Journal of Theoretics Vol.5-2 Guest Commentary Relativistic Thermodynamics for the Introductory Physics Course

Transcription

1 Journal of heoretis Vol.5- Guest Commentary Relatiisti hermodynamis for the Introdutory Physis Course B.Rothenstein I.Zaharie Physis Department, "Politehnia" Uniersity imisoara, P-ta Regina Maria, nr.1, 19 imisoara, Romania Abstrat: It is shown that the statistial deelopment of entropy and the relatiisti inariane of ounted number of stable partiles, offer an easy way to relatiisti thermodynamis for the introdutory physis ourse. Keywords: thermodynamis, speial relatiity. 1. Introdution Largely used introdutory textbooks on speial relatiity do not treat the relatiisti aspets of thermodynamis. he problem is treated in adaned textbooks and papers in a relatiely ompliated way using a four-etor approah, disouraging the teaher to use them when preparing a leture on the subet. he situation is ompliated by the fat that different transformation equations are proposed for two fundamental physial quantities: temperature and energy transferred as a result of a temperature differene (heat). Fundamental equations in thermodynamis ontain a number of N partiles, whih make up a thermodynami system. he relatiisti postulate: "all laws of physis are the same in all inertial referene frames" has the following important onsequenes: Uniersal onstants (Plank's (h) and Boltzmann's (k B )) and the ounted number of stable partiles hae the same alue in all inertial referene frames, If the quotient of two different physial quantities is a relatiisti inariant, they should transform in aordane with the same transformation equation. Let Φ and Ψ be two different physial quantities, measured in the S(xOy) referene frame. Measuring ' ' them from an inertial referene frame S'(x'O'y') they are Φ and Ψ respetiely. If the two physial quantities satisfy the ondition mentioned aboe Φ Φ' =, (1) Ψ Ψ' then we should hae Φ = f ( ) Φ', () and Ψ = f ( ) Ψ', (3) where f() represents a funtion of the relatie eloity of the two referene frames. Relatiisti kinematis, dynamis and eletrodynamis offer examples of suh funtions known as transformation equations: A dimensionless ombination of different physial quantities (mass, length, eletri harge and Kelin temperature, used in order to onstrut a system of units) is a relatiisti inariant 1.

2 If an equation is a ombination of different physial quantities, one of them being a relatiisti inariant, it enables us to onstrut an inariant ombination of different physial quantities. If a physial law is desribed as a sum (or a differene) between different physial quantities then eah of them should transform in aordane with the same transformation equation. Of ourse eah term has the same physial dimensions. Knowing the transformation equation for one of them, we know how to transform the others ones. If one of the terms is a relatiisti inariant then all the others hae the same property. If a transformation equation is deried for a partiular ase it should hold in general ases as well. A system is onsidered, ontaining a large number N of partiles, eah of whih an assume an energy from one of a disrete set of equally spaed energy leels. he energy of interation is onsidered to be negligible. If the energy of a leel is E = ε, being populated by n partiles then we should hae obiously = n N = (4) E = n E = n ε. (5) = In relation (5) E represents the total energy of the system of noninterating partiles, alled in marosopi thermodynamis internal energy. In the same relation, is the order number of a leel and ε the distane between two onseutie leels measured in units of energy. A possible representation of suh a system is shown in figure 1 below. = Figure 1. Distribution of the partiles on the energy leels Differentiating both sides of Equation (5) we obtain 3 de = = εdn + = n dε In a statistial deelopment of entropy it is onsidered that the energy that enters or leaes the system hanges n, while ε remains onstant. It is onsidered that this energy transfer takes plae from a hotter obet to a ooler one, whih is often referred to by the misnomer heat,3. Under suh onditions Equation (6) beomes de = = ε dn =. (6) (7)

