already accepted many of the values of ma they had proposed, although he did not agree at all on the values for cobalt (rightly), silver and If you have done the little exercise I suggested in tellurium (wrongly). As a matter of fact, although the introduction, checking with modern values the he was not fully convinced of the generality of the data in Fig. 1, you may have found that the most law, he concludes that “for the moment we have controversial results concern: to agree that a continuation of Dulong and Petit’s excellent work in this subject would, however, be 1. Platinum, for which the products of the data 39 in the first two columns gives 0.350, and not a vital service to science” . On the other hand, 0.37437 38With the sole exception of platinum (in this case the 37In passing, we should not blame too much Petit and ratio is about 0.92). Dulong for giving the results in the last column with four 39My translation from he first Italian edition of Berzelius’ decimals, when the second one has only three. After all, as treatise [37] (curiously, there was no contemporary English I mentioned, rigorous statistics had yet to come. . . translation of his Opus Magnus).

13 Victor Regnault, carrying on their work40, clearly about the problems with Petit and Dulong’s points out that, even using the ‘updated’ values of data, but did not make a big story about it. Berzelius, several problem persist[38]: On the contrary, they were both aware of the fundamental importance this result would have, Now, if we replace the atomic weights even if it might not be considered as an ‘absolute adopted by Dulong and Petit by those who truth’. Even more, it is far from them claiming are generally admitted now, we recognize any ‘fraud’ by two scientists whom they both that their law is far from being verified in admired. For a long time, all respected scholars such a satisfactory manner [. . . ] the spe- who have investigated the DP law seem to have cific heat of bismuth is a third too weak shared this general attitude [6, 7, 8]. Until, at to follow the law of atoms, the specific the end of the last century, a buzz of discredit heat of silver and that of tellurium are began to rise. Apparently, everything started in twice as large; the specific heat of cobalt 1985 with a radio talk of the Australian writer is too strong of about one–third; finally Peter Macinnis, followed by a letter to Chemical & platinum also deviates from the theoreti- Engineering News by Peter Schwarz [39], both of 41 cal number. which triggered the interest of Carmen Giunta [9]. I could not listen to Macinnis’ talk, broadcasted As a great experimentalist, the way he settled this on the other side of the world when I was still a issue was by performing accurate measurement that student. Yet, you can appreciate his own rather dif- rectified the values found by Petit and Dulong for ferent attitude on the ABC website, where you read Co, Te, Ag, and Bi (but also Berzelius’ values of (http://www.abc.net.au/science/slab/macinnis/story.htm): ma for the latter two elements). But he did not call into question the great value of the result they had Dulong and Petit concocted their results obtained. On the contrary, he pointed out that, when they generated their law relating spe- since the atomic weights of the substances he in- cific heat to atomic weight. Given the vestigated vary of a factor of 7 while the products fraudulent data that I can demonstrate in their results, they probably faked more cp × ma differ by no more than 10%, we should be convinced that “the law of Dulong and Petit must than half of the measurements, and fudged be adopted, if not as an absolute principle, at least the rest like a second-rate physics stu- as a result that approaches very much the truth”42. dent. But who cares? Their spurious law As we see, the two most interested parties, was more or less correct, and it allowed chemists to determine atomic weights ac- Berzelius and Regnault, knew perfectly well curately by electrolysis, ducking around 40Regnault can actually be considered as the father of problems caused by valency. modern calorimetry. In fact, by measuring the heat capacity of about 30 elements and correcting the errors made by Du- I leave out any comments, which is left to the judge- long and Petit, he gave strong support to their hypothesis ment of my readers. Similarly, I shall not waste so much that he could be considered a full–fledge coauthor time arguing against the offending letter by a rather of the DP law. obscure organic chemist. The paper by Giunta, 41“Or, si l’on remplace les poids atomiques adopt´es par Dulong et Petit, par ceux qui sont g´en´eralement admis main- however, deserves for sure much more attention. tenant, on reconnaˆıt que leur loi est loin de se v´erifier d’une So, let us start by dwelling upon the three obvious mani`ere aussi satisfaisante [...] la chaleur sp´ecifique du bis- inconsistencies they we already pointed out. muth est trop faible d’un tiers pour suivre la loi des atomes, la chaleur sp´ecifique de l’argent et celle du tellure sont deux 1. Platinum. Apparently, this is a minor prob- fois trop grandes; la chaleur sp´ecifique du cobalt est trop lem, for most reviewers including Giunta agree forte environ du tiers; enfin le platine s’´ecarte ´egalement du that it must just have been a misprint. Maybe, nombre th´eorique.” 42Very interestingly, in the final part of his investigation, but the question is, where is the misprint? Regnault tries and scrutinies why the DP law is not ‘exact’, Table 1 suggests that it should be in the pointing out the role of the “chaleur latente de dilatation” atomic weight, since this is the only value (in other words, Regnault is aware of the difference between of ma which is not a simple fraction of the cp and cv) that, although small for solids or liquid, may produce a temperature dependence of the measured specific Berzelius’ 1818 value. Using ma = 12.15, how- heat that will be different for different substances. ever, the product becomes 0.3818, which is not

