Further Analysis of the Link between Thermodynamics and Relativity

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1 Further Analysis of the Link between Thermodynamics and Relativity Jean-Louis Tane November 2008 Formerly with the Department of Geology, University Joseph Fourier, Grenoble, France Abstract: The first part of this paper is a brief summary of the reasons, previously presented in this journal, suggesting a close link between thermodynamics and relativity. The second part deals with the exchanges of heat between two systems. The subject is examined from the theoretical point of view, leading to complementary equations, and illustrated by a numerical example. Contrasting with the usual theory, that raises conceptual difficulties, the extended theory appears easily understandable, even for scientists whose main specialty is not physics or chemistry. Biologists and geologists are particularly concerned, due to the increasing use of the thermodynamic tool in life sciences and earth sciences. Keywords: Thermodynamics, energy, entropy, reversibility, irreversibility, relativity, mass-energy relation 1. Brief summary of the reasons for a close link between thermodynamics and relativity Let us consider a system defined as a given amount of water that is heated from an initial temperature T 1 to a final temperature T 2. If the heating is performed under a constant external pressure P e (for example the atmospheric pressure), the analysis of the process obeys the following peculiarities. The change in internal energy, noted du, is given by the general relation: du = dq + dw (1) where dq represents the thermal energy due to the heating and dw the mechanical energy due to the dilatation of the water. This energy dw is linked to the variation of volume dv by the equation: dw = - P e dv (2) where dv is an exact differential, since the volume is a state function. Associated with the constancy of P e, this condition shows that dw is independent of the level of irreversibility of the heating process, a conclusion that can be written: dw = Cte (3) According to the first law of thermodynamics, which postulates the relation: it can be derived from a combination of eq. 1, 3 and 4 that: du irr = du rev (4)

2 dq irr = dq rev (5) According to the second law, the change in entropy is given by the relation: ds = dq/t e + ds i (6) where ds i is positive and T e (the external temperature) too, being an absolute temperature. Multiplying each term of eq. 6 by T e leads to the corresponding energy equation: whose precise meaning is: implying the inequality: T e ds = dq + T e ds i (7) dq irr = dq rev + positive term (8) dq irr > dq rev (9) Obviously, there is a contradiction between eq. 1 and eq. 5 To solve the problem, it has been imagined (references [1] and [2]) that the positive term T e ds i represents an energy created inside the system by disintegration of matter. Such a suggestion was impossible for the creators of the thermodynamic theory, since the Einstein mass-energy relation was not known at that time. This is probably the reason why, in the usual conception, the priority has been given to eq. 1 (basic postulate concerning the first law), with the consequence that the entropy equation 6 is not considered as physically equivalent to the energy equation 7. Nevertheless, it seems that the thermodynamic theory gets clarified and extended by admitting this equivalence and substituting eq. 5 by eq. 9. This leads to unify the first and second laws under a global equation which can be written dq irr = dq rev - c 2 dm, and more generally: du irr = du rev - c 2 dm (10) In eq.10, c 2 dm is the differential of the Einstein mass-energy relation and the minus sign expresses the idea that a positive value of the energy term T e ds i is linked to a decrease in mass (dm < 0). Preliminary applications of this hypothesis, that connects thermodynamics with relativity, can be found in the references quoted above ([1], [2]). Through other ways of reasoning, addressed to scientists specialized in physics, several authors have presented arguments for the existence of a link between thermodynamics and relativity (references [3] to [8]). In the continuation of the present paper, eq. 10 is taken as a general basis of discussion and eq. 7 as its particular form, corresponding to the matter that will be discussed. 2. Detailed interpretation of the extended theory in the case of a heat exchange Let us consider an isolated system divided in two parts. Part 1 contains an amount of 2

