The First Principle of Thermodynamics under Relativistic Conditions and Temperature

Transcription

1 New Horizons in Mathematial Physis, Vol., No., September he First Priniple of hermodynamis under Relativisti Conditions and emperature Emil Veitsman Independent Researher evveitsman@gmail.om Abstrat. he first priniple of thermodynamis under relativisti onditions and with allowane for surfae tension was formulated. With the help of this law it was shown that the temperature of the system studied had varied aording to H.Ott at the adiabati and non-adiabati aeleration, i.e., it (the temperature) had inreased for the observer in the laboratory referene frame if the veloity of the objet, in turn, had inreased. A.Einstein onsidered that the temperature had, on the ontrary, to derease under these onditions when the veloity inreased. It was shown where he had made an error. Keywords: Speial relativity; relativisti thermodynamis; surfae tension; temperature. Introdution Relativisti thermodynamis was born more than entury ago in 97 when Kurd von Mosengeil s artile had been published []; he was Plank s pupil. Aording to onlusions done by K. von Mosengeil and M.Plank, the temperature of the system under study has to vary with proportion to β under relativisti onditions and at adiabati aeleration; β v / ; v is the veloity of the system studied by an observer being in the referene frame at rest; is the veloity of light. A. Einstein ame to the similar onlusion []. Studying the first priniple of thermodynamis under relativisti onditions, nobody in the th entury took into onsideration the surfae tension; meanwhile it is an important thermodynami parameter, so important as pressure p. he both M.Plank with his pupil and A. Einstein wrote down the first priniple of thermodynamis under relativisti onditions without the surfae tension as well. Besides, the both great sientists made an error integrating further Gibbs equation; we shall show it below, in setion. E.V.Veitsman was the first researher who had understood the importane of the surfae tension for relativisti thermodynamis [3 5]. He showed that the surfae tension is a Lorentz invariant [3]; he showed as well that the interfae equations of state were orret under relativisti onditions exlusively if the temperature of system under study varied in inverse proportion to β [4]. E.Veitsman also obtained expressions for speifi thermodynami funtions (J m - ) under the relativisti onditions [5] (the internal energy U, the enthalpy H, the free energy F, and free enthalpy G). hese funtions are orret under the relativisti ones if ~/ β. hus the results obtained by Veitsman are in a full aordane with a result obtained by H.Ott for the relativisti temperature, i.e., / β [6]; (here and below the symbol "" denotes that this quantity is at rest). All above results are ompletely orret for the adiabati aeleration of the system. At last H. Callen and G. Horwitz onsider that the temperature in general is a relativisti invariant [7]. M. Plank writes down the first priniple of thermodynamis du dq + da () as du dq p dv () where Q is the heat put into the system or arried off it (J); A the work done by the system or with it (J). Copyright 7 Isaa Sientifi Publishing

2 38 New Horizons in Mathematial Physis, Vol., No., September 7 Under relativisti onditions the work A is written down by M. Plank as da pdv + v dg (3) U + pv G (4) β where v is the vetor of the system veloity; G the momentum, the symbol means vetor multipliation [8]. G and i ( U + pv ) (5) form the four-dimensional vetor (4-vetor) of energy-momentum, whih has an invariant length equal to i ( U + pv ) (6) i is the imaginary unit. Evidently, we an write down the first priniple of thermodynamis in view of the surfae tension (normal ase) as du dq p dv + σ dω (7) where ω is the element of area (m ). hen the relations (3) and (4) are written down as da pdv + σdω + v dg (8) U + pv σ ω (9) β And, onsequently, the relations (5) and (6) should have a form i ( U + pv σω ) () i ( U + pv σ ω ) () Here we have to note that the quantity ω will transform as v in different ways: depending on the orientation of the surfae in spae (see Veitsman s artiles [3, 9] and Fig. and (see Appendix)). Attempting to obtain the transformation law of the temperature as v, M.Plank used dependenes (3) and (4) but not () and (). Other researhers did the same not taking aount of surfae tension, therefore the results obtained by them were inorret. So the main goals of this paper are:.the obtaining of the first priniple of thermodynamis under relativisti onditions in view of the surfae tension;.the obtaining of the temperature transformation under these onditions from the above mentioned priniple at the adiabati and non-adiabati aeleration. Solving the Problem Write down the first priniple of thermodynamis (normal onditions) in view of the surfae tension as dq du + de + p dv σ d ω da v d G () kin a where E is the marokineti energy of the objet under study; A kin a the work expended on the system aeleration up to a veloity v << ; E A. he momentum G is taken here aording to (9). kin a We will adiabatially aelerate the objet up to a relativisti veloity v now and will integrate () term by term from state of the system to state for the ase represented in Fig. (see Appendix) taking into aount (8), (9) and the equalities p p, σ σ, ω ω ; we have to note as well that here and below we do not study a whole objet only its part (a subsystem): dq du + p dv σ dω vdg, (3) Copyright 7 Isaa Sientifi Publishing