14 what they state (although still an acceptable ever, let us mull a bit more over this, taking value). In fact, the printer’s error is in the spe- into account the information I tried to sum- cific heat, for in the German translation43 of marized on the state of affairs in 1819. Pe- Ref. [22] Dulong and Petit found cp =0.0335, tit and even more Dulong were surely wor- which, multiplied by ma = 11.16, yields ex- ried about the reception of their work, where actly 0.3740. While this value (equivalent to all but one (sulfur) of the atomic weights just 25.05JK−1mol−1) is about 3% smaller than presented by the leading expert in the field44. the currently accepted value, using cp =0.0335 So, in the case of Tellurium, I really do not and ma = 12.15 they would have obtained a find any reasons why they should have halved molar specific heat 5% larger than the modern Berzelius value for ma (which they knew) dou- one, which is probably not a big issue. How- bling at the same time their experimental value ever, besides noticing that Petit and Dulong for c to make their product consistent with very likely “recycled” some of the data ob- the others. On the other hand, had they com- tained in [22], we may wonder why they made pletely ‘fabricated’ this result, why not taking such a strange change for the atomic weight of for ma a value supported by the authority of Pt. But here comes the real puzzle: If they Berzelius? Fiddling this way with data would felt that the result for Pt was suspicious, be- have been, in my opinion, a clear symptom cause it obliged them to use a rather weird of masochism, also because tellurium was one rescaling of Berzelius’ value, why did not they of the substances Berzelius was more skilled simply exclude it from their table? After all, with (it was because of his noticeable confi- they still had a list of 12 substances that sup- dence with this substance that he managed to port the law! No, they did not. Remembering discover selenium in 1818)45. Indeed, he de- how things may have gone according to Du- cidedly refused to accept the result by Petit mas, I am rather inclined to think that this and Dulong result, firmly stating that “The might have the result of the ‘compromise’ be- external properties and the specific gravity of tween Petit and Dulong who, still doubtful tellurium are also similar to those of the anti- about some of their results and convinced that mony, which convince me to take their atomic some additional work was needed, nevertheless weights as equal, regardless of the above men- accepted to present the paper provided that tioned experiments of Dulong and Petit” [37]. they report about all the substances they had The question of cobalt is different, because in investigated, sweeping no dust under the car- this case the Berzelius value was wrong (of a pet. Although no historical evidence will ever factor of two), so any experiments must have support my guess, this might have happened given a conflicting value for c × m . Again, too for the other two elements we are going to p a if they did ‘fabricate’ the experimental re- examine. sult, why not ‘hiding’ the fraud just by using Berzelius’ value? No, they did not, they in- Tellurium and cobalt. 2. When compared with cluded the value for Co anyway risking to use modern data, the Petit and Dulong results for a different (but wrong) value for ma. Berzelius these elements stand out as the most bewilder- was apparently more open to questioning the ing ones and are surely prone to rise suspicion, value he had obtained for cobalt, but in any which eventually lead Giunta to state that “In particular, the specific heats of cobalt and 44Looking at Table 1, we se that this perilous choice tellurium, which Dulong and Petit state they largely paid back, since 9 over 12 changed values of ma are measured, appear to have been fabricated”. Be- pretty close to the modern ones (exception are are again Pt, Te, and Co). fore accepting this summary judgement, how- 45Curiously both tellurium and selenium are really strange elements for what concerns heat capacity. Indeed, each 43Curiously, in the original French publication there was atoms has just two strongly bound nearest neighbors, so the another misprint, since the stated value of the atomic weight crystals resemble fibrous chain structures with weak inter- is 0.0355. This misprint, corrected by Dulong and Petit chain interactions. As a consequence, their low–temperature themselves in the German edition, was surely known to Reg- specific heats markedly deviates from the Debye T 3 limiting nault (see Ref. [38], pag. 9). law, and vanish linearly with temperature [40].

