Evidence
for a Close Link Between the Laws of Thermodynamics and the Einstein
MassEnergy Relation
Author: JeanLouis 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
massenergy 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 massenergy 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 Munoz^{1} 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 Crerar^{2}. 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 socalled "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 nonexact differential since it designates the variation of a nonstate
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 wellknown 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
= P_{e}dV (where P_{e} designates the external pressure), the expansion
of the gas inside the vacuum leads to the expression _{}W = P_{2 }dV. The value of P_{2} 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 dQ_{1} = 0
and dQ_{2} = 0
, so that for the global system, we obtain dQ = 0 .
Concerning the work exchange, the general equation dW =  P_{e}_{ }dV gives us respectively dW_{1} =  P_{2 }dV_{1}_{ }, whose value is zero as already seen, and dW_{2} =  P_{1 }dV_{2}_{ }, whose value is positive since P_{1} (the pressure of the gas) is positive and dV_{2 } (the volume variation of the vacuum) is negative. The volume change for the whole system being dV = 0 , we have necessarily dV_{2}_{ } =  dV_{1 }so that dW_{2} can also be written dW_{2} = P_{1 }dV_{1}._{}
Adding
the two contributions we are led to the conclusion that for the global system,
the energy result is dU = P_{1 }dV_{1} , 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 massenergy
relation E = mc^{2} 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 = c^{2 }dm (6)
Knowing that a decrease in mass induces an
increase in energy, and conversely equation (6) is better written under the
form:
dE =  c^{2} dm (7)
Transposed
in relations (1), (4), and (5), where the term dU has the signification dU_{e}_{ } (the
energy exchanged between the system and its surroundings), dE can receive the designation dU_{i}_{ }(the energy created or destroyed inside the system
according to the Einstein's massenergy relation).
Rewriting relations (4), (5), and (1) would therefore give them the respective forms (8), (9), and (10), that are:

rev:
_{}Q
+ _{}W = dU_{e} = TdS  PdV
(8)

irr: _{}Q
+ _{}W
<
dU_{e} + dU_{i}
= TdS
 PdV  c^{2 }dm
(9)