3 New Horizons in Mathematial Physis, Vol., No., September 7 39 () () () 3 v dv Q () v Q U U + pv pv v { U + pv σ ω } (4) Now we take the integral in (4). dv v { U + pv σω } { U + pv σω 9} β (5) 3 3 v ( β ) { U + pv σω } { U + pv σω }, β v /.(5) β β Aording to M. Plank [8], U { U + β pv }, β in view of the surfae tension we have: ( ) U + β pv + σ ω β U β hen taking into onsideration (6), we an write down (4) as { } Q () v Q U + β pv + σ ω ( β ) U + pv β () β (8) pv { U + pv σω } β If we do not input the heat in an aelerated system from its soure, then the left side of (8) equals v ; at first sight the dependene (8) zero, and we annot obtain any dependenes of the kind ( ) does not ontain the temperature. However, this relationship ontains it in a latent form. Show it. Let an objet be, e.g., the liquid moving with a relativisti veloity v. For the observer being at rest in a laboratory referene frame the temperature of this liquid depends on the ooperative veloity v of the miropartiles in the objet relative to its entre of mass. hen for the researher the veloity omponents of the total veloity w of ertain miropartile in the moving liquid equal []: w + w v (9) vw + w w β w vw + β 3 w 3 vw + w i,, 3 are the veloity omponents of the miropartile in the moving referene frame. where i ( ) aking into onsideration the dependenes (9-), it is easy to understand that the heat in the moving system inreases for the observer as v, sine for him the ooperative veloity of the miropartiles inreases in the system moving relative to its mass entre. If the heat inreases, so does dβ (6) (7) () () Copyright 7 Isaa Sientifi Publishing

4 4 New Horizons in Mathematial Physis, Vol., No., September 7 the temperature. However, S S in our ase, this means dq Sd, and (5) ontains, in fat, two parts. he first one is related to the internal energy, the seond to the heat. hen (8) takes the form Q () v Q γ () { U + pv σ ω } β { U + β pv + σ ω ( β ) } U + pv β pv () β α{ U + pv σ ω } ; α + γ ; α <. β Equation () is ompletely onsistently; it is orret in the range of veloities v for Q Q ( v). As a result, / β, i.e., in aordane to H.Ott. If it were β ( ) Q v Q β, we would сome to absurdity as v. Indeed, the left side of () here tends to zero but the right side of () does to. M.Plank as well as A. Einstein were mistaken onsidering that the temperature had to transform at adiabati aeleration aording to the law β (3), as v. he ause of this mistake will be shown below, in disussion. Now we study the seond ase represented in Fig. (see Appendix). he relation (4) should now be written down in view of the relation as ω ω β (4) dq U U + pv pv σ ω + σ ω v () () () { U + pv σ ω } (5) Sine instead of the ratio (7), we have now the ratio dv v U + β ( pv σω) U β then the formula (8) takes the form: Q () v Q γ () { U + pv σ ω } β { U + β ( pv σ ω ) } U + pv β β pv σω β + σω α { U + pv σω } β taking into onsideration (9-), we have now: Q () v { U + β ( pv σ ω ) } + pv β σ ω β β α{ U + PV σ ω } β 3 (6) (7) (8) Copyright 7 Isaa Sientifi Publishing

