Dulong-Petit’s law and Boltzmann’s theoretical proof from the Kinetic Theory of Gases

The main aspect that we will address in this work is the introductory presentation of the empirical law of the specific heats of material bodies, published by Dulong and Petit on April 12, 1819, in the magazine Comptes Rendus. The analytical demonstration of this law appeared for the first time in 1866 in the Doctoral Thesis of the Austrian physicist Ludwig Boltzmann, according to the research carried out by Cássio C. Laranjeiras and presented in his Doctoral Thesis in 2002. In the introduction we discussed how Dulong’s law and Petit played a key role in the subsequent development of a new physics (Quantum Mechanics), when Max Planck changed his research program from electrodynamics to thermodynamics, evidenced in the book: Black Body Theory and the Quantum Discontinuity, 1894-1912, published by Thomas Khun in 1978. Between 1900 and 1912, we observed that fundamental works published by Albert Einstein, Henri Poincaré and Peter Debye explained the mathematical relationship of this empirical law based on Planck’s hypothesis of the quantum of action of blackbody radiation. Despite the controversy over the authorship of the law, whether it would have been formulated in advance by Arago, we agree with Robert Fox that it was the results of Dulong and Petit’s experiments that led to the formulation of the law.


Introduction
Current research [1] [2] in the history of thermodynamics allows us to state that Dulong and Petit's law can be considered as the starting point for changes in Newton's mechanics to quantum mechanics, since the search for a theoretical formulation of this law triggered numerous researches in an attempt to explain this empirical law.Pierre Louis Dulong (1785Dulong ( -1838) ) and Alexis Thérèse Petit (1791-1820) made meticulous measurements between 1815 and 1819 that led to the determination of an important law of nature on the thermal behavior of material bodies.The determination of specific heat, both from an experimental and theoretical point of view, can be considered as a fundamental prerequisite for structural changes in the phenomenology and epistemology of physics in the late 19th century.It served as a precursor to the interpretation and mathematical modelling of microphysical phenomena, setting the stage for the introduction of Planck's hypothesis of blackbody radiation in 1900.
Boltzmann published preliminary studies on this topic in the years 1866 and 1884 [3], based on Maxwell's work in relation to the Kinetic Theory of Gases and the distribution of velocities.Thomas Kuhn [4] considers these works fundamental for the further development of Quantum Theory.In this sense, Boltzmann is, when reinterpreting the laws of thermodynamics based on the notions of probability, statistics, and entropy, one of the main theoretical contributors of Quantum Theory, as well as Poincaré [5], whose participation in the first Conseil Solvay de Physique (Solvay Council of Physics) in 1911 was fundamental to the later developments.We fully translated from French to Portuguese the 1912 article's Poincaré and published it in the Doctoral Thesis [1].Charles Galton Darwin published a commented English translation of Poincaré's article to English and it published in 1913, which 3rd World Conference on Physics Education Journal of Physics: Conference Series 2727 (2024) 012009 IOP Publishing doi:10.1088/1742-6596/2727/1/012009 2 contributed to the acceptance of Planck's theory of black body radiation, since English scientists were more likely to accept Wien's theory.
Planck introduced the constant h when starting his research on blackbody radiation, it is not possible to describe it at this time, since the beginning of the new theory (quantum theory) is established from two publications held in December 1900 [6].
Based on the Kinetic Theory of Gases, Boltzmann theoretically demonstrated the empirical law of Dulong-Petit for ideal gases.Einstein in 1907 [7], when deducing an adjustment equation for the behavior of solid bodies, proved to be valid the notion of quantization of harmonic oscillators proposed by Planck in 1900.In 1912, Peter Debye presented a theoretical formulation that, in a way, made a fundamental adjustment to the law of specific heats of material bodies [8].
In the next topic, we will address the conceptual character of the Dulong-Petit law, analyzing the original publication [9] in French, as well as some controversial aspects about whether this law was conceived by both or if it belonged to François Arago (1786-1853) [10].This law was discovered during a period when there was a fruitful debate about the corpuscular and undulating interpretation of the theory of heat [10].

The Dulong-Petit law and the controversy with Arago
According to Fox [10], what the history of physics knows as Dulong-Petit's law, in fact, would have its origin in Arago's research.The 1881 account of French chemist Jean Baptiste Dumas [11] [12] seems to cast some doubt on the discovery of the law of specific heats for solid bodies.