15 case in his treatise he does not seem to rec- ommend using specific heat to find the atomic mass of tellurium, cobalt, and of the other ‘anomalous’ elements [7].

As you see, the ‘anomalous’ results for Pt, Te, and Co may have different explanations. The fact is, we are looking at these data from the advantageous point of view of the future. I really wonder if any rumors against the Petit and Dulong work had ever been raised if they did not include the only three elements they attributed to erroneous values of ma. After all, hindsight is a well known cognitive bias. Yet, to reach his conclusions, Giunta leverages on another clue, surely more quantitative, based on Figure 6: Comparison of the values of the molar spe- comparing the statistics of the molar heat capac- cific heats cp in units of the gas constant R obtained by ities obtained by Petit and Dulong with modern Petit and Dulong (bullets) and Regnault (squares, [38]) ones. This statistical analysis leads him to con- with the modern ones reviewed by Rolla and Piccardi clude that the whole paper is basically a fraud. (open dots, [41]). The dashed lines are the averages of More specifically, he claims that all Petit and Du- the three data sets (excluding sulfur). Open squares long had measured were just the few element dis- are the results for sulfur obtained by Regnault in a fol- cussed in [22], that everything else was fabricated, lowing study (Ref. [42], pag. 344). Specifically, they that no new experiment was performed, so that refer to: 1) the same sample he had studied in 1840 even their detailed experimental description of the (crystallized from melt), measured after two years; 2) setup shown in Fig. (4) (I guess for the first time ‘natural’ crystalline sulfur (not from melt); 3) a sam- ple crystallized from melt after two months, and 4) just after more than a century) is a fake. With what we after crystallization. have seen so far, however, I think we have already good reasons to refrain from making haste to such a severe sentence. As a matter of fact, it is true iments by Petit and Dulong46. The two data sets that the results for iron, zinc, silver and copper are are contrasted in Fig. (6), together with the re- identical to those already reported in 1818. But sults for the same elements obtained by Regnault then, why changing the value for platinum? More- in 1840 [38], which I regard as a useful set for com- over, why not including antimony too, which they parison47. Before we analyze the data, however, did measure in the same work? Halving Berzelius’ let us consider the rather anomalous data for sul- m as they did for many alleged ‘fake data’, would a fur, the element with the lowest value of ma in the have given cp ×ma =0.408, which is not that much plot. In this case, the modern value is 10% lower higher than the other products. Besides, with all than the PD value and more than 18% smaller than we learned about Dulong and Petit and about the the value obtained by Regnault in 1840, discrep- way they worked from the voices of scientists like ancies are far larger than those for the other ele- Biot, Arago, Berzelius, Jamin, Dumas, and Reg- ments. Sulfur is indeed a very peculiar substance, nault, Giunta’s claim sounds to me as a jarring note presenting two main crystalline structures, rhombic in a chorus of great singers. α) and monoclinic (β), but also a spectacular poly- I admit, however, that rebutting (at least in part) morphism that derives from the tendency of this his conclusions requires to deal with statistics. So, element to associate into homocyclic rings contain- let us do it. Actually, the benchmark data used by Giunta are not that modern, since he refers to 46More recent values, such as those reported in the CRC the values of cp reviewed by Rolla and Piccardi Handbook of Chemistry and Physics [36], would anyway dif- fer by, typically, 1% or less. in 1929 [41], but I agree that this is apparently 47 ◦ in the following, I shall indicate quantities relates to the most recent collection of data obtained at 0 C, the modern, Petit and Dulong, and Regnault data with the which was the ambient temperature in the exper- subscripts 0,1, 2, respectively.