gen:
_{}Q + _{}W
dU_{e}_{ }
+ dU_{i} =
TdS  PdV  c^{2 }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* =
dU_{e} + dU_{i} (11)
dU* can be designated as the "global energy
change of the system," dU_{e}_{ } being the "external energy change"
and dU_{i}_{ }the
"internal energy change." We
must be careful that in the classical language of thermodynamics, the internal
energy change, noted as dU, corresponds to dU_{e}_{ }and not to dU_{i}_{ } (perhaps a designation such as "induced energy change" would
be more appropriate for dU_{i}_{ } 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 + dU_{i} = TdS 
0  c^{2}dm
Knowing that
dU_{i}_{ }=  c^{2}dm and that T is positive, we have for
the considered system:
dS = 0 (12)
and relation (9) is reduced to:
dU_{i} =  c^{2}dm (13)
So we are led to the idea that the condition of
evolution of the considered system would not be dS >
0 , but dU_{i} > 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  c^{2}dm under
the designation TdS (implicating by compensation that we
would write zero under the designation c^{2}dm). When applying such a procedure to the whole
system considered here, the significance of relation (9) becomes:
_{}Q + _{}W < dU_{e} + dU_{i} =
TdS  PdV  c^{2 }dm
0 + 0 < 0
+ P_{1 }dV_{1} =
TdS  0
 0
Doing so, we see that the value accepted to dS remains the same as usually given,
that is dS = P_{1 }dV_{1} /T,
where T means T_{1} (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
dU_{i} whose
value is dU_{i} =
P_{1 }dV_{1} 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 P_{1} and P_{2 },
the energized result obtained when adding the two mechanical contributions
takes the form:
dW = dV_{1} (P_{1}  P_{2}_{
})
(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 massenergy relation
that implicates:
dW = dUi =
 c^{2}dm
(15)
Observing,
in relation (14,) that the equality P_{1} = P_{2} would induce a value zero for dW, we are led to the general following
idea:
A
reversible process is characterized by the condition dU_{i}_{
}= 0 that
implicates dm
= 0 and represents an extension of the
usual expression dS_{i}_{
}=
0
.
An irreversible process is
characterized by the condition dU_{i}
> 0 that implicates dm
< 0 and represents an extension of the usual
expression dS_{i}_{
}> 0 .
Evidently,
when the relation in (14) P_{1} and P_{2} 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 = dU_{e} = TdS  PdV (8)
For an irreversible process:
They are respectively:
 C F:_{ }_{}Q + _{}W =
dU < TdS  PdV (5)
 S F: _{ }_{}Q + _{}W <
dU_{e} +
dU_{i} = TdS  PdV  c^{2}dm (9)
Let
us consider a system defined as a given mass of water which is heated from an
initial temperature T_{1} to a final temperature T_{2} . The variation of volume being negligible, the terms PdV and dW
can be eliminated. Thus _{}Q becomes equivalent to the exact differential
dU
(called dU_{e}_{ } 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
= dU_{e}
= TdS
(17)
For an irreversible process:
 C F:
dQ = dU < TdS (18)
 S F:
dQ <
dU_{e} +
dU_{i} = TdS 
c^{2}dm
(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 C_{p} and C_{v} , 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 _{}U_{e} 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 dU_{i}_{ } (equivalent to c^{2}dm )
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 T_{1 }) to a state 2 (temperature T_{2 }), 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 "spacetime 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 U_{e}), _{}
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 T_{e } (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 massenergy relation.
Since
thermodynamic theory was known long before the massenergy relation had been
discovered, it was inconceivable for its authors to give the concept dU
a larger signification than dU_{e}.
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
= dS_{e} + dS_{i} (23)
where dS_{e} =
dQ/T_{e} (24)
dQ
designating the thermal energy received (or given) by the system, and T_{e}_{ } the
temperature of the surroundings. The
reader is therefore informed, more or less abruptly, that the conditions are dS_{i} = 0 for
a reversible process and dS_{i} > 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 massenergy
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 massenergy 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/T_{e} + dS_{i} (25)
which can be written indifferently:
T_{e} dS = dQ +
Te dS_{i}
(26)
Comparing equations (26) and (11) shows immediately their close analogy. Indeed, we obtain:
T_{e }dS =
dQ + T_{e }dS_{i}_{ } (26)
dU*
= dU_{e}_{ } + dU_{i}_{
} (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 dU_{e} 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 T_{e} dS_{i}_{ },
equivalent to dU_{i}_{ },
represents the energy created inside the system by a partial disintegration of
its mass, according to the Einstein massenergy relation.
o
The term T_{e }dS equivalent to dU*
represents the global energy which is involved in the heating process.
Note that the term dQ , intercalated between T_{e}dS and T_{e}dS_{i} in equation (26) has the signification T_{e}dS_{e} as can be seen turning back to the definition of dS_{e} in equation (24). Concerning the system above, defined given mass of water that is evolving from state 1 (temperature T_{1}) to state 2 (temperature T_{2}), by contact with a thermostat of temperature T_{e}, the integration of the three terms of relation (26) gives:
 For TedS: _{ }(27)
 For TedSi: _{ }(29)
In these relations, C is the heat capacity of the system (assumed constant over the integration interval). Noting the difference between C and c (the speed of light), the total integration of relation (26) gives:
T_{e }C Ln (T_{2}/T_{1}) = C (T_{2}
 T_{1}) + T_{e} _{}S
(30)
which means T_{e} _{}S = _{}Q + T_{e} _{}S_{i} (31)
or from (11) _{}U* = _{}U_{e} + _{}U_{i}
(32)
where
_{}U_{i} =

c^{2} _{}m
(33)
Consequently c^{2} _{}m = T_{e} C Ln (T_{2}/T_{1})  C (T_{2}  T_{1}) (34)
and _{}m =  c^{2 }T_{e} C Ln (T_{2}/T_{1})  C (T_{2}  T_{1}) (35).
All these theoretical considerations are summarized in an appendix at the end of the present paper. For a more concrete representation of their practical implications, some numerical examples will now be examined.
5. APPLICATIONS OF THE SUGGESTED THEORY TO NUMERICAL EXAMPLES
A. First example:
Let
us return to the simple system defined as a given mass of water that receives
heat by contact with a thermostat. We suppose that the initial conditions are:

temperature of the water: 20 ° C (= 293 K
)

temperature of the thermostat: 60 ° C (= 333
K )