Monday, April 5, 1819, a memorable date, Petit... secretly showed his brother-in-law Arago a piece of paper, containing the reports of how the simple bodies combine, and the quantities of heat required for each to heat in the same way, having the same weight.At first, it seemed pure disorder, but when the two columns multiplied, all the products led to the same result.An hour later, the illustrious perpetual secretary, convinced that Dulong, always hesitant, could oppose the disclosure of this beautiful law, spoke of this to his colleagues, for a calculated indiscretion.Eight days later, the two collaborators announced themselves to the Academy... [12] Fox omitted the passage in which Dumas comments about the misfortune of Petit's premature death, "..., dont un an plus tard la science déplorait la mort prémature,".The correct page of Dumas's text would be xlviij, not xlviii, as it is in the footnote to Fox's article.The publication date of the text also appears to be incorrect.Dumas's text was published by the French Academy of Sciences on March 14, 1881, and Fox quotes it as being published in 1883.Perhaps Robert Fox's quotes are from publications after the original date.We are going to analyze Petit and Dulong's text of 1819 on the specific heat for solid bodies from the historical point of view, as it is an important empirical law that does not appear in basic education textbooks and that in some way is associated with theoretical modification in basics and mechanical interpretation of natural phenomena.Robert Fox considers this statement by Dumas suspicious, as he gave this account 60 years after the episode when Dumas was 18 years old and working in Geneva, moreover, all Dumas biographers claim that he came to Paris for the first time in 1823, about three years after Petit's death, and this strongly suggests that Dumas' account of the discovery of Dulong and Petit's Law had its origin in Arago.Petit was Arago's brother-in-law, and both published an article on December 11, 1815, with the title: Sur les Puissances réfractives et dispersives de certains liquides et des vapeurs qu'ils forment (On the refractive and dispersive powers of certain liquids and the vapors they form), it would be inevitable that Arago did not exert his scientific influence on his research and perhaps he also exerted it on the law of specific heats of material bodies.The period between 1810 and 1830 is the most scientifically fruitful for Arago, he was appointed professor at the L'Institut (L'Académie des Sciences) at the age of 23 and in 1810 he had made an important discovery about the polarization of light.According to Léon Foucault (1853), Arago had more ideas by himself than a full generation.
3. The Dulong-Petit law -the article from 1819 Petit and Dulong's article, Sur quelques points importants de la théorie de la chaleur (On some important points of the theory of heat), it can be considered historical, since it describes the empirical construction of a fundamental law of Physical Chemistry: the microscopic behaviour of atoms and molecules from the thermal point of view.This article is the conclusion of a series of research between 1816 and 1818, in which their measurements are reported, using mercury and air thermometers, in the measurements of copper, iron and platinum, in addition to issuing their opinions about from the works of John Dalton and John Leslie.According to Petit and Dulong, the William Irvine and Adair Crawford hypothesis would be associated with the state of molecular aggregation.Crawford admitted that this hypothesis, based on his theory of animal heat [13], is opposed to the observed facts, therefore, its application would be difficult.Even Dalton's hypothesis, according to which, the amount of heat of elementary particles, being the same in the elastic fluids, would have a difficult application, because, one would have to know the number of particles contained in the same weight or in the same volume in various gases.In the 17th and 18th centuries, elastic fluids were interpreted by Boyle according to the idea of "spring of the air".When introducing the mechanical basis of the air spring, Boyle compared the air to a ball of wool, which could be strongly compressed, but which would expand again if the compression force ceased.
Dulong and Petit rejected the theory proposed by Dalton because it presented only theoretical results, in addition to observing the inaccuracy of most measurements of physicists and chemists, except for those made by Laplace and Lavoisier, through their calorimeter, being very accurate, but in small amount.On the other hand, using the theory of elastic fluids, Delaroche and Bérard [14], obtained expressive results for the calculation of specific heats in gases.J. L. Heilbron [15] describes in detail the experimental path of determining the specific heats of material bodies and experimental apparatus of various scientists, in determining this important constant in the thermal behaviour of materials.In this sense, when historians of science seek to critically describe the theoretical and experimental development in the work of scientists, they are seeking to dialogue with the past, so that it can build a rational image to be used as a methodology in the teaching and learning of natural science.So, Natural History Museums constitute a fundamental element of knowledge appropriation by students and teachers but visiting as observers of the built apparatus is not enough.For scientific concepts to become clearer, students and teachers must also learn to reconstruct past experiments, as well as propose new laboratory experiments.