16 ing up to 20 sulfur atoms or even into long–chain the total deviation from the average, I did not re- “living” polymers of indefinite length [43, 44]. Al- gard as appropriate to include its Cp value in the though the α structure is only the low–pressure sta- statistical analysis that follows51. ble phase (probably made of rings containing eight Compare first the means of the three data set, sulfur atoms), fast cooling from the β phase or from which show that the values obtained by Regnault the liquid usually leads to the formation of amor- are on the average 6% higher than those by Petit phous sulfur with a ‘plastic’ texture, which has no- and Dulong. Regnault himself pointed out this dif- ticeable influence on the thermal properties of the ference, arguing that some details of the experimen- material. In fact, it took a long time before an tal protocol used by Petit and Dulong, in particu- accurate value for the heat capacity of pure rhom- lar for those elements that they also measured with 48 bic sulfur was obtained . Notably, however, the the method of mixtures, may have lead them to un- problem was already well know to Regnault, who derestimate the heat capacity52. Unfortunately, he in a following paper briefly discussed the difficulty does not seem to be right. While the values he he met in measuring cp for this element. The open obtained are on the average higher than the mod- squares in Fig. (6) show that a sample just crystal- ern ones by 5%, this figure decrease to a skimpy lized from melt displays a consistently higher heat 0.7% (in the opposite direction) for the Petit and capacity, which slowly decreases in time reaching, Dulong’s data. after two years, a value very close to the one he ob- But what Giunta mostly cares about are the fluc- 49 tained for ‘natural’ (not melt-crystallized) sulfur . tuations about the mean. Admittedly, the situation Note in particular that the value for the freshly here is suspicious, since the relative standard devia- crystallized sample is rather close to what Petit Du- tion (the coefficient of variation) CV = s/hci of the long found, suggesting that the sample they used data of Petit and Dulong (CV1 = 0.013) is about was similarly prepared. Even the ‘natural’ sulfur twice larger than that of modern data (CV0 = studied by Regnault has a molar heat capacity that 0.027)53. To an experimentalist like me, this surely is 3% higher than the modern value, which is so smells of data adjustment, although the fact that low with respect to the other elements in the plot the relative standard deviation of Regnault’s data because sulfur has a rather high Debye tempera- (CV2 = 0.032) is only marginally higher than the ture, ϑD = 527 K [46]. In fact, sulfur contributes modern one would also imply that his experimen- to about 1/3 to the fluctuations about their av- tal precision was comparable with that reached 80 erage of the modern data. Obviously, these fluc- years later, which is also a bit strange54. While tuations are not due to any ‘measurement error’, 51 but rather reflect physical differences among the Giunta rises a question of allotropy for tin too, which turns from the usual ductile ‘white’ phase to the brittle ‘grey’ elements, mostly (but not exclusively) due to the tin by lowering the temperature below 13.2◦. However, this different values of their Debye temperatures, which is true only for very pure tin, since even a small amount 50 vary from 87K for Pb up to 386K for Co . What of impurity lowers a lot the transition temperature. Be- I shall call for brevity ‘averages’ and ‘standard de- sides, the transition kinetics is very slow even for pure tin (namely, white tin is highly metastable). Therefore such a viations’ are not therefore statistical estimators of phase change, known in France as la l`epre d’´etain, may well an underlying simple statistical distribution, but have ruined the tin buttons of the uniforms of Napoleon sol- are just parameters quantifying the mean and the diers during the bitter winter of the long Russian campaign, r.m.s. of intrinsic fluctuations: this is a trivial but as recently suggested, but it would have hardly took place within the limited time of a cooling experiment. Hence, very useful observation to comment Giunta’s inferences. likely Petit and Dulong measured white tin. On account of the rather uncontrollable behavior 52Regnault, however, had no doubt on the fact that Petit of sulfur, which gives it a predominant weight in and Dulong did perform the measurements presented in 1819 with the method of cooling. He also points out one of the main cause of errors of this technique, namely the possible 48Even the figure in Ref. [41] differs by 3% from the ac- condensation or vapor on the blackened inside of the vessel, curate value at 0◦C obtained by Eastman and McGavock 8 which would reduce its absorbance. His attentive description years later [45]. suggests that he may well have seen the apparatus shown 49Interestingly, Regnault already suggests that this is due in Fig. 4, arguably when he was a student at the Ecole,´ to ‘incomplete crystallization’. directed at that time by Dulong. 50For the modern data set, the correlation coefficient be- 53If we include sulfur, this figure rises to about 3. 54 tween cp and ϑD (excluding again S) is about 0.57. Including sulfur, Regnault’s data dispersion becomes