5 New Horizons in Mathematial Physis, Vol., No., September 7 4 { } Q () v U + β ( pv σ ω ) + pv β σ ω β β α{ U + PV σ ω } β he equation (9) is orret as and in ase, if Q ( v) Q (9) ; it is inorret if Q ( v) Q β. β here are more ompliated ases than ones represented in Fig. and (see Appendix), in partiular, the ases of the droplet or bubble, however we do not study them in this paper. We should here add that we ome to the absurdity, if we onsider that the temperature is the Lorentz-invariant. Now determine, how the temperature and heat have to transform under relativisti onditions when the heat is input into the system from the outside. o solve this problem, we should take into onsideration that the thermodynamial state of the system is independent of the way of its transition to it. Let there be two states of the system ( and ) and two its intermediate ones ( and ). In state the system is at rest and ontains some quantity of heat whih is equal to Q. We input into the system an additional quantity of the heat Δ Q ; now the system is in the. he heat in the system equals Q +Δ Q. Aelerate adiabati the system up to veloity v. Now the system is in state ; the heat of the system is Q +ΔQ Q (3) β However, the system an go to state otherwise: through state. o do that, we must aelerate adiabati the system being in state up to the veloity v. hen the system will be in state. Its heat equals Q Q (3) β Further, we input a quantity of heat into the system ΔQ and transfer it to state. hen the quantity of heat in the system Q equals Q Q Q + Δ Q + ΔQ (3) β Evidently, Q +ΔQ Q + ΔQ (33) β β From equation (33) it follows that ΔQ ΔQ (34) β Now we an formulate the first priniple of thermodynamis under relativisti onditions (3-D formalism): du dq + dq + da + v d G, (35) s d s where Q is the heat hanging the entropy of the system (entropi heat). i.e., dq ds ; Q the heat s not hanging the entropy of the system (relativisti heat), i.e., dq Sd ; A the work of deformation, d e.g., da pdv + σdω ; v dg d s the inrement of the work expended on the inreasing of miropartiles veloity of the system moving relative to its mass entre minus the inrement ( dq ) of the work transformed into heat for the observer being in the laboratory referene frame (see above). If v, the law (35) will be transformed in the law (). Copyright 7 Isaa Sientifi Publishing

6 4 New Horizons in Mathematial Physis, Vol., No., September 7 aking into onsideration the relations (9) (), we an write down: γ ( U + pv σω ) iγ ( U + pv σω) G v ; x (36) β α ( U + pv σω ) iα( U + pv σω) G v ; s x (37) β he quantities in braes of (36) and (37) form, in Minkowski, spae vetors of energy-momentum ( v v ) having invariant lengths equal to x y ( + ) iζ U pv σ ω ; ζ α, γ. (38) We do not exlude the ase when α, then γ and the ratio (34) disappears but the formulas ontaining α are simplified. In [] C.MØller represents a quantity Δ Q as s Δ Q Δ G ΔQ} (39) ΔQ where Δ G (h) v ; ΔQ is here the amount of heat transferred to the system during the proess; in fat. Δ Q dq (see above the law (35)). s ( ) i { h, he relation (39) was obtained by MØller with taking into aount the influene of the vessel walls, ontaining (the vessel) our substane, i.e., in fat, for a system onsisting of two subsystems: a substane and the vessel with its walls. It is important to note that the law (3) was obtained without the influene of the walls of the vessel. he law (35) is orret in Eulidian spae up to a veloity of the system motion max v. If v > v, the max miropartile veloities for the observer in the laboratory referene frame begin to prevail in the diretion X over the miropartile ones in the diretions X and X 3 with aordane to the relativisti law of the veloity omposition. Now both the heat and temperature are not already salars they are vetors. hen we an write down the law (35) as where k 3 Disussion du dq k + d Q k + da + v d G (4) ks () k ( ) d s is the dimensionless vetor; Q and Q ks () k ( ) are the vetors; k,,3. As seen above, the temperature of the system is present in the first priniple of thermodynamis in a latent form by means of the seond priniple of thermodynamis δq δ S, (4) where S is the entropy, whih is Lorentz-invariant aording to Plank, i.e., S S. (4) If we write down the transformation of heat Q under relativisti onditions as Q Q β, (43) then the temperature has to transform under these onditions aording to (3), i.e., aording to M.Plank. However, if we write down the transformation of Q as Q Q (44) β then the temperature has to transform under these onditions aording to the following formula Copyright 7 Isaa Sientifi Publishing