In the first paragraph, it is observed that the objective of Dulong and Petit is to establish a law for the specific heats of material bodies as a function of measurements of their temperatures (Les atomes de tous les corps simples ont exactement la même capacité pour le chaleur (The atoms of all simple bodies have exactly the same capacity for heat), describing the proportion of chemical compounds and, consequently, the molecular constitution of these compounds.An important aspect to be considered by French physicists that may have influenced the later work of Boltzmann and Poincaré in thermodynamics is the mention of the use of the corpuscular theory of probability.In fact, Boltzmann when describing mathematically in 1866 the law of Dulong and Petit gives evidence of the introduction of this theory in the universe of molecular structures.In 1816 Arago was appointed professor at l'École Polytechnique, and in 1818 he added a course in social arithmetic, a new discipline that applies the calculus of probabilities developed by Laplace to demography and economics.
The success which we have already obtained makes us hope not only that this kind of considerations will be able to contribute powerfully to the further progress of physics, but that the corpuscular probability theory, in its turn, will receive from it a new degree of probability, and that she will find in them certain means of discerning the truth between equally plausible hypotheses.[9] The statement of the empirical law of the specific heats of material bodies appears on page 405 of the 1819 article and can be stated as: Les atomes de tous les corps simples ont exactement la meme capacite pour la chaleur (The atoms of all simple bodies have the same capacity for heat).According to the measures developed by Dulong and Petit: "The law which we have just stated appears to be independent of the form which the bodies affect, provided, however, that we consider them under the same circumstances".According to Dulong and Petit, Irvine and Crawford considered that the amount of heat contained in bodies was proportional to their capacity, that is, the sum of their elements.With this, Irvine and Crawford concluded that whenever the specific heat of a compound is higher or lower, an absorption or release of heat must occur at the time of combination.On the other hand, as previously described, John Dalton admitted the idea that the amounts of heat attached to the elementary particles of elastic fluids would be the same for each one of them, which was rejected by Dulong and Petit as they showed in their experiments.
Dulong and Petit considered that when carefully established, temperature measurements in the processes of melting ice or mixing material bodies with water can lead to very accurate results, but when experiments were carried out with many substances, success makes up harder.In this case, they adopted a new method of cooling.According to this method, first applied by Meyer, it was known that there were different rates of cooling when bodies were subjected to the same circumstances, allowing to obtain the thermal capacities as a function of cooling, since they differed very little for homogeneous substances in relation to the mixtures.However, they observed that the ideal would be to use small volumes, eliminating uncertainties and imprecisions, because, for small volumes, heat losses occur more slowly -"puisqu'en diminuant la masse d'un corps on augmente la vitesse avec laquelle sa chaleur se dissipe" (since by decreasing the mass of a body we increase the speed with which its heat is dissipated).They concluded with this adjustment that the heat conductivity measurements of substances no longer have any perceptible influence on the measurement of heat capacities.
After describing the adjustments in the experiment (paragraphs 7 and 8), Dulong and Petit present a table with three columns: one with the values of the specific heats of the pure substances, another with the relative weights of the atoms and a third with the products of the weight of each atom by corresponding capacity.Below is a description of Dulong and Petit's adjustments and comments, an important point of experimental change and obtaining the empirical law.
To this first means of slowing down the cooling, without causing the measurements to lose the precision which they must preserve, we added another, the influence of which we could calculate by knowledge of the laws of the communication of heat.It follows from these laws that the rate of cooling of a body can, all other things being equal, be considerably diminished when its surface possesses only a very weak radiant power, is that it is immersed in an air extremely dilated [hot].To realize these circumstances, we have decided to operate on solid bodies only after having reduced them to very fine powder.In this state, they were enclosed and tightly packed in a very thin cylindrical silver vase, of very small capacity, the axis of which was occupied by the reservoir of the thermometer which served to indicate the progress of the cooling.The numbers (1) (La chaleur spécifque de l'eau est prise pour unite) and (2) (Le poids de l'atome d'oxigène est supposé égal à un) in parentheses refer, respectively, to the specific heat of water, which is taken as a unit and to the weight of oxygen atom equal to one.
When comparing the results of the specific heat measurements for oxygen and nitrogen gases with the measurements of Laroche and Berard, they conclude that the difference is minimal and that in relation to the hydrogen atom, even if the measurements present a value a little below, with the new corrections established by Laroche and Berard, a consensus result is reached.The text below expresses the confidence that its results were satisfactory, and that the announcement of the new law could not be postponed.
