On the Thermodynamics of Moving Systems
|On the Thermodynamics of Moving Systems (1908)
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|Zur Thermodynamik bewegter Systeme, Zur Thermodynamik bewegter Systeme (Fortsetzung), Sitzungsberichte der mathematisch-naturwissenschaftlichen Klasse der kaiserlichen Akademie der Wissenschaften, Wien. Bd. 116, Abt. IIa, Heft 9, S. 1391-1405; and Bd. 117, Abt. IIa, Heft 2, S. 207-215In German:|
On the Thermodynamics of Moving Systems
Dr. Fritz Hasenöhrl
(Presented in the session of October 31, 1907.)
A specific electromagnetic momentum is connected to radiation in a moving cavity. Because the heat content of each body partly consists of radiating energy, each body has a specific electromagnetic mass that depends on its energy content, thus e.g. also on its temperature. This assertion was demonstrated by me in earlier papers. Since then, a paper by v. Mosengeil on the radiation in a moving cavity has been published, in which (besides other things) the energy content of the moving cavity has been calculated by the aid of the relation between energy and momentum. Furthermore, Planck has studied the dynamics of an arbitrarily moving system, where he presupposes the existence of the mentioned electromagnetic momentum.
Planck presupposes the validity of the so called relativity principle in the form of Einstein, then he uses the moving cavity as test body, and so he obtains theorems that are valid for each body. In the present paper I have attempted to work out a theory of an arbitrarily moving body as well. The chosen way differs essentially from the method of Planck. Only the thermodynamic theorems as well as the definition of electromagnetic momentum are presupposed. If one then requires that a co-moving observer shall not notice his motion, then the Fitzgerald-Lorentz contraction hypothesis is given.
Thus we consider an arbitrary body, whose state is given by the inner energy and the volume when at rest. If it is adiabatically brought to the velocity , then it has the specific momentum which must be representable as a function of . We put
There, the performed work of the translation forces is
The energy of the body has increased by this amount; if we denote it by , then
Furthermore, we introduce the quantity
that we can also consider as function of and . Then it is:
In the differentiation of one of the quantities , the two others have to be kept constant. We also emphasize, that we understand as the energy which it obtains when the body is brought adiabatically and isochorically to rest; the way by which the body has acquired its momentary (moving) state, is completely irrelevant.
1. Computation of the pressure.
We denote the pressure of the resting system by , that of the moving one by . In order to compute the latter, we consider the following circular process:
A. The initial state shall be that of rest; shall be the values of the relevant state variables. We change the volume adiabatically from to ; then the energy assumes the value .
B. We bring the body to the velocity . The energy assumes the value
The work of the external forces is
C. We change (at constant velocity) the volume adiabatically by . The external forces perform the compression work and the translation work (in order to keep the velocity constant). Thus the increase of energy is
D. We bring the body to rest in an adiabatic and isochoric way. There, the work
is performed. The state of the body is now given by the variables . The total work of the external forces is:
According to the first thermodynamic main-theorem, the work must be equal to . The second main-theorem additionally requires, that this difference is zero. Otherwise this circular process (or the reverse one) would represent a thermal perpetual motion machine. Thus we put in the previous expressions; then we notice that
thus we obtain:
or according to (2):
Thus we obtain the theorem: If an arbitrary body (in the state of rest) is under the pressure , then it assumes (at adiabatic-isochoric acceleration) the value as given by equation (6).
This theorem can be easier derived in the following way (even though it is less clear from the physical standpoint): At adiabatic state changes, the amount of only depends on the momentary values of the quantities and . Thus when (at arbitrary velocity) is adiabatically changed by , then changes by . Then the energy increase, which is equal to the work of the external forces here, is:
and therefore also
is a complete differential, thus
Here, is to be understood as a differentiation at adiabatic state change; thus if is given as an explicit function of and , we have:
Furthermore, since we have according to (4)
the previous equation can be integrated towards and we obtain
This constant can also be a function of and ; it reduces to zero, since we have
2. The differential of the supplied heated.
is increased (as the energy increase) by the performed work of the considered body, thus
If we again introduce and as independent variables, then it becomes:
If we consider (1), (2) and (6), then it becomes:
This expression is valid in full generality.