17 the two sets of data, which Fig. (6) seems to sug- gest. The extent to which the molar specific heats by Petit and Dulong are linearly correlated with those in Ref. [41] can be estimated from Fig. 7, where Regnault data are also plotted for compar- ison. While the slope s of the linear fit to the P&D data basically vanishes within the error bar for s (which of course means that the average of the P&D data is pretty close to the modern one), the value of the correlation coefficient r ≃ 0.34 witness a moderate degree of correlation. Testing the significance of r for a set of n × n data pints {x,y} is usually done by transforming to the vari- able t = [(n − 2)r2/(1 − r2)]1/2 that, provided that x and y have a bivariate normal distribution, has a Figure 7: Molar specific heats obtained by Petit and Student’s t-distribution with n − 2 degrees of free- Dulong (bullets) and by Regnault (squares), pltted ver- dom. It seems therefore that here we are incurring sus the values given by Rolla and Piccardi [41]. The slopes s and correlation coefficients r of the linear fits in the same problem (non-Gaussianity) we pointed to the two sets of data are shown in the legend. out before. However, in this case stating approxi- mate confidence levels is relatively safer. Baudinet– Robinet has indeed shown that for a sample of un- data adjustment is a reasonable hypothesis, data correlated data of size n ≥ 10, t has approximately fabrication, however, is much less credible: Im- the same Student’s distribution, regardless of the proving data precision by adjusting them is one parent distributions of x and y [47] A simple ap- thing, another one is increasing their accuracy, if plication to our case (n = 12, t ≃ 1.146) shows you do not have a benchmark reference. Namely, that, in spite the relatively small value of r, the how could Petit and Dulong ‘divine’ their data so null hypothesis (namely, that the two set of values well as to agree much better than Regnault with a are fully uncorrelated) can still be rejected with a modern data set that was long to come? level of confidence larger than 70%. Which is not To support his claim that the Dulong and Pe- very high, but far from being negligible. Paying tit data have not been adjusted but truly ‘fabri- Regnault his due, it is however important to no- cated’, Giunta compares their variance σ2 with that tice that the correlation of his own data with the one of modern values by means a standard F -test, modern ones (r ≃ 0.78) is extremely high, which which seems to support a null hypothesis, namely definitely vouches for his scientific fairness. that there is no relationship between the two sets That Petit and Dulong adjusted their data, or of data. However Snedecors’s F -test applies only at least selected among those they obtained the to populations that are normally distributed. As I ‘best’ ones, possibly fearing that someone could warned before, this is far from being the case55. At have ‘stolen’ their result56, is nevertheless a con-

the cost of sounding pedantic, let me stress again 56 that the differences in the observed values of c at a That Dulong and Petit feared plagiarism quite a lot is p evident from a letter of Berzelius to Alexandre Marcet, dated given temperature are not due to ‘errors’: their are 27 April 1819 (just 8 days after Dulong’s presentation at the physical facts, they cannot be reduced by improv- Academy), where he writes “Although I am very close to Mr. ing measurements! Hence, using an F -test (but Dulong, I did not want to get an in–depth knowledge of his even more refined statistical approaches) to test a work, the details of which have not yet been communicated (the memory read at the Academy was only a preview for null hypothesis makes little sense. deterring the thieves whom Paris is supposed to be full of We can nevertheless test correlations between and to preserve the priority of the discovery), I avoided it because, being myself engaged in the publication of a little even smaller than the modern one. work on corpuscular theory, it might well be suspected to 55Giunta claims to have tested that the two data sets are have taken advantage of the advice given by Mr. Dulong, normally distributed. I guess none of my colleagues would and although the contrary is not difficult to prove, since my be so daring with only 13 data points. memory has already been published a year ago in Swedish,

