A Revised Interpretation of the Thermodynamic Theory Including the Einstein Mass-Energy Relation E = mc 2

Thermodynamics being among the most synthetic theories of physics and the mass-energy relation E = mc 2 among the most general equations of science, it is somewhat surprising that this latter is not explicitly present in the laws of thermodynamics. Coupling this observation with the conceptual difficulties often felt in learning thermodynamics leads to the idea that both situations may have the same cause. On the basis of these clues, this paper is intended to provide complementary arguments to a hypothesis already presented. It consists of showing the existence of an imperfect compatibility between the conventional formulations of the first and second laws of thermodynamics and suggesting the need of the mass-energy relation to solving the problem.


Brief Review of the Concepts of Reversibility and Irreversibility
The system we are considering ( Figure 1) is a gas contained in a cylinder permeable to heat (i.e. diathermal), and equipped with a mobile frictionless piston, which is also diathermal. In state 1, the external pressure P e and the internal pressure P i (pressure of the gas) have the same value P 1 . Then, due to the deposition of a mass of sand on the piston, P e passes from the value P 1 to a value P 2 , and the piston moves down with the consequence that, in state 2, P i also reaches the value P 2 . The sand can be deposited very rapidly (i.e. all at once) or very slowly (i.e. in several successive small batches).
In the first case, P e reaches immediately the value P 2 , while P i reaches this value only when the piston stops. It is well known that this situation is designated irreversibility in the sense that P e and P i are always different during the process.
In such conditions, the work done on the gas by the piston, for an elementary change in volume noted dV, must be written In the second case, by contrast, the pressure exerted by the piston passes very gradually from the initial value P 1 to the final value P 2 , with the consequence that, at each moment of the process, P e and P i can be considered as equal. This situation is designated reversibility and, in such conditions, the elementary work done on the gas, by the piston, is given by the relation: For a given value of dV, the difference between dW irr and dW rev takes the form: that can also be written: Referring to Figure 1, it appears that we have dV > 0 when P i > P e and dV < 0 when P i < P e . (this second situation being the one illustrated by the diagram). As a consequence, the term ( ) meaning itself that we have always: Another possible expression of Equation (4) is therefore: where dW add can be regarded as an additional energy representing the difference between dW irr and dW rev and obeying the condition: In practice, it is well known that the additional energy, noted here dW add , mainly appears in the form of heat, classically designated dQ, that is evacuated towards the surroundings. This heat evacuation explains that the final temperature of the gas has in fact the same value as its initial temperature, that is also the temperature of the surroundings.

Remarks Concerning the First Law of Thermodynamics
For the matter that will be discussed in this paper, the information we need about the first law is summarized in Equations (9), (10), (11), (12), and (13) given below. The first law of thermodynamics is a postulate which consists in admitting that when a system passes from a determined state 1 to a determined state 2, its internal energy variation, noted dU, always obeys the relation: If the energy exchanges are limited to work (W) and heat (Q), the detailed meaning of Equation (9) takes the form: If the system taken in consideration is isolated, the complementary precision given by the first law is that Equations (9) and (10) become respectively: Knowing that an isolated system has no exchanges of energy with its surroundings, all the terms of Equations (11) and (12) refer to energy exchanges occurring inside the system.
We have seen with Equation (6) that dW irr and dW rev obey the general relation: Combining this data with the information given by Equations (9) and (10), leads to the conclusion that the terms dQ irr and dQ rev must themselves obey the relation: In thermodynamic textbooks, the considerations concerning dQ are generally discussed in the chapters dealing with the second law. We will now enter in this context in order to see if the relation dQ irr < dQ rev given by Equation (13) can really be verified.