3. The temperature of the moving body.
We first consider a system of bodies, all of them moving with the same constant velocity. Experience tells us that is then a complete differential. Equation (7) shows that this condition is satisfied, when we put
is the integrating denominator of , when we (analogous to the preceding) understand under the temperature assumed by the body when it is adiabatically and isochorically brought to velocity zero. The arising function of plays here the role of a constant, so it is irrelevant. Of course, it must have the same value for all bodies.
Since we see as constant in this case, it is
In a system whose velocity isn't changing, plays the role of the inner energy for a co-moving observer; between the quantities the same relations exist, which follows from the thermodynamic main-theorems for .
Now, let a body pass through Carnot's circular process, in which the two reservoirs have different velocities; namely, are the temperature and velocity of one reservoir; are the relevant quantities for the other one. If the theorem of the impossibility of a thermal perpetual motion machine is also valid when it assumes different velocities in its different stages, then the ratio of the heat quantities given off to the reservoirs cannot depend on the nature of the circular process. Then it must be
One can easily see, that this function must have the form
must be the case as well, because the ordinary definition of temperature must hold for bodies of same velocity, so the form for is given by
We want to set this function , as well as function in (8a), equal to one; then it becomes
Though we have to emphasize, that there is a certain arbitrariness in this. Even when we don't set these functions equal to one, we neither come into contradiction with the theorem of the impossibility of a thermal perpetual motion machine, nor with the ordinary definition of temperature which is indeed only related to bodies of same velocity. The criterion of equality of temperature is not applicable to bodies of unequal velocity, since we cannot directly bring them in reversible heat exchange, but only with the aid of an auxiliary body which assumes different velocities. Though if we don't set equal to one, then also the entropy of the adiabatic acceleration changes.
Anyway, it is the easiest way to define by equation (8); then is a complete differential and the entropy remains constant at adiabatic acceleration.
4. The entropy of a moving body.
We arrived at the result that pressure and temperature assume the values
at isochoric-adiabatic acceleration. plays the role of the inner energy in a system moving with constant velocity.
Let the entropy of the resting system be , that of the moving one can be expressed by . The relations hold
since it is indeed (at constant )
Since the system was adiabatically brought from the state of rest to that of motion, the entropy has the same value in both cases, thus:
From that, also equations (6) and (8) are immediately given.
Thus far we have presupposed the existence of momentum, without making a special assumption concerning its value. Now we want to assume in agreement with the theory of Lorentz and Abraham, that momentum is equal to the space integral of the (absolute) energy flow, divided by the square of the speed of light. If we assume that it's about the flow of the total energy, i.e. that the total inner energy is of electromagnetic nature, then momentum can be calculated by the following simple consideration.
We consider a cylindrical body of cross-section unity, moving in the direction of its axis (a differently deformed body can be imagined as divided in cylindrical parts). By an arbitrary cross-section that shares the motion, the (relative) energy flow shall flow in the direction of motion, the energy flow in the opposite direction. Since the body is imagined as homogeneous, these quantities are independent of the location of the cross-section; thus the backward (in the sense of motion) base surface will emanate the energy quantity in unit time, and will get the energy quantity . The difference must be equal to the work performed at this surface in unit time. The force acting here is the pressure ; the pressure work in unit time is ; thus
In order to calculate the absolute energy flow, i.e. the energy flow through a cross-section imagined as at rest, we have to add the product of the energy density multiplied with the translation velocity, to the relative energy flow in the direction of motion, i.e. to the quantity . If we denote the first one with for the moment, then the absolute energy flow through the cross-section is given by the quantity
If we multiply this quantity with the volume and divide by , then
Thus the types of inner energy don't matter, as long as they are only of electromagnetic nature (we imagine that they are composed of radiating energy and energy of arbitrarily moving electrons). Also the relative velocity and the energy flow don't matter as well; the individual energy types can of course flow with different velocities.
Of course, one comes to the same result when the individual energy flows are taken into account. For instance, let be the density of a specific energy type, moving in a relative direction that encloses an angle between and with the direction of motion. Then the total energy of this type is
We obtain the momentum when we multiply the absolute flow, i.e. (where is the flow-velocity) with , where is the angle between the absolute flow direction and the direction of motion. Thus:
However, now it is
The first summand is equal to ; the second one gives the surplus of the energy emanating from the basis surface over the inflowing energy, it is thus connected with the pressure work , by which we come to equation (10) again.