3 with representing the energy transferred under the onditions mentioned aboe. he statistial deelopment of entropy leads to the following onepts useful in our approah to relatiisti thermodynamis: Number of mirostates, W N! W = n! n1! n! n3!..., (8) whih is a relatiisti inariant, being expressed as a funtion of ounted numbers of stable partiles, entropy S = k B ln W, (9) whih is for the same reason a relatiisti inariant as well, hange of entropy ds = k B d ln W =, (1) with representing the Kelin temperature of the system and the transferred energy whih hanges n without to hange ε. he transferred energy whih hanges ε without to hange n is alled work. As we see, is a relatiisti inariant, and transforming in aordane with the same transformation equation f(). he fundamental physial quantities used in thermodynamis are Kelin temperature, heat, work, L and internal energy, U, related by the first law of thermodynamis = du + dl, (11) with the sign onentions: heat introdued into the system is positie and the work done by the system is negatie. Equation (11) suggests that, du, and dl should transform in aordane with the same transformation equation. he inariane of requires that du dl and are relatiisti inariants as well. It is assumed that the student knows lassial thermodynamis, relatiisti kinematis, relatiisti dynamis and relatiisti eletrodynamis as well.. ransformation equations for heat Q and Kelin temperature, A. Exhange of heat at onstant eloity We onsider a transfer of energy between two bodies 1 and performed in a way alled heat (radiation or existene of a temperature differene). S'(x'O'y') is the rest frame of the two bodies, both loated on its O'x' axis (figure ). Figure. Energy exhange between bodies 1 and at onstant eloity in the S' referene frame where both of them are in a state of rest. Let m be the rest mass of body 1 and Q the energy (heat) it reeies from body. During a gien time

4 interal its mass hanges with Q m =. (1) At a gien instant of time the rest mass of body 1 is m Q (m < m < m + ). We onsider the same heat transfer proess from the S(xOy) referene frame relatie to whih S' moes with onstant eloity = β in the positie diretion of ommon Ox(O'x') axes. he momentum of the partile in the S frame at a gien instant of time is g x gien by m g x =, (13) resulting that the partile is ated upon by a fore dg x dm Fx = =, (14) dt dt d where we hae taken into aount that the transfer of energy takes plae at a onstant eloity ( = ). he dt work done by that fore is dl = Fxdt = dm. (15) Newton's seond law applied in S leads to d dm 1 dm ( m ) = =. (16) dt dt dt In relation (16) m represents the instantaneous mass of partile 1 at a gien instant of time. he hange in the total energy of partile 1 is dm de = dm =. (17) Combining Equations (1), (14) and (17) we obtain de = δl +. (18) First law of thermodynamis is in S(xOy) de = δ L +, (19) with representing the transferred energy (heat) when measured from S. Comparing Equations (18) and (19) and taking into aount the sign onention the result is =, ()

5 a transformation equation for this energy transferred as heat and performed at a onstant eloity of the heated body. Inariane of entropy requires that in this ase temperature transforms as =. (1) he results obtained aboe are in aordane with those proposed by Einstein 4 and Plank 5. B. Heat transfer at onstant momentum Body 1 (spherial) at rest in S' and reeies energy (heat) isotropially from the surrounding medium (Fig.3). Figure 3. Energy transferred at onstant momentum in the rest frame of the system of partiles. An inrease in its energy () ' ' E = Q takes plae, its momentum, before and after the energy transfer being equal to zero. When deteted from S the transferred energy is Q Q = (3) the inariane of entropy requires that in this ase the temperature hanges as =. (4) he results obtained aboe are in aordane with those obtained by Ott 6 and Arzélies 7.

6 3. he ideal gas and speial relatiity he state of an ideal gas in thermodynami equilibrium is desribed by the equation PV = k B N, (5) in the rest frame of its enter of mass S', P, V and representing its pressure, olume and temperature respetiely. Changes in its internal energy, heat transferred and work are related by Equation (19) owing that they are reersible. Reersibility requires that the hanges in the state of the gas take plae ery slowly. his means that in S no hange in the total momentum takes plae, it being equal to zero at eah moment of time. Figure 4. Senario for deriing the equation of transformation for olumes. A possible approah to the relatiisti behaior of an ideal gas goes as follows: onsider the essel in whih the gas is kept as it is shown in figure 4 as long as all the walls of the essel are in a state of rest, the fore F,x whih ats upon wall 1 normal to the axes Ox(O'x') as well as the surfae of the wall are relatiisti inariants and so is the pressure i.e. P = P (6) and is easy to shown that Equation (6) holds in the ase of all the walls in aordane with the fat Equations (5) and (6) lead us to the following relatiisti inariant V V =, (7) owing that the olume of the gas transforms in aordane with the length ontration effet we hae resulting that temperature should transform as V = V, (8) V 1 β = = β. (9) V Considering that work is done at onstant temperature, work and heat are related by and the result is that work and heat transform as temperature does i.e. and V L = Q = k N ln, (3) V L Q B 1 ' = β L, (31) ' = β Q. (3)

