Evidence for a Close Link Between the Laws of Thermodynamics and the Einstein Mass-Energy Relation

Author:   Jean-Louis Tane  TaneJL@aol.com

Formerly with the Department of Geology, Joseph Fourier University,  Grenoble France

 

Abstract:   After recalling the conceptual difficulty that is encountered in thermodynamic theory, the aim of this paper is to show that solving the problem requires a close correlation between the classical laws of thermodynamics and the Einstein mass-energy relation.  The resulting idea is that the condition of evolution within a system, usually interpreted as an increase in entropy, must be extended to an increase in internal energy, itself related to a disintegration of mass. This new concept gives the theory a better coherence and opens a bridge between the field of thermodynamics and that of gravitation.

 

Keywords:  laws of thermodynamics, energy, entropy, Einstein's mass-energy relation, gravitation.

 

1.  INTRODUCTION

 

         It is well known that while being very efficient in practice, the thermodynamic tool remains difficult to understand from the theoretical point of view. It is also well known that the difficulties encountered are not mathematical, but rather conceptual, and that they are perceived by those who have to learn thermodynamics as well as  by those who have to teach it.  Geology being now among the sciences that use thermodynamics for solving some specific problems, earth scientists have discovered, after others, the reality of this situation. They can greatly appreciate that in some books of thermodynamics, especially written for them, the reality of the conceptual difficulty is openly and rapidly evocated rather than cancelled as a forbidden subject.  One of the best examples is that given by Nordström and Munoz1 who related, in the preface of their book, the following opinion of the great physicist Arnold Sommerfeld about thermodynamics:

        

         "The first time I studied the subject, I thought I understood it except for a few minor points.

         The second time, I thought I didn't understand it except for a few minor points.     

         The third time, I knew I didn't understand it, but it did not matter, since I could use it effectively."

 

         Another book intended for geologists and emphasizing the same conceptual problem is that of Anderson and Crerar2. Amid the quotations they relate is one of Reiss (page 3) recording his conviction that nobody understands thermodynamics completely. Another one is from Dickerson (page 295) who also notices the possibility of knowing thermodynamics without understanding it.

         Coming from renowned specialists, these remarks must evidently be regarded as messages of high scientific signification.

 

2.  THE PRECISE LOCATION OF THE CONCEPTUAL DIFFICULTY

 

         To avoid complications that are not directly linked to the given problem, let us consider a thermodynamic system where the energy exchanges are limited to heat and the so-called "PV  work".  As a starting point, we recall that the main relation we have to examine, and which is a synthesis of the first, second, and third laws of thermodynamics is given by the expression:

          

Q + W  =  dU    TdS  -  PdV                                (1)

 

         In this formula (often presented in two or three separated relations) the  symbols refer to the system which is considered and has the following meanings : U  (internal energy), S  (entropy), V  (volume), T  (absolute temperature), P  (pressure), Q  (heat exchanged), W  (work exchanged). The letter d  indicates an exact differential since it designates the elementary variation of a state function such as U, S  or V.  To the contrary, the letter d  indicates a non-exact differential since it designates the variation of a non-state function (except in particular conditions) such as Q  or W.

         In conditions of reversibility, the energized quantities Q  and W  are respectively equivalent to TdS  and - PdV, and can therefore be represented by the expressions:

 

Q    =   dQ   =   TdS                                                     (2)

W   =   dW  =  - PdV                                                    (3)

 

            The conceptual difficulty of thermodynamics comes from the fact that in order for it to be used with success, equation (1) must be worked in a way that does not seem logical. Theoretically, since dU  is the exact differential of the state function U, the sum Q  + W would have the same numerical value as the sum TdS - PdV, whether the process involved is reversible or irreversible. Practically, it does not seem true, since the efficiency of the tool  -  that is its ability to confirm or predict experimental results  -  implicates that Q +W  would be less than TdS - PdV  for an irreversible process.  Such a particularity explains that, in relation (1), the sign that appears between dU  and TdS - PdV  is written "" and not "=", but the physical reason for this situation remains mysterious. 

         In order to see what kind of solution can answer the problem, let us reexamine the interpretation of very simple processes and try to locate precisely the litigious point of our classical reasoning. For the clarity of the discussion, relation (1) is better separated in two possibilities whether it corresponds to the theoretical expression or to the practical one. They are respectively:

 

Theoretical expression:   Q   + W       =    dU      =    TdS  -  PdV            (4)

 

Practical expression:       Q   + W       =     dU      <    TdS   - PdV            (5)

 

 

3. RECALLING THE INTERPRETATION OF A WORK  EXCHANGE

        

         We consider the well-known Joule experiment, which is an isolated system divided in two parts, 1 and 2. A gas is initially confined in part 1, while part 2 has been evacuated. After liberating the piston separating the two parts, the gas expands into the whole system.  The thermodynamic interpretation of this process is very classical and generally regarded as the only possible solution. We recall it briefly hereunder.