6. The change of volume.
A resting system in mechanical and thermal equilibrium is given, i.e. in which all bodies have the same pressure and the same temperature. If this system is adiabatically set into motion (each body adiabatically for itself), then pressure and temperature of every single body is changing, namely in different measure for every single body as we must assume from the outset. Thus the equilibrium is disturbed; if it is restored again, then the individual bodies must change their volumes. If these volume changes are different for different bodies, then they are principally observable. However, if the mechanical and thermal equilibrium is restored again by the change of dimensions of all bodies in the same way, then an influence of the common translatory motion is not observable.
This is indeed the case; first it can be shown that the pressure of a body remain unchanged when is adiabatically changed by and by at the same time. Thus it must be
(the latter relation generally holds for adiabatic state changes, according to (7)).
We notice that according to (10) and (2)
If we also insert for its value from (5), then it is given
Then it is
If we insert this value into (11) and consider (12), then we see that indeed .
The simultaneous change of is:
However, it is now
If we insert herein
a relation known as following from the thermodynamics of resting bodies, then it becomes:
The change of temperature is thus the same for all bodies.
This expression as well as the expression for (12) can be immediately integrated. Then we obtain:
When the volume decreases with velocity according to the previous law, i.e. when the dimensions of matter are contracting in the direction of motion in the ratio
then the pressure of every body remains unchanged at adiabatic change of velocity, while the temperature of all bodies decreases in the same measure. Then, no influence of a common translatory motion is observable.
This is in agreement with the contraction hypothesis of H. A. Lorentz, as well as with the theorems derived by Planck from the so called relativity principle.
While Planck assumes the validity of the relativity principle from the outset, we arrived to a certain extent at a proof of the contraction hypothesis, by postulating the theorem that a common translatory motion is not observable for a co-moving observer; or additionally by demonstrating that a volume change must arise in the previously given way at constant pressure.
(Presented in the session of February 6, 1908.)
7. Calculation of quantity H.
In order to express by the variables , we insert the value for from (6) into (13) and obtain
This partial differential equation assumes a simpler form, when the quantities are chosen instead of as independent variables. Namely, shall be the value of entropy again, when the system is adiabatically brought to rest; of course . Thus we think of as being expressed by entropy and volume; if for example
Then it is:
because according to (7), is changed by at adiabatic volume change. If we furthermore introduce the variable
instead of , then (15) becomes
or when is different from zero:
It follows from this equation, that must be a function of , which of course must also depend on . Furthermore, must be identical with for . We satisfy these requirements when we put
is evidently the energy amount of the resting system, when it is adiabatically expanded from to ; if we denote this energy value with , then
Now, if we assume in accordance with Lorentz's hypothesis, that the velocity change is accompanied with a volume change proportional to , then is the energy of the resting body; then we remove the prime and thus put:
8. Summary of results.
With the aid of equations (2) and (10), momentum and total energy () can be expressed by the state variables of the resting system. (We have to consider here, that equations (1), (3), (4) and (5) may not be applied now; they only hold for velocity changes at constant volume.) We obtain
from which it is given under consideration of the first equation (14):
Finally, the momentum is according to (10):
If we summarize everything, we come to the result:
If a body whose state at rest is given by the variables , is adiabatically brought to velocity , then the state variables assume the value:
These equations are in agreement with the results of the paper of Planck. Besides thermodynamics, Planck used the relativity principle, while stating equation (10) for momentum is essential in our work.
9. Application to cavity radiation.
We base our calculation on the relative ray path. We consider radiation that encloses angles between and with the direction of motion; it carries – in unit volume through the unit surface of a perpendicular (co-moving) plane – the energy amount:
We call the intensity of the total (relative) radiation. If this radiation is incident upon an absorbing surface, it performs the pressure work:
where is the angle between the absolute radiation direction and the direction of motion. The difference:
we call the true (relative) radiation. The true radiation intensity
is crucial for the heat transport between bodies of equal velocity.
We employ the standpoint of Lorentz's contraction hypothesis and introduce the angle by the equation
then is the true radiation intensity observed by the co-moving observer, whose measuring rods have experienced the mentioned contraction.