3 water of mass m 1 and initial temperature T 1, part 2 an amount of water of mass m 2 and initial temperature T 2. We suppose that the two parts are separated by a diathermic wall. It is well known that the natural evolution of the system consists of an exchange of heat between the two parts, each of them reaching the same final temperature T f, whose value can be predicted by the equation: T f = C 1 T 1 + C 2 T 2 C 1 + C 2 (11) where C 1 and C 2 are the heat capacities of part 1 and part 2, respectively. We have noted above that the precise meaning of the energy eq.7 is: dq irr = dq rev + positive term (8) This is a way to say that the precise meaning of the entropy eq. 6 is itself: ds = dq rev /T e + ds i (12) Eq. 12 corresponds to the general case of an irreversible process and can equally be written: For a reversible process, it takes the particular form: whose meaning becomes: ds = ds e + ds i (13) ds = dq rev /T i (14) ds = ds e (15) Similarly, it can be noted that the precise meaning of the classical equation dq = mcdt is: dq rev = mcdt (16) where the c is the massic heat capacity and mc the heat capacity, corresponding to the term C (detailed as C 1 and C 2 ) of eq.11 Keeping in mind these precisions, the following peculiarities can be observed. When the temperature of the water contained in part 1 increases from T 1 to T f, its change in heat (assuming that the product mc is constant over the interval of integration) is: and its change in entropy is: ΔQ 1rev = m 1 c 1 (Τ f - T 1 ) (17) ΔS 1 = m 1 c 1 Ln Τ f / T 1 (18) 3

4 that is independent of the level of irreversibility of the process since S is a state function. Correlatively, the temperature of the water contained in part 2 decreases from T 2 to T f, so that its change in heat is: and its change in entropy is: ΔQ 2rev = m 2 c 2 (Τ f - T 2 ) (19) ΔS 2 = m 2 c 2 Ln Τ f / T 2 (20) Since ΔQ rev1 and ΔS 1 have definite values, we can say that, during the process, the average temperature of the water contained in part 1, designated by the symbol T 1, is: T 1 = ΔQ rev1 / ΔS 1 (21) Similarly, the average temperature of the water contained in part 2 is T 2 = ΔQ rev2 / ΔS 2 (22) Although they are not taken into account in conventional thermodynamics, the concepts T 1 and T 2 (that can be considered as space-time parameters) are of great usefulness, as will be shown under. Indeed, eq. 21 and 22 can be written in the differential forms: ds 1 = dq rev1 / T 1 ds 2 = dq rev2 / T 2 (23) (24) Knowing that ΔQ rev1 and ΔQ rev2 are linked together by the equality: ΔQ rev2 = - ΔQ rev1 (25) we have the correlative equality: dq rev2 = - dq rev1 (26) Therefore, the change in entropy of the whole system, defined as: ds syst = ds 1 + ds 2 (27) is given by the formula: " 1 ds syst = dq rev1 T - 1 % $ ' (28) # 1 T 2 & According to eq.13, the total change in entropy of a system obeys the general relation: 4

5 ds syst = ds e syst + ds i syst (29) where ds e syst is the external component of entropy and ds i syst the internal component. Since the system presently considered has been defined as isolated, we are in the particular case corresponding to: and therefore: ds e syst = 0 (30) " 1 ds syst = ds i syst = dq rev1 T - 1 % $ ' (31) # 1 T 2 & The condition T 2 > T 1 leading to dq rev1 > 0 and the condition T 2 < T 1 to dq rev1 < 0, we see that ds i syst is always positive, so that in the present case ds syst is positive too. Although this conclusion is a well-known data of the thermodynamic theory, the reasoning gets more easily understandable when the expression giving the value of ds i is written with the precisions just mentioned rather than in the less explicit form, often encountered: " 1 ds i = dq 1-1 % $ ' (32) # T 1 T 2 & Referring to eq. 32, an important point concerns the fact that the factorized term dq 1 means dq rev1 not dq irr1. In conventional thermodynamics, the difference is not taken into account because, in the context presently examined, the equality dq rev1 = dq irr1 is implicitly admitted, with the consequence that the equality dq irr2 = - dq irr1 is admitted too. In the extended theory we are dealing with, the factorization is impossible if the term dq 1 of eq. 32 represents dq irr1. The reason is that according to eq.7, 8 and 9, dq irr is not equal to dq rev and consequently the equality dq irr2 = - dq irr1 is not true. In a similar way, interesting relations can be evidenced by the use of the concepts T 1 and T 2 which are explicitly present in eq. 31 (extended theory) but not in eq. 32 (classical theory) A first kind of relation concerns the equalities: ds e1 = - ds 2 (33) ds e2 = - ds 1 (34) which can be derived as follows. From eq. 12 and 13, we see that ds e = dq rev /T e (35) 5