         

 

A.  Classical Interpretation

 

         By reference to equation (1) and defining the system as the gas itself, the classical interpretation is the following:  Q  = 0   because the global system is isolated, and inside it, the gas cannot exchange heat with a vacuum. And W  = 0  for the same reasons. We observe effectively that according to the general relation dW =  -PedV  (where Pe designates the external pressure), the expansion of the gas inside the vacuum leads to the expression W = -P2 dV.  The value of P2  being zero, since it designates the pressure of the vacuum, we obtain W =0.

         Having noted that Q,  W,  and consequently dU are zero, the classical interpretation consists in writing TdS - PdV = 0  and concluding  dS = (PdV)/ T.  Since all the terms of this last equation are parameters of the system (none of them refers to the vacuum), the values of  P,  T,  and dV  are all positive, so that dS  is positive too.

         While this result is considered in good accordance with the second law of Thermodynamics (which states the condition dS  > 0  for an isolated system concerning an internal irreversible process) there is a point not perfectly clear regarding the way it has been obtained.  Crossing from dU = 0    to  TdS - PdV = 0   implicates the use of equation (4), that is of a thermodynamic tool that theoretically would be appropriate, but practically is known as not being such, since gas expansion in a vacuum is an irreversible process.

      

B.  Other Possible Interpretation

 

         We define the system as being part 1 (the gas) and part 2 (the vacuum), instead of part 1 only.  Concerning the heat exchange and for the reasons recalled above, we can write  dQ1   =  0  and  dQ2   =  0 , so that for the global system, we obtain dQ   =  0 .

         Concerning the work exchange, the general equation  dW  = - Pe dV  gives us respectively dW1   =  -  P2  dV1 , whose value is zero as already seen, and dW2   =  - P1 dV2 , whose value is positive since P1  (the pressure of the gas) is positive and dV2   (the volume variation of the vacuum) is negative. The volume change for the whole system being dV  = 0 ,  we have necessarily  dV2    =  - dV1    so that  dW2    can also be written dW2   =  P1  dV1.

         Adding the two contributions we are led to the conclusion that for the global system, the energy result is dU  = P1 dV1 , which  consequently  has a positive value.  Of course, a question that may be asked is how can we conceive a positive value for dU  in the case of an isolated system? The only possible answer is given by the Einstein mass-energy relation E  = mc2  which provides the possibility that an energy would be created in an isolated system by disintegration of its mass. In such a case, dQ , dW, and dU have indeed a value zero (at the scale of the whole system), and the positive energized quantity that must be taken into account is produced within the system itself.

         By differentiation and since c  (the speed of light) is a constant, the Einstein relation gives:

 

dE  =  c2 dm                                                         (6)

 

Knowing that a decrease in mass induces an increase in energy, and conversely equation (6) is better written under the form:

 

dE  = - c2 dm                                                       (7)

 

         Transposed in relations (1), (4), and (5), where the term dU has the signification dUe  (the energy exchanged between the system and its surroundings), dE can receive the designation dUi (the energy created or destroyed inside the system according to the Einstein's mass-energy relation).

         Rewriting relations (4),  (5), and (1) would therefore give them the respective forms  (8), (9), and (10), that are:

        

- rev:    Q  + W     =   dUe                   =    TdS   -  PdV                               (8)

- irr:    Q  + W    <      dUe   + dUi       =    TdS   -  PdV   -  c2 dm             (9)

- gen:  Q  + W      dUe   +  dUi     =    TdS   -  PdV   -  c2 dm              (10)

 

where "rev," "irr," and "gen" mean reversible, irreversible and general.

         For convenience, it may be useful to introduce the concept dU*  which is defined as:

 

dU*  =  dUe  +   dUi                                                 (11)

 

dU* can be designated as the "global energy change of the system," dUe  being the "external energy change" and dUi   the "internal energy change."  We must be careful that in the classical language of thermodynamics, the internal energy change, noted as dU,  corresponds to dUe   and not to dUi    (perhaps a designation such  as "induced energy change" would be more appropriate for dUi  in order to avoid confusion).

        Comparing relations (1), (4), and (5) with their respective homologues (10), (8), and (9) gives an answer to the conceptual problem evocated above. The understanding of this answer is easy when we reexamine the evolution of the system evocated before (the Joule experiment).