Remarks Concerning the Second Law of Thermodynamics
As was the case for the first law, the concepts of reversibility and irreversibility remain of fundamental importance for the understanding of the second law, but they are associated here with the complementary concept called entropy. Its presentation is often done in the following condensed form.
In conditions of reversibility, a change in entropy (noted dS) is linked to a change in heat (dQ) and to the absolute temperature (T) by the relation: In conditions of irreversibility the corresponding relation becomes: When we discover these equations for the first time and receive the complementary information that dS has the same value in both cases (because S is a state function, as well as V, the volume), we are tempted to think that dQ represents dQ rev in Equation (14) and dQ irr in Equation (15). With this idea in mind, we are led to deduce that these terms obey the relation: which is the one suggested by Equation (13), at the end of the previous section. At this stage of the presentation, the reasoning seems coherent and easily understandable. The problem is that the relation dQ irr < dQ rev is not confirmed when we examine the numerical applications presented as examples of the use of these equations (one of them, very simple, is mentioned below in paragraph 2.2).
The reason is due to the fact that, in this context, we discover that the precise meaning of Equation (14) is really: but the precise meaning of Equation (15) is interpreted as being: complemented by the information that the term dS i , designated as the internal component of entropy, has always a positive value and therefore can always be written: A related piece of information is that the term dS = dQ/T e is designated dS e and called the external component of entropy, so that a condensed writing of Equation (17) is: Although the term dQ of Equation (15) is sometimes designated dQ irr in thermodynamics textbooks, the numerical value given to dQ in Equation (17) is really dQ rev , as will be seen further with Equation (28) that refers to a typical numerical example. The conclusion which follows from this observation is that the term dQ irr is not present in the equations going from number 14 to number 19, so that its comparison with the term dQ rev is not possible at the moment.
The situation was different for the comparison between the terms dW irr and dW rev introduced in paragraph 1 because their definitions, given by Equations (1) and (2), were directly presented in the form of energy equations.
To obtain the conversion of the entropy equations (16) and (17) into energy equations, the solution consists in multiplying both terms of Equation (16) by T i , and the three terms of Equation (17) by T e . The results are: For Equation (16): In the same manner as, in Equation (7), we have designated dW add the term representing the difference dW irr -dW rev , we can here adopt the designation dQ add for the term T e dS i , so that an alternative writing of Equation (22) is: Having noted with Equation (17) that dS i is always positive and knowing that the same is true for T i and T e (because, in thermodynamics, temperatures are always expressed in Kelvin), the term T e dS i , also noted dQ add is itself positive.
Consequently, it can be derived from Equation (23) that dQ irr and dQ rev are linked together, not by the relation dQ irr < dQ rev presented above as the expected one (Equation (13)), but by the inverted relation, that is:

Preliminary Remark
If we accept the proposal dQ irr > dQ rev given by Equation (24) and at the same time the proposal dW irr > dW rev given by Equation (6), their addition leads to the proposal: (10) as the conventional understanding of the first law of thermodynamics. The situation is similar if the addition of Equations (24) and (6) is entered in Equations (11) and (12), because the result obtained would mean that an isolated system can be concerned by a positive variation of its internal energy (dU irr > 0).
Despite the fact that such a proposal appears impossible in the conventional approach of thermodynamics, it becomes easily understandable if the mass-energy relation is taken into account. Two reasons may explain why it has not been done previously. The first one is that at the time of the discovery of the laws of thermodynamics, the theory of relativity was not known. The second is that even without the help of the mass-energy relation, the efficiency of the thermodynamic tool is indisputable, so that no urgency was felt to revise the interpretation of its laws. Nevertheless, it seems that the theory gets more easily understandable when this complement is added and perhaps the scope of its applications may be extended.