6 Applying eq.35 to the water contained in part 1 leads to: ds e1 = dq rev1 / T 2 (36) Having from eq. 26 dq rev2 = - dq rev1, it appears that eq. 36 can also be written: ds e1 = - dq rev2 / T 2 and since the definition of ds 2 is: ds 2 = dq rev2 / T 2 we obtain the conclusion formulated by eq. 33 A similar derivation leads to eq. 34. A second kind of relation concerns the equalities: ds i1 = ds 1 + ds 2 (37) ds i2 = ds 2 + ds 1 (38) which imply themselves the equality: ds i1 = ds i2 (39) The explanation of eq. 37 is the following: By definition, ds i1 = ds 1 - ds e1. Having noted with eq. 33 that ds e1 = - ds 2, we get immediately the result ds i1 = ds 1 + ds 2 expressed by eq. 37. Applying the same reasoning to ds i2 leads to eq. 38. All these results appear as entropy equations. If they are inserted in eq. 7 and 8, the new obtained results take the form of energy equations. Indeed eq. 7 and 8 correspond to the respective expressions: T e ds = dq + T e ds i (7) dq irr = dq rev + positive term (8) Giving to the "positive term'" of eq. 8 the designation dq add (i.e dq additional ), and applying eq. 7 to the two amounts of water which exchange heat, we can write: For part 1: T 2 ds 1 = dq rev1 + T 2 ds i1 (40) whose meaning is dq irr1 = dq rev1 + dq add1 (41) 6

7 and implies the equality: dq add1 = T 2 ds i1 (42) For part 2: T 1 ds 1 = dq rev2 + T 1 ds i2 (43) whose meaning is dq irr2 = dq rev2 + dq add2 (44) and implies the equality: dq add2 = T 1 ds i2 (45) For the whole system dq irr syst = dq irr1 + dq irr2 (46) that is: dq irr syst = dq rev1 + dq add1 + dq rev2 + dq add2 (47) Knowing from eq. 26 that dq rev2 = - dq rev1, eq. 47 reduces to: Then it can be seen from eq. 42 and 45 that: dq irr syst = dq add1 + dq add2 (48) dq irr syst = T 2 ds i1 + T 1 ds i2 (49) Remembering from eq. 39 that ds i1 = ds i2, we get as a final result: dq irr syst = ds i1 [ T 2 + T 1 ] = dq add syst (50) To illustrate the use that can be made of these equations, a numerical example is examined hereafter. 3. Numerical example Let us consider an isolated system divided in two parts, each of them containing a sample of water. For sample 1, the mass is m 1 = 10 g, the initial temperature T 1 = 293 K (= 20 C) For sample 2, the mass is m 2 = 100 g, the initial temperature T 2 = 333 K (60 C) The massic heat capacity of water being c = J g -1 K -1, the exchange of heat between the two samples obeys the following relations. 7

8 The heat capacity of sample 1 is C 1 = J K -1 The heat capacity of sample 2 is C 2 = J It can be derived from eq. 11 that, the temperature reached by both samples at the end of the heat exchange is: T f = K Entering this value in eq. 17 and 19 gives (rounded results): From eq. 18 and 20, we get: ΔQ rev1 = 1522 J ΔQ rev2 = J ΔS 1 = 4.89 J K -1 ΔS 2 = J K -1 The change in entropy for the whole system is therefore: ΔS syst = ΔS 1 + ΔS 2 = 0.29 J K -1 From eq. 21 and 22, the values obtained for the space-time parameters T 1 and T 2 are: T 1 T 2 = K = K Applying eq. 35 to both samples (as shown by eq. 36 for sample 1) gives: ΔS e1 = 4.60 J K -1 ΔS e2 = J K -1 As predicted by eq. 33 and 34, it appears that these results are consistent with the ones obtained above, respectively for ΔS 2 and ΔS 1. At this point of the discussion, we can be tempted to write that the external change in entropy of the whole system is given by the relation: ΔS e syst = ΔS e1 + ΔS e2 = 0.29 J K -1 Such an idea would be erroneous, because the whole system being isolated, we have necessarily: 8