         We have defined the system as including both part 1 (the gas) and part 2 (the vacuum), the whole being isolated. At the scale of the global system we may therefore write: Q = 0, W = 0, dUe = 0 and  dV = 0.  Introducing this information in relation (9), leads to:

 

0    +   0    <    0    +    dUi       =      TdS   -  0   -   c2dm

 

Knowing that  dUi    =  - c2dm   and that T  is positive, we have for the considered system:    

dS    =   0                                                              (12)

 

and relation (9) is reduced to:

dUi       =     -  c2dm                                             (13)

                                              

So we are led to the idea that the condition of evolution of the considered system would not be  dS   >  0 , but  dUi   >  0 ,  implicating  dm   <  0.

         The introduced modification concerns the interpretation given to the process, but not the usefulness of the thermodynamic tool, which remains unchanged and uncontested. For the latter reason, it is not necessary to introduce modifications in the usual conventions of language that give dS  a positive value.  Referring to relation (9), the solution for maintaining this result would consist of counting the energized quantity  - c2dm   under the designation TdS  (implicating by compensation that we would write zero under the designation -c2dm).  When applying such a procedure to the whole system considered here, the significance of relation (9) becomes:

 

Q    +  W          <      dUe    +   dUi      =     TdS    -  PdV   -  c2 dm

0    +     0            <        0       + P1 dV1    =     TdS    -    0       -    0

 

Doing so, we see that the value accepted to dS remains the same as usually given, that is  dS  = P1 dV1 /T,  where T  means T1  (since the concept of temperature has no significance in the vacuum).  The important point is in taking into account the reality of the energized quantity dUi   whose value is  dUi    =  P1 dV1   and not  dUi   =  0  as  usually admitted.

          In a more general way, for an isolated system composed of two gaseous parts separated by a thermostatic mobile piston and having initial pressures P1  and P2 , the energized result obtained when adding the two mechanical contributions takes the form:

                    

dW  =  dV1  (P1  -  P2  )                                               (14)

        

         The natural evolution of the system being - at least in our near universe - a reduction of volume for the part with the lower initial pressure, we have dW  > 0, and the only available explanation, as already seen, is given by the Einstein mass-energy relation that implicates:

 

dW   =  dUi  =  -  c2dm                                                  (15)

 

         Observing, in relation (14,) that the equality P1  = P2  would induce a value zero for dW, we are led to the general following idea:

 

                   A reversible process is characterized by the condition dUi = 0 that implicates  dm  = 0   and represents an extension of the usual expression  dSi = 0 . 

                   An irreversible process is characterized by the condition dUi  > 0 that implicates  dm < 0  and represents an extension of the usual expression dSi > 0 .

 

         Evidently, when the relation in (14) P1  and P2  are not equal, the temperature increases in the gaseous part having the lower initial pressure and decreases in the other part. Therefore, the work exchange is itself completed by a heat exchange and the problem lies in knowing whether they necessarily balance one another as admitted in the classical interpretation of the first law of thermodynamics.   For examining the matter in question, we shall consider next the interesting case of a heat exchange that is not accompanied by a work exchange.

         

4.  RECALLING THE INTERPRETATION OF A HEAT EXCHANGE

                                      

         The discussion concerning a heat exchange has the same general basis as those recalled - or proposed - when analyzing the previous example.

 

         For a reversible process:

The classical formula  (C F) and the new suggested formula (S F) are respectively:

 

- C F:                                Q     +  W     =       dU          =        TdS    -   PdV                                (4)

- S F:                          Q     +  W     =      dUe         =         TdS    -   PdV                               (8)

 

         For an irreversible process:

They are respectively:

 

- C F:               Q    +   W    =    dU                    <     TdS    -  PdV                                     (5)

- S F:                Q    +   W    <   dUe    +  dUi    =    TdS    -  PdV   -  c2dm                        (9)

        

         Let us consider a system defined as a given mass of water which is heated from an initial temperature T1  to a final temperature T2 .  The variation of volume being negligible, the terms PdV  and dW  can be eliminated. Thus Q  becomes equivalent to the exact differential dU  (called dUe  in the new suggested formulation) and may be written dQ. Relations (4), (8), (5), and (9) therefore take on these respective reduced forms:        

 

         For a reversible process:

                 

- C F:         dQ          =       dU                        =     TdS                                      (16)

- S F:          dQ          =       dUe                      =     TdS                                      (17)

 

         For an irreversible process:

        

- C F:            dQ          =       dU                        <     TdS                                   (18)

- S F:           dQ  <    dUe   +   dUi       =      TdS    -   c2dm                              (19)

 

         Since dQ , here, is an exact differential, the total thermal energy received by the system has the same numerical value, which can be noted Q, whether the process of heating is reversible or irreversible. Knowing that  dQ  =  C dT,  where C  designates the thermal capacity of the system (without a significant difference between Cp  and Cv ,  since the system is condensed), Q  is given by the relation:

                           (20)

In relation (20) C*  is the mean value of  C  on the interval of integration and Q can be identified with U  (which would then receive the designation Ue in the new suggested formulation).