Analysis of a Heating Process Concerning a Closed System
For an easy approach of the matter, let us examine the situation below, presented in a basic thermodynamic course as an example of use of the concept of entropy. The system is defined as 1000 g of water (1 liter) that are heated from 20˚C (293 K) to 80˚C (353 K), by contact with a thermostat at 80˚C (353 K), under atmospheric pressure. The asked question concerns the determination of the terms ΔQ, ΔS, ΔS e and ΔS i , knowing that the specific heat capacity of water is cp water = 4,184 JK −1 •g −1 that can be admitted constant over the temperature range.
The answers, equally given in the course, are as follows: So far, these results do not raise ambiguity whether the hypothesis to which we refer is the classical interpretation or the new suggested one. The divergence begins when the asked question is in knowing whether the term ΔQ, mentioned in Equation (28) means ΔQ irr or ΔQ rev . The interest of the question comes from the fact that in the classical approach, the relation between both terms is the inequality dQ irr < dQ rev given by Equation (13), while in the new suggested hypothesis, it is the inequality dQ irr > dQ rev given by Equation (24).
In the first hypothesis, the question is considered useless for the example presently examined, because the formulation of the first law being dU irr = dU rev , it reduces to dQ irr = dQ rev when the energy exchange is limited to heat. This is practically the case here, since the work due to change in volume of the water heated from 20˚C to 80˚C can be considered negligible, compared to the change in heat, so that we can write dW = 0.
The situation is not the same in the second hypothesis because the conversion of the Entropy Equation (17) into the Energy Equation (21) introduces a real difference between dU irr and dU rev and consequently between ΔQ irr and ΔQ rev although the change in volume is always considered negligible.
To know the exact meaning of the term noted ΔQ, the criterion to take into account is that its value is independent of the degree of irreversibility of the heating process when it represents ΔQ rev , while it is not the case when it represents ΔQ irr . To make a test using this criterion, we may imagine a division of the heating process in two steps noted A and B, the first one from 293 K to 223 K, the second from 223 K to 253 K. Reproducing the calculation presented above and adding the obtained results, we are led to the following table: Through these results, it can be effectively observed that the division of the heating process in two steps has no effect on the values of ΔQ and ΔS, but increases the value of ΔS e (change from 711 to 744 JK −1 ) and decreases that of ΔS i (change from 69 to 36 JK −1 ). If the number of steps is multiplied, we easily conceive that the value ΔS i tends progressively towards zero, meaning that the heating process tends towards reversibility. The question asked above is therefore answered in the sense that ΔQ represents here ΔQ rev and the corresponding term ΔQ irr can be calculated by Equation (22), whose differential and integrated formulations are respectively: Applying Equation (34) to the considered system leads to the following observations: When the heating process is done in a single step, we have: Being lower than the previous one, this last value concerns effectively the term ΔQ irr which decreases progressively and tends towards ΔQ rev (itself remaining constant) when the number of steps is multiplied. Thanks to this difference of behavior, the problem of the identification of ΔQ irr and ΔQ rev is solved.
The reason why this procedure is not used in thermodynamic courses is evidently that the conclusion ΔQ irr > ΔQ rev to which it leads it is not in accordance with the usual understanding of the first law. As mentioned above, this last one implies the general equality ΔU irr = ΔU rev which becomes ΔQ irr = ΔQ rev when the energy exchange is limited to heat, as it is the case here. Referring to this situation, it can be observed that the strict obedience to the conventional understanding of the first law reduces the information given by the second law, because it does not allow the conversion of the entropy Equation (17) The entropy being a state function, as the volume, the differential dS has the same value in both equations. Therefore we can write: The terms T e and T i being positive (absolute temperatures), we also see from Equations (37)

Analysis of a Heat Exchange Occurring inside an Isolated System
Let us imagine now, the case of an isolated system made of two parts 1 and 2 separated by a diathermic wall. Supposing that part 1 is defined as 1 liter of water at 20˚C and part 2 as 1 liter of water at 80˚C, our objective is the analysis of their exchange of energy. A first idea that comes to mind is that, just after the end of this exchange, both parts will be at the same final temperature T f = 50˚C, (= 323 K). If a thermometer is placed in each of them, this result can be verified directly, but can also be predicted by the well-known equation: which gives T f = 323 K (taking into account that C 1 = m 1 c 1 and C 2 = m 2 c 2 ). Journal of Applied Mathematics and Physics Designating T i the initial temperature and T f the final temperature, the main elements of the calculation are given below.
For the terms ΔQ rev1 , ΔQ rev2 and ΔQ rev syst , we get: For the terms ΔS 1 , ΔS 2 and ΔS syst : In the conventional approach of the theory, the only results taken in consideration for the whole system are ΔQ syst = 0 (Equation (47)), where ΔQ syst means indifferently ΔQ rev syst or ΔQ irr syst and ΔS syst = 36 JK −1 (Equation (50)).
In the revised approach that is proposed here, we see that a distinction is made between the term ΔQ rev syst = 0, and the term ΔQ irr syst > 0, whose calculated value is ΔQ irr syst = 23328 J.
The reader accustomed to the usual approach of the theory may be tempted to consider this last result as irrealist, because it seems impossible that an energy can be created inside an isolated system. Before examining below the possibility of a link between thermodynamics and relativity, the attention is called to the following remark, briefly evoked in the abstract.
Taking into account that thermodynamics is among the most synthetic theories of physics and the relation E = mc 2 among the most general equations of science, we may be astonished that this latter is not explicitly present in the laws of thermodynamics. Perhaps it is hidden behind our habit of giving the first law the meaning dU irr = dU rev instead of dU irr > dU rev ? It is on the basis of this idea that the considerations discussed in the following paragraph are based.