The quantity must be constant, i.e. independent of angle , i.e. the true radiation is uniformly distributed into all directions in the contracted system; it obeys Lambert's -law. Then two arbitrary oriented, equally moving surface elements are radiating the same amount of heat to each other. A mirror brought into a cavity doesn't change the distribution of radiation, since the ordinary reflection laws hold for the relative ray path in the contracted system. (This was shown in the most general way by H. A. Lorentz and can be directly proven in this case.)
From (20) it easily follows:
thus it is:
where means the relative velocity:
If we set in the previous integral according to (19):
then it becomes:
The second summand is equal to:
as one can most simply recognize by comparison with the penultimate equation of p. 11 of my first report. In consequence of (2) it is therefore:
However, according to (16) it is:
where is the radiation intensity in the resting cavity. Thus it must be
or, since :
This is in agreement with the generally valid theorems of the theory of H. A. Lorentz.
If we set in accordance with the Stefan-Boltmann law
then it follows (see. (14)):
The constant of the Stefan-Boltzmann law is thus to be divided by .
The energy density of the true radiation is:
thus it has the same value as in the resting cavity.
The total energy follows from (17): it has the value:
In any case, the concept of a mass depending on inner radiation and thus on temperature was first stated in my earlier papers, and also the term of the mass which is independent of velocity was calculated for the cavity. Thus it is incomprehensible to me, why Planck – who discussed those things in detail in the introduction of his last publication – doesn't mention my papers at all.
- F. Hasenöhrl, these proceedings, CXIII, p. 1039, 1904; Ann. d. Phys. (4), 15, p. 344, 1904, and 16, p. 589, 1905.
- K. v. Mosengeil, Berlin Dissertation 1906; Ann. d. Phys. (4), 22, p. 867, 1906. – Later, I will discuss v. Mosengeil’s criticism of my papers.
- M. Planck, Berliner Berichte, 1907, p. 542.
- Only reversible processes shall be discussed, is the speed of light in the aether.
- Compare the following section 2.
- Planck was the first who alluded to the fact, that the translation work must be considered in the determination of the temperature of a moving cavity.
- This was also concluded in the papers of v. Mosengeil and Planck at the determination of the temperature of a moving cavity.
- Compare for instance M. Abraham, Theorie der Elektrizität, II., p. 108.
- The method given here is based on the consideration already given by me in an earlier paper (these proceedings, CXIII., p. 1039, 1904). Equation (10) was already derived by Planck. The method of Planck, however, has nothing at all to do with the one used here.
- For instance, compare F. Hasenöhrl, Ann. d. Phys., 15, p. 347, 1904 (however, only radiating energy is considered there. We now have to replace the quantities denoted as and by and . Since the relations are purely geometrical, this replacement is allowed without further ado.
- Compare these proceedings, CXVI, p. 1391 (1907).
- Berliner Berichte, 1907, p. 542.
- M. Abraham, Boltzmann-Festschrift, p. 90, 1904. Compare for instance F. Hasenöhrl, Jahrb. d. Radioaktivität, 2, p. 281 (1905).
- This terminology agrees with the one used in an earlier paper (Ann. d. Phys., 15 ). There, and was written instead of and . See also Jahrb. d. Radioaktivität und Elektronik, 2, p. 283 (1905).
- H. A. Lorentz, Versl. kon. Akad. v. Wetensch. Amsterdam, 7, p. 507 (1899) and 12, p. 886 (1904). – See also M. Abraham, Theorie der Elektrizität, II, p. 282 (1905).
- The absolute radiation intensity can be calculated from this equation. It is equal to
(see. F. Hasenöhrl, Ann. d. Phys., 16, p. 589 ), where
because (see F. Hasenöhrl, Ann. d. Phys., 15, p. 347, Gl. 7 ). That the absolute radiation intensity is changing with direction proportional to , was already demonstrated by v. Mosengeil in another way (Ann. d. Phys., 22, p. 875, eq. 11 ).
- See for instance, M. Abraham, Theorie der Elektrizität, II, p. 282 (1905)
- So when v. Mosengeil says at the end of his paper, that I considered the dimension change as necessary, then this is based on a misunderstanding.
- Ann. d. Phys. (4), 15, p. 350 (1904).
- Ann. d. Phys., 22, p. 791 (1907).
- Berl. Ber., 1907, p. 542.
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