mass of water: 0.24 g, that gives for
the thermal capacity C = 1, a value that will be considered constant
over the temperature interval
Recalling
that the dilatation of the system is negligible, the interpretation of the
heating process presents some differences whether referring to the suggested
theory or to the classical one. We obtain
respectively:
With the suggested theory:
Calculating the two first terms of equation (30) gives:
T_{e }C Ln (T_{2}/T_{1}) = 333. 1. Ln(333/293)
= 333 .
0. 127 969
=
42. 61 J
C (T_{2 } T_{1}) = 1( 333  293) = 40 J
Consequently equations (30) to (34) lead to:
 c^{2}
_{}m
= T_{e }_{}S 
_{}Q = 42.
61  40 = 2.
61 J
This
last result (2. 61 J) represents the
energy that is created inside the system, by a (very) partial disintegration of
its mass according to the Einstein relation.
With the classical theory:
The successive steps of the calculation are:
_{}Q = 1(333  293) = 40 J
This
last result is usually considered as outside the context of the Einstein
massenergy relation, but that is only an impression. As already emphasized, the "missing quantity" that
represents the difference between _{}Q and T_{e }_{}S
has the necessary dimensions of an energy. Thus, _{}S_{i} must be multiplied by T_{e}_{ }, which leads to the energized quantity:
333 . 0. 007 848... = 2.
61 J.
This result is the same as found earlier and
represents the energy T_{e} _{}S_{i} created
inside the system, according to the theory of relativity.
B.
Second example:
Let us consider an isolated system consisting
of two parts, 1 and 2, that can exchange heat but not work. We suppose, for instance, that each of them
is represented by a mass of water of 0.24
g and that the initial
temperatures are T_{1 } = 293 K
and T_{2} =
333 K.
As
can be observed experimentally (or obtained by thermodynamic calculation) the two temperatures will evolve toward a final
value 313 K. To calculate how
much energy is created inside the system, we apply to part 1 and part 2 the
previous procedure, which will give us:
For part 1:
_{}U_{1} = C_{1 }(313  293) = 20 J
_{}S_{1 }= C_{1 }Ln (313/293) = 0. 066 030 J.K^{1}
_{}U_{1 }= T_{1}^{* }_{}S_{1}_{}_{}
_{ }
where
T*_{1}_{ } (as
already seen) is the mean temperature of part 1 during the process. Its value
is T*_{1}_{ } = _{}U_{1}/_{}S_{1} = 20/0. 066 030 = 302. 89 K.
For part 2:
Proceeding
in a similar way gives us:
_{}U_{2 }=  20 J, _{}S_{2 }=  0. 061 939 J.K^{1}
T*_{2 } = 322. 89
K
For the whole system:
The
term T_{e }_{}S has the following values:
For
part 1: T*_{2 }_{}S_{1 }= 322.89
. 0. 066 030 = + 21.32
J
For
part 2: T*_{1} _{}S_{2} = 302.89
.  0. 061 939 =  18.76 J
Rewriting
equations (31) and (32), we can note, just under them, the numerical values
corresponding successively to part 1 and part 2. Then by adding the two
contributions, we have the result concerning the whole system. This procedure yields:
T_{e} _{}S = _{}Q + T_{e} _{}S_{i }_{ }
(31)
_{}U* = _{}U_{e } + _{}U_{i }_{ } (32)
part 1
: 21. 32 =
20 + 1. 32
part
2 :
 18. 76 =  20 + 1. 24
Whole
system : 2. 56 = 0 + 2. 56
This
result can be condensed under the form:
_{}U* = _{}U_{e} + _{}U_{i}
2. 56 = 0 + 2. 56
The value _{}U_{e} = 0
corresponds to the expression of the first law of thermodynamics in the
case of an isolated system. In the
classical concept, where the existence of _{}U_{i}_{ }is not
recognized, it is simply written _{}U = 0
as was done in the first pages of the present paper. The positive value of _{}U_{i} represents the energy which is created
inside the system according to the theory of relativity.
Taking c = 3.
10 ^{8 }ms^{1},
we obtain for the considered system:
_{}m =  2. 56 /( 3. 10 ^{8}) ^{2 } =  2. 84 .
10 ^{17} kg
While
not measurable in practice, this loss of mass appears as a necessary
consequence of the suggested theory. As
seen just above, the corresponding increase in energy is far from being
negligible. Another interesting information
lies in the fact that the energies which are created by loss of mass are
positive for both part 1 and part 2, but their values are not the same. The value found for part 1 (whose initial
temperature is the lowest) is higher than that created in part 2.
Returning
to the previous example, if we consider that the water and the thermostat
together constitute an isolated system, we can proceed as follows to calculate
how much energy has been created by disintegration of mass inside the
thermostat.
According
to the first law of thermodynamics, giving to the thermostat the symbol th, we
can write _{}Ue_{th} =  40 J. We also have T*_{th} = 333 K, since the temperature of the
thermostat can be considered constant in time and space. Then we can calculate _{}S_{th} using the
relation T*_{th} = _{}Ue_{th}/_{}S_{th} , giving _{}S_{th} = 40/333 =  0.
120 120 J.K^{}^{1}
.The next step consists in writing T*_{e }_{}S_{th} where T*_{e}_{ } is the mean temperature of the mass of
water. Turning back to the vbalues