The Link between Thermodynamics and Relativity
The first evocation of a link between thermodynamics and relativity seems to have been advanced by R. C. Tolman in 1928 [1]. At the sight of a photo nowadays presented in Wikipedia [2], one can imagine that he had the opportunity to discuss this subject with A. Einstein, all the more that they were co-authors of several articles. In the following years, many other authors have published on the same topic, their contributions being generally addressed to readers having a strong background in physics and mathematics. Some recent ones, quoted below in the section References, were written by H. Callen and G. Horwitz [3], Y.
Rengui [4], S. A Haywards [5], C. A. Farias, P. S. Moya and V. A. Pinto ( [6], M. Requart [7], R. C. Gupta, R. Gupta and S. Gupta [8], T. Yarman, A. L. Kholmetskii and J-L Tane [9], V. Krasnoholovets and J-L. Tane [10]. From the observations reported in the present paper, the first information to keep in mind is that there is always a difference between the terms dQ irr and dQ rev , even when the energy exchange concerning the considered system is limited to heat. The second information is that this difference always obeys the condition dQ irr > dQ rev . Added to the condition dW irr > dW rev (Equation (6)), we get the relation dU irr > dU rev (Equation (43)) which seems more appropriate than the proposal dU irr = dU irr (Equation (9)) in the role of fundamental postulate attached to the first law.
The hypothesis already suggested in several papers (J-L. Tane [11] [12] [13] [14] [15]) is that the inequality dQ irr > dQ irr , written above (Equation (41) where c 2 dm is the differential form of the mass-energy relation E = mc 2 . The sign ± means than the choice between + and − is not done for the moment and will be examined further.
The term dU add having been described above as omnipresent in thermodynamic processes, its assimilation to the term c 2 dm is a way to say that thermo- Anderson [16], the mass-energy relation E = mc 2 is mentioned, with the remark that its practical use is mostly impossible, since the changes in mass are too small to be measurable. Referring to the context evoked above (isolated system made of two liters of water designated 1 and 2 that have a mutual heat exchange) there is no doubt that the change in mass is not detectable. Despite this state of affairs, the problem seems of great interest from the theoretical point of view.
The starting reason is that the change in mass that can be predicted is not the same, depending on whether we adopt the classical approach of the theory or the new suggested one. In the first case, the heat exchange will be characterized by the relation ΔQ 1 = −ΔQ 2 , (Equations (45) and (46)) implying Δm 1 = −Δm 2 and consequently Δm syst = 0. In the second case, having obtained ΔQ irr1 ≠ −ΔQ irr2 (Equations (53) and (54)), we will get Δm syst ≠ 0 and the remaining question is the choice between Δm syst > 0 and Δm syst < 0 for the corresponding change in mass.
In the same way as, in paragraph 1, the energy initially designated dW add (Equation (8)) was not really work, but heat, the energy noted ΔQ irr syst = 23328 J (Equation (56)) is not really heat but another kind of energy. It is not heat because the experimenters never reported that an exchange of heat between two parts of an isolated system have led to a final temperature exceeding the value given by Equation (44). Being generated inside the system, the possible solution is that ΔQ irr syst is a potential energy of gravitation, whose origin is a decrease in mass of the system. Referring to the mass-energy relation this explanation corresponds to the formulation dE = −c 2 dm which means that Equation (56) would itself be written: This equation would keep the same form for a system containing other materials than water. Transposed at the scale of a very large system, the presence of the term c 2 dm, implies a possible gravitational effect, which cannot be taken into account when the basis for discussion is the equality dU irr = dU rev instead of Equation (57). Preliminary comments concerning this question are presented in reference [15], with the possibility that the decrease in mass contributes to increase the Earth-Moon distance. Correlatively, the possibility is evoked in reference [17] that, for living systems, Equation (57) could be substituted by equation: Remaining in the context of systems made of inert matter (objects of the experiments that have inspired the laws of thermodynamics), another interesting question is the relation between c 2 dm and the function Gibbs Free Energy. Designated G, in honor of its discoverer Josiah Willard Gibbs, its definition is: general equations of science while, in the same time, it is almost never used in thermodynamics textbooks. Being limited to the macroscopic approach of thermodynamics, and the use of a basic mathematical tool, the hypothesis suggested in this paper seems easily accessible to a wide scientific readership. My hope is that it can be recognized useful by specialists and that the conversion of the entropy Equation (17) into the energy Equation (21) can be accepted in the future by authors of thermodynamics textbooks.