18 crete possibility, which would also explain why a it became evident that the specific heat actually careful experimentalist like Dulong was, according vanishes as T → 0, which of course was totally to Dumas, so reluctant at making the big step. incompatible with classical statistical mechanics58 Conversely, that they cheat to the point of mak- The vanishing of cp low temperature behavior pre- ing up 8 results over 13 sounds speculative, not dicted by the Einstein model, a triumph of the early supported by any serious proof, and definitely in- quantum physics, the introduction of collective vi- congruous with the point of view of valuable con- brations by Debye that provided the correct T 3 lim- temporary witnesses. Hence, if I were a judge called iting behavior, the subsequent refinement by Born upon to decide whether on 19 April 1819 Petit and and van K´arm´an that paved the way to the modern Dulong perpetrated a gross ‘scientific fraud’ by fab- investigation of phonons in solids, is a known story ricating most of their data, I would surely set them to physicists (for an accurate review, see [50]. free for insufficient evidence, although I might not, Our community is probably less acquainted with in good conscience, fully acquit them for having not the at least equally important role played by the committed the crime. The trial, to me, is over. DP law in chemistry, in particular as a key tool to unravel the nature of the atomic weights. We have already seen how Berzelius used the law to 6 Legacy correct some, but not all, of the atomic weights he had measured. What Berzelius could not accept The fate of the DP ‘law’ and its effects on the devel- at all, however, was the law proposed in 1811 by opment of physics are well known. By the middle Amedeo Avogadro [51], stating that a given vol- of the XIX century Regnault had already shown ume of any gases, for fixed values of temperature that elements with a low atomic weight and high and pressure, always contained the same number of melting temperatures like boron, carbon, and sil- molecules. When put together with the gas atomic icon had exceptionally low specific heats at room weights obtained by Dumas (see footnote 31), Avo- temperature. Yet, in 1875 Heinrich Weber showed gadro’s law implied indeed that even simple gases that even the heat capacity of these elements ap- like hydrogen or nitrogen had to be made of di- proaches at high temperature the value predicted atomic molecules. For Berzelius, who believed that by the DP law, which should then be regarded as bonds between atoms always derive from electric 57 a limiting law, . In his words [49], forces, this was clearly untenable and almost pre- posterous: how could two identical atoms with the The three curious exceptions to the same charge bind? As a matter of fact, Avogadro’s Dulong-Petit law which were until now a 59 ideas remained in oblivion for a long time until cause for despair have been eliminated: they were given the place they deserve by Stanislao the Dulong-Petit law for the specific heats Cannizzaro, the greatest Italian chemist of the XIX of solid elements has become an unexcep- century who had studied calorimetry with Regnault tional rigorous law. 60 at the Coll`ege de France . In a letter to the secre- Il Nuovo Cimento Yet, it became very soon clear that Weber’s at- tary of Salvatore De Luca [52], Sunto di un corso di Filosofia Chimica61 tempt to ‘rescue’ the DP law was just a way of get- entitled , ting round the real problem. After all his own data, 58Although a moderate temperature dependence could be obtained by cooling with dry ice, showed that the justified because of anharmonic effects, as already pointed out by Richarz [34]. specific heat of diamond went down by more than 59 one order of magnitude by decreasing T from 1000 Actually, the nomenclature used by Avogadro, who al- ◦ ways refused to use the word ‘atom’, did not help. For to −50 C. In 1905, when Dewar managed to reach instance, what we now call an atom was an ‘elementary temperatures as low as 20 K using liquid hydrogen, molecule’, while ‘constituent’ and ‘integral’ molecules were respectively the molecule of a pure element and of a com- I prefer not to be put in the condition of raising any suspi- pound of different atoms. cions” [32]. 60Cannizzaro went to France to escape from a death 57Weber’s observation was crucial for Thomas Humpidge penalty he had been sentenced to for having actively partic- to find in 1885 the correct atomic weight of Beryllium ipated to the 1848 Sicilian revolution against the Bourbon (at that time also called ‘glucynum’), which shows a very rulers. 61 anomalous value of cp too[48]. Cannizzaro’s seminal letter is meant to be just the sum-