The numbers they report for oxygen gas and nitrogen gas only differ from what they should be to strictly agree with our law, by an amount less than the probable errors in such experiments.The number relating to gas it is true, a little too low; therefore, by carefully examining all the corrections that the authors have been obliged to subject to the immediate results of observation, we soon recognize that the rapidity with which the hydrogen gas puts itself in temperature equilibrium with the surrounding bodies, compared to the other elastic fluids, must necessarily have brought, in the determination relative to this gas, an inaccuracy from which they did not seek to guarantee themselves; nor have the surrounding bodies, compared to other elastic fluids, must not have sought to guarantee themselves; and by evaluating this cause of error as much as possible, we explain the difference in question without being obliged to make any forced assumptions.[9] After presenting the table, Dulong and Petit admit that there is no precise way to determine the total number of atoms that combine during molecular construction, but, according to the experimental data, it is possible to build the empirical foundations of the specific heats of the solids.It is possible that Boltzmann had some direct or indirect reading of this statement, which motivated him to find a theoretical expression to justify the number of atoms involved in the molecular structure.

Boltzmann's theoretical proof of the Dulong-Petit law -1866
We will approach subject III of the Boltzmann's Doctoral Thesis (Über die mechanische Bedeutung des zweiten Hauptsatzes der Wärmetheorie -About the mechanical meaning of the second law of the theory of heat) with the title: Begründung des Ampèreschen, Dulong -Petitschen und Neumannschen Gesetzes für Gase -Justification of Ampère's, Dulong-Petitschen and Neumann's laws for gases.Ludwig Eduard Boltzmann (1840Boltzmann ( -1906)), apparently he was the first physicist-mathematician to theoretically demonstrate, in his doctoral thesis, the empirical law of Dulong and Petit [2].The mathematical discussion of the proof can be observed in topic III, where Boltzmann uses the expression P from item II as the ratio between the Kinetic Energy Integral of the molecule and the Time Integral.
In previous topics, Boltzmann used a series of arguments regarding the principles of conservation of energy and molecular linear momentum to define the concept of temperature as a living rational state of the ratio between kinetic energy and time, according to the kinetic theory of gases.Furthermore, Boltzmann based his argument on the concept of molecular disaggregation introduced by Clausius in 1862 [17].Mathematically, molecular disaggregation is defined as the quotient between external and internal work by the temperature of the state, that is, it would be the measure with which molecules are dispersed during interaction processes within the system.Clausius concluded that the effect of heat always tends to weaken the connection between molecules and increase the average distance between them.Boltzmann defines the energy (vis viva) of molecules by the relationship mc 2 /2, and eliminates for the first time the presence of the ether from the intermolecular medium of gases but keeps it in the interior space of the atom, which seems to contradict its final conclusions on this topic.
It is necessary, for the first time, to abstract the presence of the ether in the gas, whereas previously the ether filled the space inside the atom.The product of the pressure for the volume of the gas would then be given by pv = 2NP/3 = Nmc 2 /3, according to a formula derived from Krönig, Rankine and Clausius.
This position adopted by Boltzmann can be considered complex.Why does it keep the intermolecular space unoccupied by the ether, while inside the atom the ether is present?Perhaps, to facilitate the collision processes and consequently the heat transfer and the infinitesimal variation of the temperature in the thermodynamic equilibrium.Poincaré, in his 1912 article, will deduce Planck's law in three different ways without making any conjecture about the presence of ether in the processes of mechanical shocks, however, he does not abstain from trying to develop a theory that would lead to the same results in the presence of ether, according to the Doppler-Fizeau theory.Historically, it would be necessary to consider this momentary "abandonment" of ether by Boltzmann and to see if this had implications for Albert Einstein's theory, influencing him to abandon it also in favour of the Special Theory of Relativity.
On the other hand, Boltzmann was aware that the molecules should not be rigid and that their elastic deformation would imply emission of radiation, just like the collisions between rigid bodies that emit sound.If these collisions were to occur constantly between the gas molecules, at some point they would totally lose kinetic energy to the ether.Boltzmann imposed the condition of thermal equilibrium between the molecules and the ether partially justifying the excerpt mentioned above.A justification for the elastic molecules in equilibrium with the ether was given by Marcel Brillouin and it may have been one of the factors that led Louis de Broglie to conceive of the electron as a pulsating membrane.Marcell Brillouin's attempt to justify the presence of ether in mechanical movements took place around 1893-94 [1], a period in which he published several articles on the subject.