already obtained, we find for the
water, T*_{e }= _{}U_{e}/_{}S =
40/(0.127 969) =
312, 57 K and therefore T_{e} _{}S_{th} =
 37. 54 J. The energy
created inside the thermostat is thus:
_{}U* = _{}U_{e} + _{}U_{i}_{}
T_{e} _{}S
= _{}Q + T_{e}_{}S_{i}
 37.
54 = 
40 + 2. 46
We
see that the result is 2. 46 J, a
value which is positive but less than that obtained for the water (2. 61 J). While the initial mass of the thermostat is evidently much
greater than that of the water, we note once again that the part of the system
where the created energy is higher, is that with the lower initial
temperature. Note that knowing the mass
of the thermostat is not necessary for obtaining the value of the energy
created inside it. Inversely, if the mass of the thermostat as well as
that of water is known, the energy
created in each part of the system can be expressed per unit of mass,
emphasizing the difference between the value obtained for the water and that
obtained for the thermostat.
6.
CONCLUSION
We have suggested in this paper that to be solved, the conceptual
difficulty of thermodynamics needs a synthesis of the classical laws of
thermodynamics with the Einstein massenergy relation. The main resulting idea
is that the condition of evolution within a system, usually assimilated to an
increase in entropy, should be extended to an increase in energy that is linked
to a correlative decrease in mass. Far
from being reserved to exceptional phenomena, the theory of relativity appears
to be omnipresent around us.
In
classical theory, as in our everyday life, an increase in energy is
recognizable by a symptom such as an elevation of temperature for a supply of
heat or a reduction in volume for a supply of work. What can be the concrete
symptom of a decrease in mass, in the sense of the Einstein massenergy
relation?
Independently
of the very indirect one that can be seen in the better coherence of the
theory, the physical symptom of a decrease in mass is closely related to
gravitation. The thermodynamic analysis presented above has led to the
conclusion that when two systems (or two parts of a system) exchange energy,
the global result is an increase in energy, linked to a decrease in mass.
According to Newton's general law of gravitation, it can be expected that the
symptom of the decrease in mass is an extension of the distance separating the
two systems. In the very local scale of
the systems that were considered in part 5, the loss of mass is too small for
having a visible effect, but integrating similar effects at the scale for two
astronomical objects or systems, each of them undergoing internal irreversible processes, the reality
of an extension of their mutual distance can be easily conceived. On that score, the idea of an expanding
universe is in good accordance with the conclusions of the thermodynamic
analysis presented here. We must have in mind however, that these conclusions
concern what can be seen in our near part of the universe. It does not exclude the possibility that, in
some other parts, the thermodynamic conditions of evolution could be reversed,
implicating dU_{i} < 0 and dm> 0.
Acknowledgments
I would like to thank the authors mentioned
in the text  and some others too  for having stimulated the attention of
their readership upon the conceptual difficulty that is encountered in
thermodynamics. I hope the suggestions
presented in this paper can be a useful contribution in dealing with the
problem they have evoked, and that specialists in the fields of mechanics and cosmology can
go further in analyzing its gravitational significance.
References :
^{1}D. K. Nordstrom and J. L. Munoz, Geochemical Thermodynamics, Blackwell
Scientific Publications 1986.
^{2}G. M. Anderson and D. A. Crerar, Thermodynamics in Geochemistry, Oxford
University Press, 1993.
Appendix (theoretical summary):
The
main difference between the new suggested theory and the classical theory can
be summarized as follows:
New suggested theory:
Reversible:
_{}Q + _{}W =
dU_{e} = TdS
 PdV
Irreversible: _{}Q + _{}W =
dU_{e} = TdS
 PdV
so
that _{}Q + _{}W <
dU_{e} + dU_{i} =
TdS  PdV  c^{2} dm
equivalent to _{}Q + _{}W <
dU_{e} + dU_{i} =
T_{e}dS_{e}  PdV + T_{e}dS_{i}
or _{}Q + _{}W < dU*
=
T_{e}dS  PdV
Note that T_{e}dS_{e}
= TdS; T_{e}dS_{i} =  c^{2} dm and T_{e}dS_{e} + TedS_{i} = TedS
In the
case of chemical or mineralogical reactions (which have not been examined in
the present paper and implicate dP =
0 and dT = 0) the new suggested
thermodynamic tool would take the form:
dG
=  dUi = + c^{2} dm
where G is the Gibbs free energy of the considered
system.
Classical theory:
Reversible: _{}Q + _{}W = dU =
TdS  PdV
Irreversible: _{}Q + _{}W =
dU < TdS
 PdV
where dU has the signification dU_{e} and where the sign "<" means that dU_{i} is implicitly taken into account while not identified as such (which explains that the sign "<" is placed after dU, not before).