19 Cannizzaro, granting Avogadro’s hypothesis, made Acknowledgements extensive use of the DP law to evaluate accurate atomic and molecular weights. This led him to This investigation would not have been possible formulate his fundamental result, a major step to- without the precious support of free online repos- wards giving physical reality to atoms: itories like Gallica, the Internet Archive, and the Hathi Trust Digital Library, which provide an in- valuable service not only to the community of pro- The various quantities of the same ele- fessional scholars, but also to anyone like me who ment contained in different molecules are is just curious of the way our science blossomed. all multiples of the same quantity that, al- ways entering as a whole, must be called atom.62 References

Cannizzaro presented his ideas at the first global [1] A.T. PETIT, P.L. DULONG, Ann. Chim. Phys. 10 conference of chemists, held in Karlsruhe in 1860. , 395 (1819) One of the conference’s participants was a young [2] A. PAIS, Subtle is the Lord: The Science and the Russian chemistry student, Dmitrii Ivanovich Life of Albert Einstein (Oxford Univ. Press, USA, Mendeleev, who, along with everyone else in atten- 1982) dance, received a copy of the paper by Cannizzaro. Immediately after reading the paper, Mendeleev [3] M.A. AMUSIA et al., Theory of Heavy-Fermion Compounds: Theory of Strongly Correlated Fermi- wrote an enthusiastic letter to his teacher A. A. Systems (Springer Int. Publ., Switzerland, 2015) Voskresenskii in St. Petersburg, mentioning him that he found that all Cannizzaro’s values satisfied [4] R. FOX, The Caloric Theory of Gases from the DP law [53]. The appreciation of Cannizzaro’s Lavoisier to Regnault (Clarendon Press, Oxford, work and of the importance of the DP law had a 1971) crucial consequence in the construction of the pe- [5] S.G. BRUSH, The kind of motion we call heat, riodic table. Mendeleev indeed used the DP law Vol. 2 (Elsevier, Amsterdam, 1976) to correct the atomic weights for indium, cerium, and uranium, which were wrong in his first 1869 ta- [6] P. LEMAY, R.E. OESPER, Chymia 1, 171 (1948) ble63 allowing him to produce the remarkable table [7] J.V. VAN SPRONSEN, Chymia 12, 157 (1967) of 1871 that lasted basically unchanged for almost 70 years [53, 54]. [8] R. FOX, Br. J. Hist. Sci. 4, 1 (1968) As I already stated, I am strongly convinced [9] C.J. GIUNTA, Bull. Hist. Chem. 27, 62 (2002) that they can be charged at most of misdemeanour rather than of a scientific ‘crime’. But even if I am [10] A.T. PETIT, P.L. DULONG, Ann. Phil. 14, 189 totally wrong, even if they really deceived, this has (1819) been one of the most fruitful frauds of the history of science, because the legacy of the work jointly per- [11] R. PIAZZA, Statistical Physics. A Prelude and Fugue for Engineers (Springer Int. Publ., Switzer- formed by an exquisite experimentalist like Dulong land, 2017) and a gifted theorist like Petit is truly invaluable. [12] J.A. DEVILLE (correspondent), Precis Anal. mary of the chemistry course he taught at the University of Travaux Acad. Roy. (Rouen) pp. 138–146 (1844) Genova. 11 62“Le varie quantit`adello stesso elemento contenute in [13] J.C. JAMIN, Revue des Deux Mondes, 2e s´erie , diverse molecole son tutte multiple intere di una medesima 375 (1855) quantit`ala quale, entrando sempre intera, deve a ragione chiamarsi atomo”. [14] F. ARAGO, Œuvres - Notices Bigraphiques, Tome 63 The 1869 values, ma(In) = 75.6, ma(Ce) = 92, Troisi`eme (Gide et J. Baudry, Paris, 1855) ma(U) = 116 were changed to ma(In) = 113, ma(Ce) = 138, ma(U) = 240, which, albeit not perfect, allowed him to [15] P.L. DULONG, Ann. Chim. (Paris) 82, 275 (1812) position these elements in the correct groups of the periodic table. [16] F.L. HOLMES, Chymia 8, 105 (1962)