7 his is in aordane with the formulas obtained onsidering that heat transfer ours at a onstant eloity of the reeier. Equations (9) and (3) also sae the inariane of entropy. In an adiabati proess we should hae U P =. (33) V Inariane of pressure imposes the following transformation equation for olumes dv = β dv. (34) In a seond approah to the same problem we inole the fat that all the thermodynami proesses we onsider are reersible and if the total momentum of the gas is initially equal to zero in its rest frame, then it should remain equal to zero during the proess (during the hange of its marosopi parameters) i.e. G =. In aordane with the transformation equations for energy and momentum we hae when deteted from S frame U = γu (35) G = γ U. (36) U and U representing the internal energies. he relatiisti inariane of the first postulate requires that heat and work should transform as internal energy suh as L = γl (37) Q = γq, (38) where inariane of entropy requires that = γ. (39) Equation (33) leads in that ase to the transformation equation for elementary olumes dv = dv. (4) β Of ourse the two approahes are not free of ritiisms. he weak point of the first approah is Equation (3), whih implies for the internal energy an equation of transformation du = γ 1 du, (41) whih is not in aordane with the fat that the proess is reersible. An apparent weak point of the seond approah is Equation (4) whih is onfliting with the length ontration effet. But suh a transformation equation is not strange to the physiist who knows that length ontration an lead sometimes to embarrassing paradoxes. Consider the senario presented in figure 4. Let 1' and ' be the initial and the final positions of a piston moing with onstant eloity u' in the positie diretion of the O'x' axis. Its initial position is assoiated with the eent 1'(x 1 ', y 1 ', t 1 ') = 1'(,,) whereas its final position is assoiated with the eent '(x ', y ', t ') = '(u'dt',,dt') resulting that displaement of the piston as measured from the S frame is ' ' ' u dt + dt dx =. (4) he inariane of the surfae of the piston leads to the following transformation equation for olumes dv = dv, (43) β

8 owing that the inoled time interals are ery small. A length measurement proedure proposed by Streltso 8, inoling loks and light signals leads to Equation (43) as well. aking into aount the fat that the first approah is not in aordane with the onept of reersibility and inoles formulas whih hold in the ase in the ase of energy transfer at onstant eloity and not at onstant momentum we inline to the iew that the seond approah is orret. 4. Conlusions Contraditory formulas enountered in the field of relatiisti thermodynamis are disussed showing the onditions under whih they hold. he onlusion is that the formulas proposed by Plank 4 and Einstein 5 hold in the ase when a transfer of energy takes plae at onstant eloity whereas those proposed by Ott 6 and Arzélies 7 hold in the ase of a transfer of energy at onstant total momentum. It was shown how they work in the ase of an ideal gas underlining the adantages of the seond approah. Referenes [1] J.Futterman, "Inariane of harge and ation", Am.J.Phys. 8,59 (196). [] D.C.Shoepf, "A statistial deelopment of entropy for the introdutory physis ourse", Am.J.Phys. 7, (). [3] M.ribus, "hermostatis and hermodynamis (D.Van Nostrand Company, In. Prinenton 1961) pp [4] M.Plank, "Ann.Phys.",6,1(198). [5] A.Einstein, "Jahrb.Radioaktiität und Eletronik",4,411(197). [6] H.Ott, "Lorentz-ransformation der Währme und der emperatur", Zeitshrift für Physik, 175,7-14 (1963). [7] H.Arzélies, "hermodynamique Rélatiiste et Quantique", Gauthier-Villars, Paris, (1968). [8] V.N.Streltso, "On the relatiisti length", Found.Phys. 6,93-97 (1976). Journal Home Page Journal of heoretis, In. 3