Boltzmann considered that if the product pv=T, is constant when applied to the first law of thermodynamics, it can also be considered constant when applied to the second law, with the temperature being adjusted proportionally to the average living force -mittleren lebendigen Kräften (medium vital forces) of each atom.Considering that the system is composed of N molecules, containing atoms from a simple substance, forming the first element and atoms ܽ ଵ , as the second element, successively, Boltzmann wrote pv as a function of T = mc 2 /2, (German energy is also spelled as Tatkraft, maybe that's why Boltzmann used the letter T), according to the expression below.
getting pv/T = 2N/3, which is Ampère's law to calculate the number of molecules for any gas, and writing the heat according to the expression below: The equation (2) above, is a derivation already obtained by Krönig, Rankine and Clausius.It shows that Boltzmann still preserves the essential parts of the "old" thermodynamics.The equation (3) represents the amount of heat [lebendige Kraft -vis viva] that would be equivalent to the work that is added to the system by providing a variation of external heat.
In the sequence, Boltzmann works algebraically on the relations and makes conjectures about the addition of heat in the system until reaching the formula for calculating the number of atoms present in the molecule at constant volume and temperature.Boltzmann knew Loschmidts result of 1865 which was the first estimate of the measurement of the size of the air molecule.The heat capacity (specific heat) of the molecule was in accordance with the relationship below.
3rd World Conference on Physics Education Journal of Physics: Conference Series 2727 (2024) 012009 γ is therefore constant as long as we assume that the volume units were obtained at the same temperature and pressure, proportional to the number of atoms in the molecule, therefore, according to the previous law, proportional to the sum of the volumes for simple gases, which form the volume of the substance.Masson considered this law in his treaty "Sur correlation des propriétés physiques des corps" -("On the correlation of physical properties of bodies") confirmed experimentally, but with a few exceptions.It also follows that the specific heat at constant volume, in relation to weight, is proportional to the number of atoms in the molecule divided by the equivalent [gram] of the body.[3] Boltzmann quotes the work of M. A. Masson on the correlation between the physical properties of bodies, certainly, paragraph II of topic II -Détermination du rapport des chaleurs spécifiques des gaz (Determination of the relationship between the specific heat of gases) and the paragraph III -Notions sur la théorie méchanique de la chaleur (Understanding the mechanical theory of heat).A more detailed discussion of the consequences of the correlation of the speed of sound in gases proposed by Masson would require another article.Let's just use these results to justify Boltzmann's proof.
After this brief account of the consequences of Masson's experiments and Dulong's research, we emphasize that Boltzmann was in possession of some important theoretical and experimental results to develop his own theory on the determination of specific heats in gases.Boltzmann found an expression that made it possible to determine the number of atoms in the molecule.Probably he was already aware of the article by Loschmidt that allows determining the size of the molecule of the substance, although he did not quote it in his thesis.The relationship below was obtained analogously as Masson, who determined Dulong's law for vibrations in gases: where: shows that the ratio of pressure to temperature is proportional to the number of atoms in the molecule.γ is, therefore, constant, since we assume that the volume measurements were obtained at the same temperature and pressure, proportional to the number of atoms in the molecule, therefore, according to the previous law, proportional to the sum of the volumes of the simple gases that form the volume of the substance.[3] The final part of the topic III Boltzmann's thesis presents relevant discussions from the point of view of the foundations of molecular mechanics based on the kinetic theory of gases.Boltzmann makes considerations about the importance of the presence of ether in the molecular construction process from the absorption of heat by the substance.The negligence of this substance, according to Boltzmann, would not be decisive to alter the experimental results and consequently the theoretical construction of the law of specific heats of gases.Adjusting Ampère's law and observing that Masson's law, determined from the propagation of sound in a closed tube, containing a certain gas is an exception, that is, even with the experimental results updated in relation to the previous ones, Boltzmann admits being premature the establishment of an atomic-molecular theory for gases; the Clausius entropy hypothesis remaining, as an alternative to the previous ideas of Dulong-Petit and Masson.