20 [17] J.B. BIOT, J. Phys. (Paris) XCII, 241 (1821) [38] A.V. REGNAULT, Ann. Chim. Phys. 73, 5 (1840)

[18] A.T. PETIT, J. Ec.´ Polytech. 16, 1 (1813) [39] P. SCHWARZ, Chem. Eng. News 65, 3 (1987)

[19] B. BELHOSTE, Augustin-Louis Cauchy: A Biog- [40] W. DESORBO, J. Chem. Phys. 21, 764 (1953) raphy (Springer-Verlag, New York, 1991) [41] L. ROLLA, G. PICCARDI, International Critical [20] F. ARAGO, A.T. PETIT, Ann. Chim. Phys. 1, 1 Tables (McGraw–Hill, New York, 1929), Vol. 5, (1816) pp. 92–94

eme [21] P.L. DULONG, A.T. PETIT, Ann. Chim. Phys. [42] A.V. REGNAULT, Ann. Chim. Phys. 3 serie 2, 240 (1816) 9, 322 (1843) 76 [22] P.L. DULONG, A.T. PETIT, Ann. Chim. Phys. [43] B. MEYER, Chem. Rev. , 367 (1976) 7, 113 (1818) [44] R. STEUDELL, B. ECKERT, Top. Curr. Chem. 230 [23] D.J. TRITTON, Physical Fluid Dynamics, 2nd ed. , 1 (2003) (Clarendon Press, Oxford, 1988) [45] E.D. EASTMAN, W.C. McGAVOCK, J. Am. 59 [24] I. NEWTON, Phil. Trans. 22, 824 (1701) Chem. Soc. , 145 (1937)

[25] J. LESLIE, An Experimental Inquiry into the Na- [46] C.Y. HO, R.W. POWELL, P.E. LILEY, J. Phys. 3 ture and Propagation of Heat (Printed for J. Maw- Chem. Ref. Data , I10 (1974) man, London, 1804) [47] Y. BAUDINET-ROBINET, Nucl. Phys. A222, 525 (1974) [26] P. PREVOST, J. de Physique 18, 314 (1791) [48] T.S. HUMPIDGE, Proc. Roy. Soc.(London) 39, 1 [27] U. BESSON, Sci & Educ 21, 1085 (2012) (1885) [28] R.C. DOUGAL, Phys. Educ. 14, 234 (1979) [49] H.F. WEBER, Phil Mag., ser. 4 49, 161 (1875) [29] J.B. DUMAS, Mem. Acad. Sci. 42, 37 (1883) [50] M. BLACKMAN, Rep. Prog. Phys. 8, 11 (1941) [30] K. NASH, Harvard Case Histories in Experimen- [51] A. AVOGADRO, J. Phys. Chim. Hist. Nat. 73, 58 tal Science (Harvard Univ. Press, Oxford, 1957), (1811) Vol. 1, chap. 4 [52] S. CANNIZZARO, Il Nuovo Cimento 7, 321 ´ [31] J.C. JAMIN, Cours de physique de l’Ecole poly- (1858) technique, Tome Deuxi`eme (Gauthier–Villars, Paris, 1886) [53] M. KAJI, Bull. Hist. Chem. 27, 4 (2002)

[32] R.A. LEHFELDT, Jac. Berzelius Lettres [54] N.M. BROOKS, Found. Chem. 4, 127 (2002) (Almquist & Wiksells, Uppsala, 1915)

[33] L. BOLTZMANN, Sitzung. Akad. Wissen. (Wien) 63, 712 (1871)

[34] F. RICHARZ, Ann. Phys. Chem. 48, 708 (1885)

[35] R.A. LEHFELDT, A List of the Chief Memories on the Physics of Matter (Taylor and Francis, Lon- don, 1894)

[36] C.R. HAMMOND, Handbook of Chistry and Physics (CRC Press, Boca Raton, 2007), chap. 9

[37] J.J. BERZELIUS, Trattato di Chimica (transla- tion by G. Guarini), Vol. 5 (Puzziello Tipografo Libraio Editore, Napoli, 1839)

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