7 New Horizons in Mathematial Physis, Vol., No., September 7 43 β i.e., aording to H.Ott. As we have seen above, relations (44) and (45) do not lead to the absurdity, relations (3) and (43), on the ontrary, lead to it. Where were Plank and Einstein mistaken? We make an attempt to show it making a start from (). Einstein proeeds from Gibb s equation taken in the form (normal state): ds de + pdv vdg, (46) where E is the internal energy. Aelerate adiabatially our system up to the veloity v. Further, integrate (43) term by term from state (v) up to state (v ): (45) ds de + pdv vdg (47) Sine S S, then in view of (6) E E + pv pv ( U + pv ). (48) β However, Einstein meanwhile operates with ds, when there is not any input of heat in the system! We annot use the above Gibbs ratio here, we are to use only the first priniple of thermodynamis under ondition that dq S d (see above)! If we input the heat in our system aelerated up to a veloity v, then the inrement of the heat dq will ontain the term ds, however the heart of the matter has not to hange. As before, we ome to absurdity onsidering that the temperature will be transformed aording to Plank-Einstein s law. aking the transformation of the temperature aording to H.Ott under these onditions, we have no ontraditions. he transformation of heat and temperature under relativisti onditions is onsidered to depend on the veloity of the soure of heat oming to the system (see, e.g., van Kampen work []). In this regard the author of [] examines two ases:. when the objet under study and the soure of heat are moving with the same veloity,. when they have different veloities. he problem is solved in Minkowski spae under some assumptions. In partiular, van Kampen proeeds from the assumption that the internal energy U will not transform under relativisti onditions of the system by formula (6), but as U U (49) β hen in the first ase the heat had to transform in an adiabati aeleration of the system aording to (44) and the temperature had to transform aording to (45) if entropy S S. In the seond ase the heat transforms under relativisti onditions aording to ompliated laws. In order to find them, van Kampen uses an imaginary model. here are two blak bodies a and b separated by a thin metalli sheet. Relative to the laboratory frame, a and b have veloities u a and u b parallel to the sheet. he heat may be leaking from the subsystem a to the subsystem b. Of ourse, suh a system does not exist in nature and annot be reated artifially. Aording to van Kampen, the heat Q in the subsystems a and b has to transform under relativisti onditions as dq a ρ + γρ (5) a b dq b ρ + γρ (5) where ρ and ρ are the energy density in the subsystems multiplied by, a b 4 A t γ u ; ΔA is the area of a small hole through whih the heat goes from the subsystem a to the subsystem b, Δ t the interval of time when the hole is open; u v, is the veloity of light adopted equal to. hen we have с for the whole system dq + dq ( γ ) ( ρ + ρ ) > (5) a b a b and should note the following. Using the above model, van Kampen has obtained a ompliated law of the heat transformation under relativisti onditions. b a Δ Δ ( ) / Copyright 7 Isaa Sientifi Publishing

8 44 New Horizons in Mathematial Physis, Vol., No., September 7 However, using of different imaginary models, we an obtain some speial relations not having any fundamental importane. 4 Conlusions.he first priniple of thermodynamis was obtained under relativisti onditions in view of the surfae tension..using the first priniple of thermodynamis, it was shown that the temperature varied under relativisti onditions in adiabati aeleration aording to H.Ott, i.e., in inverse proportion to β. 3.It was shown where A.Einstein made the mistake whih led afterwards to the inorret Referenes dependene β.. Mosengeil, K. von: heory der stationaren Strahlung in einem gleihformig bewegten Hohlraum. Annalen der Phys., (97).. Einstein, A.: Ueber das Relativitaetprinip und die aus demselben gezogenen Folgerungen. Jahrbuh deutshe Radioaktivitaet und Elektronik. 4, 4-46 (97). 3. Veitsman, E.V.: On Relativisti Surfae ension. Colloid Interfae Si. 65, (3). 4. Veitsman, E.V.: Relativisti State Equations for the Interfae. Ibid, 9, 3-34 (5); Corrigendum, 95, 59 (6); 333, 4 (9). 5. Veitsman, E.V.: Speifi thermodynamial Potential on Surfaes under Relativisti Conditions. Ibid, 337, (9). 6. Ott, H.: Lorentz ransformation der Waerme und der emperatur. Zeitshrift f. Phys. 75, 7-4 (963). 7. Callen, H. and Horwitz, G.: Relativisti hermodynamis. Am. J. Phys. 39, (97). 8. Madelung, E.: Die matematishe Hilfsmittel des Physikers. Springer-Verlag, Berlin (957). 9. Veitsman, E. V.: Some Problems in Relativisti hermodynamis. Journal of Experimental and heoretial Physis, 5, (7).. Levih, V.G.: he Course of heoretial Physis, V. (in Russian). Fizmatgiz, Mosow (96).. MØller, C.: Relativisti hermodynamis. A strange Inident in History of Physis. Det Kongelige Danske Videnskabernes SelskabMatematisk-fysiske Meddelelser. 36, 6 (967). See also C. Moller s artile he hermodynamis in the Speial and the General heory of Relativity. In: Old and New Problems in Elementary Partiles, G.Puppl, G. (eds) pp. (968).. Van Kampen, N.G.: Relativisti hermodynamis of Moving Systems. Phys. Rev., 73, 95-3 (968). Appendix Figure. he moving flat interfae; the veloity v of its motion is perpendiular to the flat one. ΔL is the thikness of the surfae. Copyright 7 Isaa Sientifi Publishing

9 New Horizons in Mathematial Physis, Vol., No., September 7 45 Figure. he moving flat interfae; the veloity v of its motion is parallel to the flat one. ΔL the surfae. is the thikness of Copyright 7 Isaa Sientifi Publishing