This formula would allow an immediate calculation of the number of atoms in the molecule; here, however, a peculiar difficulty arises in which the calculation, assuming γ'/γ for air and most simple gases = 1.411, provides a number η slightly greater than 1 and 1/2, a result that is generally it is not correct, but it is conceivable that we have to assume that the gases in question consist of two parts, one of which contains 1 atom in the molecule and the other part, 2 atoms in the molecule.This disagreement is not very likely to result from the neglect of the etheric mass present in gases [...].If we disregard the absolute value of these numbers, we find that they do not behave like the atomic numbers of the molecule, calculated according to Ampère's law, but as normally 1 or 2 larger integers, although it seems strange that they do it this way.Masson's values for (γ'-γ)/γ, determined from the sound propagation in a tube containing gas, is an exceptional behaviour, as is the number of atoms.If we add that the numbers found by Dulong behave very differently, not to mention previous attempts, it must be admitted that the experimental data are not yet 3rd World Conference on Physics Education Journal of Physics: Conference Series 2727 (2024) 012009 IOP Publishing doi:10.1088/1742-6596/2727/1/0120098 in agreement, to provide reliable theoretical conclusions about the composition of the gases.However, the Clausius hypothesis seems to be the most likely, for simple gases, at least 2 atoms, when combined to form a molecule.[3] 5. Contemporary Perspectives on the Dulong-Petit Law An interesting and important question to be considered is how the Dulong-Petit-Gesetz law [18] is interpreted, applied and/or modified contemporaneously in engineering, but mainly in relation to its teaching and learning at basic and higher levels.In this case, we will analyze some recent articles, where discussions of this nature can show us how the Dulong-Petit empirical law, meticulously determined by French physicists in 1819, because of their research between 1815 until the publication of the final article, continues to be valid in the molecular level from the new physics, Quantum Mechanics.
The theoretical implications presented in Boltzmann's demonstration in 1866 of the Dulong-Petit law preserve arguments from Classical Mechanics and the application of probability and statistics has not yet been observed, however, with the introduction of the quantum of action by Planck in 1900, followed by the Einstein's 1906 article made it possible to construct a rational explanation for this law within the scope of solid-state physics.Would it be possible for it to be valid at the molecular level for gases and liquids?
From the point of view of technological application, the Dulong-Petit law is preserved when obtaining new materials, whose role is fundamental in relation to the interaction of radiation with matter, for example, in the science and industrialization of thin films.Kholder et al (2010) in the article: Second Wind of the Dulong-Petit Law at a Quantum Critical Point, demonstrates that the contribution to the specific heat coming from the boson is part of the free energy due to the "zero sound" mode (name given by Lev Landau in 1957 to unique quantum vibrations in Fermi quantum liquids) transverse follows the Dulong-Petit's law.In the case of the two-dimensional liquid of 3 He, the specific heat becomes independent of the temperature that characterizes a state, within a few millikelvins.They further comment that renewed interest in the physics of 3 He has been stimulated by the experimental observation of the non-Fermionic liquid behavior of dense 3 He films at low temperatures [19].
The fundamental question, based on the report above, is: how to create teaching and learning strategies for new physics concepts, perhaps much more complex today, for high school students and for students who are not in a regular physics course?We argue that the History of Science can create learning objects from the investigation of primary sources and thus allow students and teachers to understand how natural science concepts evolved.
Brazilian physicist César Lattes (1924-2005) was a researcher in particle physics and responsible for detecting the π-Meson (particle carrying nuclear force), whose theoretical model was proposed by Japanese physicist Hideki Yukawa in 1935.Lattes discovered this particle based on his modifications to the technique of Cecil Frank Powell  during his stay in Bristol.Lattes carried out his experiments in Bolivia, at the peak of Midi in Chacaltaya in 1947.The Brazilian physicist argued that without history there is no objective reality and that every theory is provisional, and a better one may appear, but an empirical result is not provisional, and may one appear with greater precision.In this case, laboratory experimentation in many cases obtains more refined results, requiring a bibliographic review of the theory based on the analysis of the mathematical foundations used in its construction.Understanding these new ideas requires modifications in learning strategies during the didactic transposition process.
In the article, Entransy analysis of irreversible heat pump using Newton and Dulong-Petit heat transfer laws and relations with its performance [20], E. Açıkkalp proposes a heat transfer analysis model based on the concept of entransy (a physical quantity that describes the heat transfer capacity).This concept was introduced into physics on February 26, 2007, by a research group led by professor Jianfeng Guo from Tsinghua University, its main purpose would be to study optimization processes during heat transfer between material bodies [21].The Dulong-Petit empirical law appears again in another scientific article in the context of thermodynamic applicability.In the article: Thermal stability analysis of nuclear and fossil fuel power plants including the Dulong-Petit heat transfer law and economic characteristics [22], Ortega et all discuss the Dulong-Petit law as an application in several technological areas, preferably because there is a growing interest in achieving low-cost electricity production.In this sense, if on the one hand we try to establish an optimization of analysis criteria with economic parameters, on the other hand, it is necessary to study how the thermal stability of plant models can be more efficient from a thermodynamic point of view when heat transfers occur.
Scientific articles of this nature have the merit, when applied to teaching and learning new concepts, during a didactic presentation by the teacher in the classroom, to show that science and its application in technological engineering can contribute to economic and social development.In the contemporary world, it is increasingly necessary for students in formal education, and society as a whole, to understand the motivations of scientific research and its applications.
Again, we realize that as the concepts of natural sciences are introduced as elements to be grasped by students and teachers, the History of Science becomes necessary and the investigation of primary sources can produce learning objects that make them clearer and, prevent conceptual error during the didactic transposition process.However, it is necessary for the Historian of Science to take precautions in relation to errors in conceptual understanding when introducing a new concept by the scientist.Anachronism cannot be used as a practice -which basically consists of using the concepts and ideas of one era to analyse the facts of another historical period.In other words, anachronism is a mistaken way in which we try to evaluate a certain historical time (potentially in the history of science) in the light of values that do not belong to that same historical time.Scientists when reading scientific articles (primary sources) in their respective areas may make historical errors or when reading secondary sources, they may also be misled.Emilio Segrè (1905Segrè ( -1989)), Italian physicist, and the Austrian philosopher of science Karl Popper , when commenting on the concept of wave-particle duality, made a historic error, attributing to Einstein a concept that he did not construct theoretically [ 23] [24].
Detailed studies carried out by scientists who became historians of science through the practice of historical research have proven to be very reliable examples; This is the case of the excellent scientific bibliography on Boltzmann, published by Olivier Darrigol [25].In this detailed description of Boltzmann's scientific publications, Darrigol clearly exposes the evolution of the Austrian physicist's scientific thought.In many cases, the analysis of letters exchanged by scientists can reveal important chronological information from the point of view of research in History of Science and, having more detailed information can be important when transferring new knowledge to students and teachers who are not professionals in History of Science.Historical curiosity is an element that is present in the social majority and can motivate students to investigate in detail.
Another equally important scientific biography, like the one written by Olivier Darrigol about Boltzmann, is the excellent book by British professor and historian of mathematics Jeremy Gray [26].In it, the professor reports in an exceptional way the process of scientific construction developed by Poincaré, considered the last classic physicist-mathematician. Poincaré made numerous contributions to various branches of science.In 1889 he received an important prize from the Swedish crown for the "solution of the three-body problem", in fact, Poincaré highlighted that the problem was not correctly established, and proved that the complete solution could not be found.Poincaré participated in the first Solvay Council in 1911 and was fundamental for the acceptance of Planck's hypothesis of body radiation on the European continent, but mainly in England, where Wien's law was more accepted.His main arguments were based on Thermodynamics.

Final considerations
Dulong and Petit's law was established through meticulous experiments between 1815 and 1819 and the experimental results obtained by French physicists provided important elements for the tentative theoretical explanation proposed by Boltzmann in 1866.He used the kinetic theory of gases to explain the law theory of specific heats, Clausius's idea of entropy and the consequences of correlations between the speed of sound and the properties of gases proposed by Masson, reinterpreting the second law of the mechanical theory of heat.Although he still had his reservations about a final theory of molecules, structures, and interactions, mainly due to the momentary abandonment of the ether, Boltzmann initiated a new way of interpreting the interactions between radiation and matter, paving the way for Newton's mechanics to be completely modified.
We consider that during the teaching and learning process, didactic transposition requires a certain amount of care with the use of the History of Science as a learning object [27].The student will not always obtain better learning results with this teaching situation.The teacher needs to have good training in this area to understand how scientific concepts evolve and know exactly when he can use the History of Science as a learning situation.The important thing is to emphasize that science is another form of social construction and that the figure of scientific genius is a mythological figure.The current reality of science "requires" teamwork or research groups, as the amount of information is increasing.