9 ΔS e syst = 0 The explanation is that both ΔS e1 and ΔS e2 represent entropy variations due to the exchange of heat between part 1 and part 2, an exchange that occurs necessarily inside the system. As a consequence, the value 0.29 J K -1 just obtained represents ΔS i syst, not ΔS e syst This proposition (already checked with the result ΔS syst = 0.29 J K -1 obtained above) can be verified by eq. 37, 38 and 39, which show that the common value of ΔS i1 and ΔS e2 is actually: ΔS i1 = ΔS e2 = 0.29 J K -1 The consistency of this data is also confirmed by the fact that we get the same result using the relations of definition of the internal component of entropy, that is: ΔS i1 = ΔS 1 - ΔS e1 ΔS i2 = ΔS 2 - ΔS e2 As concerns the additional energy, which represents the difference between ΔQ irr and ΔQ rev, the results given by eq. 42, 45 and 50 are: ΔQ add 1 = 96 J ΔQ add 2 = 90 J ΔQ add syst = 186 J The positive value of this last result is the sign that the exchange of heat between part 1 and part 2 has really generated an additional energy. 4. Conclusions The analysis presented in this paper lies on the idea that eq 7, whose expression is: T e ds = dq + T e ds i represents the real physical meaning of eq. 6 whose expression is: ds = dq/t e + ds i Referring to the process evoked in section 1 (heating of a given mass of water, under a constant external pressure), this is a way to say that the inequality dq irr > dq rev seems to be a more accurate interpretation of the first law of thermodynamics than the equality dq irr = dq rev usually admitted. It can be noted that the discussion presented in sections 2 and 3 does not require the use of the Einstein mass-energy relation to introduce new theoretical equations 9

10 (section 2) and to calculate the value dq add = dq irr - dq rev corresponding to a concrete example (section 3) Nevertheless, the Einstein relation appears as the simplest - an perhaps the only one - solution to explain the origin of the additional energy. This is the reason why the basic hypothesis adopted in section 1 takes the general form of eq. 10, that is: du irr = du rev - c 2 dm and the reduced form dq irr = dq rev - c 2 dm when the exchanges of energy are limited to heat. Eq.10 can be looked as an attempt to gather thermodynamics and relativity in a single formula. As noted in reference [1], it is not excluded that the minus sign preceding c 2 dm needs to be substituted by a plus sign for systems made of living matter. Acknowledgments I would like to thank the readers who sent me comments (generally positive) about my previous papers on this matter. My specialty being geology, not physics, I have limited the discussion to the macroscopic aspect of the subject. If this preliminary step is recognized as valid, my hope is that physicists can go further in this analysis, extending the problem to the microscopic point of view. References [1] J-L Tane. Thermodynamics and Relativity: A Condensed Explanation of their Close Link arxiv.org/pdf/physics/ (2005) [2] J-L Tane. Thermodynamics and Relativity: An attempt to explain their close link in one page. The General Science Journal. (2008) [3] V. Krasnoholovets and J.-L. Tane. An extended interpretation of the thermodynamic theory, including an additional energy associated with a decrease in mass, Int. J. Simulation and Process Modelling 2, Nos. 1/2, (2006). [4] R. C. Tolman. Relativity, Thermodynamics and Cosmology (1934, 501 pages). Reprinted by Dover Publications 1988 [5] Ye Rengui, The logical connection between special relativity and thermodynamics Eur. J. Phys (1996) [6] R.C. Gupta, Redefining Heat and Work in the right perspective of Second-law-ofthermodynamics arxiv.org/abs/physics/ (2006) [7] C. A. Farías, P. S. Moya, V. A. Pinto. On the Relationship between Thermodynamics and Special Relativity. arxiv.org/abs/ (2007) [8] M. Requardt. Thermodynamics meets Special Relativity -- or what is real in Physics? arxiv.org/abs/ v1 (2008) 10

A Simplified Interpretation of the Basic Thermodynamic Equations

A Simplified Interpretation of the Basic Thermodynamic Equations Jean-Louis Tane (tanejl@aol.com). October 2009 Formerly with the Department of Geology, University Joseph Fourier, Grenoble, France Abstract: