Adolfo Bartoliís Derivation

of

The Kirchhoff-Clausius Law

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    In a previous essay I commented that the Kirchhoff-Clausius Law, to which Max Planck referred in his 1900 paper on blackbody radiation, had been influenced, at least in part, by the work of Italian physicist Adolfo Bartoli. Since that time I have translated into English a paper that Bartoli had published in Il Nuovo Cimento (third series, Volume VI) in 1879, in which paper he derived what we now call the Kirchhoff-Clausius law and did so in a way that I did not expect. In this essay I begin by presenting my translation of that paper and then add my own commentary.

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Elementary Demonstration of a Theorem Relative to the

Theory of Radiation Given by Prof. R. Clausius

by Dr. ADOLFO BARTOLI

Professor of Physics at the Technical Institute of Florence

    I propose to demonstrate with very simple elementary calculations a theorem relating to the theory of radiation established by Clausius and well known to all physicists, with the sole aim that such a modest work would not be completely useless for teaching.

    But first I believe it opportune to set out a brief historical mention of the origins of the theory of the movable equilibrium of temperature.

    Since 1791 Pietro Prevost, regarding caloric as a discontinuous fluid (with considerations analogous to those with which Daniel Bernoulli in Section X of his Hydrodynamics yielded a reason for the law of Mariotte and for the law of equilibrium and for the movements of atmospheric air) tried to explain a famous experiment of Pictet, known under the name of reflection of cold.

    Prevost was thus drawn to formulate that hypothesis that he himself named hypothesis of the movable equilibrium. It was expressed by him in the following terms:

    "Caloric is a discontinuous fluid; each and every element of caloric follows constantly the same direction as long as it does not encounter any obstacles. Every point in a hot space is traversed continually by heat rays.

    "If this constitution of caloric is admitted, the following consequences are necessary.

    "1. Free caloric is a radiant fluid; and since it comes back free to the surfaces of bodies; every point of the surface of bodies is a center to which arrive and from which leave in all directions the heat rays.

    "2. The equilibrium of heat through two close free spaces consists of the equality of the heat replacements.

    "3. When the thermal equilibrium is destroyed, it is re-established by means of unequal replacements of heat, etc."

    Prevost deduced, therefore, the principle of the movable equilibrium from an hypothesis on the nature of heat. Then Fourier, Laplace, and Poisson covered with geometric form those slightly vague notions that are found in the writings of Prevost and thanks to the works of these distinguished geometers the hypothesis of the movable equilibrium was found well postulated and without contrast with the theory of heat. After the hypothesis of emission fell and that of undulations remained victorious, the theory of the movable equilibrium remained and still remains unshaken and acknowledged by all who have treated the theory of heat.

    The hypothesis of the movable equilibrium, as is now generally accepted, can be stated so: "Each and every part of the surface of a body emits heat in all possible directions, whatever be its temperature and its relations with bodies with which it returns heat; and any part of the surface is found in an equilibrium of temperature when, with its exchanges of heat with other bodies, it emits and absorbs in the unit of time exactly equal quantities of heat."

    It is useful here to recall some of the consequences to which this hypothesis leads. If it is believed, as proven by the experiment that in a closed container all of the parts and the walls of which are on the outside kept at a constant temperature or, it is supposed, preserved from any loss to the outside, the thermal equilibrium would be established with time, whatever were the conditions of the container, then we must admit, at least for a surface deprived of reflective and diffusive power, the law of Lambert.

    Kirchhoff, accepting the existence of absolutely black bodies (that is, such that with the thinnest, even infinitesimal, thickness) that absorb the totality of the incident heat rays, has demonstrated theoretically the law of the proportionality between the emissive and absorbent powers.

    What is known to all as the second principle of thermodynamics rests on an axiom introduced into science by Clausius in 1850, an axiom that is itself briefly stated: "It is not possible to pass heat from a colder body to a hotter one without the consumption of work". The connection of this principle with the theory generally admitted of radiation is such that the same Clausius had deduced about it to be necessary "that the emissive power of a body depends not only on the nature of the body and on its temperature, but also on the quality of the medium in which the body is found: and that the emissive powers in different media are in direct proportion to the squares of the indices of refraction of those same media".