         Returning to relations (5) and (9), more precisely to their right hand terms, it can be easily expected that the interpretation of an irreversible process of heating would not be exactly the same whether the existence of dUi  (equivalent to -c2dm ) is recognized or not. The difference can be summarized as will be discussed next.

 

        A. Classical interpretation

 

         We must recall what has been seen in part 2 concerning the conceptual difficulty of thermodynamics. Theoretically, relation (4) would constitute an appropriate expression for both reversible and irreversible processes, but practically it is not true and the study of irreversible processes needs the use of relation (5).

                   Transposed upon the heating process examined here, relation (16) is the one which, theoretically, would be appropriate for both reversible or irreversible conditions and relation (18) is the one which is practically needed for irreversible conditions.

         When the considered system is heated from a state 1 (temperature T1 ) to a state 2 (temperature T2 ), the integration of equations (16) and (18) gives  respectively:        

- rev:                       (21)

- irr:                       (22)

 

          In (21), the term T*  can be called the mean value of T during the heating process and may be regarded as a state parameter of the system, being the ratio U/S, where both U  and S  are state functions. More accurately, T*  is a "space-time state parameter" in the sense that it represents the mean value of local temperatures that not only change with time, but for a given instant are not homogeneous in space.

         Looking at relations (21) and (22) and knowing that U  is a state function (having the signification Ue), has necessarily the same value whether the conditions of heating are reversible or not and a similar observation can be made concerning .  Consequently, the coherence between the two relations implicates that the signification and the value given to T  in (22) are not the same as that given to T* in (21).  For anyone having some practice in thermodynamics, this is evidence, since T in (22) means Te  (the external temperature) as recalled with the numerical examples considered further. Is it sufficient to conclude that therefore there is no problem? The answer is no, because the obtained coherence is mathematical, not physical. When admitting that the numerical value of T  in (22) is higher than that of T*  in (21), we admit implicitly that the energized quantity T   evolved in the case of an irreversible heating, is higher than the thermal energy which is received by the system, but we don't explain the origin of the complementary energy defined as the difference between T and.   Here is the litigious point of the classical theory and as was seen when analyzing the previous example (the Joule experiment), the only possible solution for solving such conceptual problem is given by the Einstein mass-energy relation.

         Since thermodynamic theory was known long before the mass-energy relation had been discovered, it was inconceivable for its authors to give the concept dU  a larger signification than dUe. Consequently, and while very powerful in practice, the thermodynamic tool they have invented is not perfectly coherent from a theoretical point of view. The usual concept of thermodynamics does not provide an explanation as to why relation (16) must be substituted by relation (18) in the case of real processes, nor in a more general way, as to why relation (4) must be substituted by relation (5).      

         To get around this difficulty, recent books of thermodynamics have often presented entropy in an axiomatic way.  Entropy is defined as a state function whose variation dS  is given by the relation:

 

                                dS   =   dSe  +  dSi                                                (23)

where                          dSe  =   dQ/Te                                                   (24)

 

dQ  designating the thermal energy received (or given) by the system, and Te  the temperature of the surroundings.  The reader is therefore informed, more or less abruptly, that the conditions are dSi   = 0   for a reversible process and  dSi   > 0   for an irreversible process.

                   The undisputable efficiency of the thermodynamic tool is evidently linked to the fact that while not identified as such, the effect of the Einstein mass-energy relation is being taken into account. As will be shown below with numerical examples (part 5), it is implicitly present in equation (18) for instance. Nevertheless, reconciling theoretical coherence with practical usefulness requires that the effect of the mass-energy relation is explicitly present in the equations.

                                                          

         B. New suggested interpretation

        

         It is well known that combining (23) and (24) leads to the relation:

dS   =   dQ/Te  +  dSi                                                (25)

which can be written indifferently:

 

Te dS     =   dQ      +   Te dSi                                       (26)

 

Comparing equations (26) and (11) shows immediately their close analogy.  Indeed, we obtain:

 

Te dS      =     dQ      +     Te dSi                                             (26)

dU*      =     dUe     +       dUi                                                (11)

 

         Each term of equation (26) has the physical significance given by the corresponding term of equation (11), the one written on the same vertical.  Referring to the heating process of a given mass of water as considered above, the detailed significance of the terms is the following:

 

o      The term dQ, equivalent to dUe  represents the thermal energy given to the system by the thermostat. Its value, as already seen, is the same whatever the level of irreversibility of the heating process.                

o      The term Te dSi , equivalent to dUi , represents the energy created inside the system by a partial disintegration of its mass, according to the Einstein mass-energy relation.        

o      The term Te dS equivalent to dU* represents the global energy which is involved in the heating process.

 

         Note that the term dQ , intercalated between Te