    Now I come to demonstrate with a most elementary calculation the truth of this proposition.

Fig 1

Created with GeoGebra (see note below)

    In Figure 1 A, B, C would be three indefinite, parallel planes. The space contained between plane B and plane C is filled with a diathermic medium of index of refraction n for the species of heat rays that are considered; in the space contained between plane B and plane A is an absolute vacuum. The plane A and the plane C belong to the surface of two perfectly black bodies (giving to the word the same significance that Kirchhoff gave it) that have the same temperature. On plane A we consider an element or a minuscule portion dSA; from any point whatsoever of dSA we draw the perpendicular khgf common to the three planes. In the plane B with the center in point g (the point of meeting of the normal with plane B) we describe two circles of different, infinitely small radii gl, gm; and we consider the rectangular element lm, mílí (which is called dSB) contained between the two circles and the two radii gm, gmí making an infinitely small angle. Taking the point h as a vertex and the line lmmílí as directrix we describe a cone; all of the heat rays that come from the point h to the element dSB will be contained in this cone. Any heat ray hl contained inside this cone arrives at the surface of separation of the two media, is refracted, changes direction and turns into that of lp. We encompass the width of the portions of the surface where the heat rays emanating from the point h and contained in the cone hlmmílí intercept plane C. We suppose

lm=r; pq=ρ

Angle lhm=ω; angle lhg=θ

gh=α; gf=β;

i is the angle that the ray hl makes with the normal to plane B after it leaves the first medium: that made by ray hm after it leaves the first medium; we have

    K is the point in which the direction of the refracted ray lp meets the normal fk, and e the point at which the directions of the refracted rays lp, mq meet. In order to determine the ratio ρ/r the following relations are had:

so that

Observing that the angles ω and (ií-I) are infinitesimal, we get

so that

To the circular element mmí=r1, corresponds the element qqí=ρ1 and we have ρ1/r1=fp/gl with gl=αtanθ.

fp=gl+βtani=αtanθ+βtani

so that

But

dSB=rr1; dSC=ρρ1

so that

(1)

 

Equally we obtain for the expression of the element δSB that the heat beam going from the point p to the element dSA intersects on the plane B.

and observing that

we get

(2)

 

    If it is supposed that the angle θ is infinitesimal, the two preceding expressions become

and, calling vα and vβ the velocity of propagation of a heat ray in the two media, n=vα/vβ so that

    We indicate with QA the quantity of heat that the element dSA sends in one second to the element dSC through the intermediary dSB and with QC the quantity of heat that the element dSC sends in one second to the element dSA through the intermediary δSC and get

in which the quantities EA, EC would be called the emissive powers in the normal direction.

    (With regard to the law of cosines (law of Lambert) there is a most simple relation between the emissive power in the normal direction and the quantity of heat emitted from a unit of surface in one second. Calling the first E and the other E, we have through the same body etc.

where the integral is extended over the half sphere, so that Eπ=E.) [This is a footnote in the original. Ė D.A.]

    Substituting the values obtained for dSB, šSB we obtain

But in two bodies A and C having the same temperature it will have to be QA=QC; that is

that is, in the case considered here, the emissive powers in the normal direction from two black bodies having the same temperature etc. etc. have to be in inverse ratio to the squares of the velocities of propagation of the heat rays in the media that surround those elements; or proportional to the squares of the indexes of refraction of those same media.

    The preceding theorem is easily extended to the case of any θ. In such a case, conserving the preceding notation, it will be

or recalling that

where EA and EC indicate the quantities of heat emitted in the unit of time from the unit of surface area which belongs to the elements dSA, dSC

but the two bodies having the same temperature, that has to be

QA=QC

and with regard to (1) and (2) and observing that

it comes to

or

(3)

that is, "the radiation of completely black bodies is different in different media: and the emissive powers are in inverse proportion to the squares of the velocities of propagation of heat rays in the two media".

    This result is obtained supposing that the medium that surrounds the surface SA would be an absolute vacuum; but it is evidently extended to the case in which this medium is anything. It is supposed also, for simplicity, that the heat rays would not have suffered a real loss in traversing the media that surround the bodies A and C and that the heat rays emitted from bodies A and C would have had the same wavelength. As what is true for one species of ray has to be equally so for any another species, so the result is general and it will apply equally to heat composed of different kinds of rays etc.

    Note also that the last relation (3) doesnít contain either α or β, so the result is independent of the substance of the medium that surrounds the bodies A and C, provided this substance is not too small.

    It remains now to demonstrate how the relation (3) is a necessary consequence of the second principle of thermodynamics.

Fig. 2

Created with GeoGebra (see note below)

    We suppose that the element A (Fig. 2) belongs to a body ending at two black and plane surfaces: B and Bí are two planes parallel to plane A; and C, Cí two planes also parallel to A that end in the surfaces of two absolutely black bodies. Between A and B, as also between A and Bí, is drawn a vacuum, and between B and C, as also between Bí and Cí, is a medium of index of refraction n. On C and on Cí we determine two elements equal and symmetric with respect to plane A, with the method described previously. t would be the temperature of body A and of body C and T that of body Cí, and EC and E the emissive powers of the bodies C and Cí. The hypothesis of the movable equilibrium of temperature wants that the emissive power of body A does not depend on the temperature of the bodies with which it exchanges heat: it will therefore be equal as much from one as from another face of body A. We indicate (that) with EA and with α the distance ab and with β the distance bc. We have for the quantity of heat QA that the element dSA sends in a unit of time to the element dSC

and analogously

M being a factor that is the same in the two expressions. We put T=t+h. In the theory of radiation the emissive power is a continuous function of the temperature; it would be therefore

where φ(h) is a continuous and growing function of that is annulled for h=0. If therefore it would not be true that

then it would be true that

We suppose at first that

then

so that

and there would exist a valueh1 of h, finite and different from zero, such that for all values of h<h1 it would be true that

that is

so that QA>Q and therefore it would be possible to make heat pass (in quantities as large as would have been wanted) from a colder body to a hotter one, without compensation from inverse transformations.

    If instead it would be true that it would be enough to suppose h negative, that is T<t and it would be demonstrated with reasoning analogous to the preceding that there exists for h a value h2 different from zero, such that for all values of h inferior to h2

so that

Q>QA

so that again it would be possible to make heat pass from a colder body to a hotter one, without compensation from an inverse transformation.

    The preceding demonstration is valid whatever the inclination of the rays that go from the element dSA to the element dS.

    It is therefore demonstrated that the postulate of Clausius, and also the second principle of thermodynamics, yield necessarily the relation (3).

    Florence, 30 September 1879

Appendix: Commentary

    This essay surprised me when I first translated it. In my essay on the Kirchhoff-Clausius law I used the optical analogue of Joule-Thompson throttling to derive the law that Max Planck referred to in his 1900 paper on blackbody radiation. But Bartoli had a different vision.

    In this little essay Bartoli used classical geometric optics to derive one of the laws of radiation thermodynamics. In so doing he gave us a clue, I believe, to a means of connecting thermodynamics to Newtonís dynamic geometry. Such a connection, once worked out in appropriate detail, would take us a step closer to realizing Einsteinís dream of devising a theory that expresses all of physics in terms of classical geometry.

Note on GeoGebra

    I created the two diagrams in this essay with GeoGebra, a free program that you can find on the Internet at www.geogebra.org. The program was created by Judith Hohenwarter and Markus Hohenwarter and posted on the Internet for public use. The program consists of a manual, an exercise book, and the interactive mathematics program itself, all of which you can copy into your computer (indeed, the creators encourage you to do so). This a great program; itís easy to use; and I recommend it to anyone who needs to create mathematical diagrams for their Internet files.

habg

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