Adolfo Bartoli’s Derivation

of

The Kirchhoff-Clausius Law

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.

*****

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 dS_{A}; from any point whatsoever of dS_{A} 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*, *m’l’ *(which is called
dS_{B}) contained between the two circles and the two radii *gm*,
*gm’* making an infinitely small angle. Taking the point *h* as a
vertex and the line *lmm’l’* as directrix we describe a cone; all of the
heat rays that come from the point *h* to the element dS_{B} 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 *hlmm’l’* 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: *i’* 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 (i’-I) are infinitesimal, we get

so that

To the circular element mm’=r_{1}, corresponds the element qq’=ρ_{1}
and we have ρ_{1}/r_{1}=*fp/gl*
with *gl*=αtanθ.

*fp*=*gl*+βtan*i*=αtanθ+βtan*i*

so that

But

dS_{B}=*rr*_{1}; dS_{C}=*ρρ*_{1}

so that

(1)

Equally we obtain for the expression of the element
δS_{B}
that the heat beam going from the point *p* to the element dS_{A}
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 v_{α}
and v_{β}
the velocity of propagation of a heat ray in the two media, n=v_{α}/v_{β}
so that

We indicate with Q_{A} the quantity of heat that
the element dS_{A} sends in one second to the element dS_{C}
through the intermediary dS_{B} and with Q_{C} the quantity of
heat that the element dS_{C} sends in one second to the element dS_{A}
through the intermediary δS_{C}
and get

in which the quantities E_{A},
E_{C} 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. – D.A.]

Substituting the values obtained for dS_{B},
äS_{B} we obtain

But in two bodies A and C having the same temperature it will have to be Q_{A}=Q_{C};
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 E_{A} and E_{C} indicate the quantities of heat emitted
in the unit of time from the unit of surface area which belongs to the elements
dS_{A}, dS_{C}

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

Q_{A}=Q_{C}

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 S_{A} 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) doesn’t 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 B’ are two planes parallel to
plane A; and C, C’ 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 B’, is drawn a
vacuum, and between B and C, as also between B’ and C’, is a medium of index of
refraction *n*. On C and on C’ 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 C’, and E_{C}
and E_{C’} the emissive powers of the bodies C and C’. 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 E_{A} and with
α
the distance *ab* and with
β
the distance *bc*. We have for the quantity of heat Q_{A} that the
element dS_{A} sends in a unit of time to the element dS_{C}

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 valueh_{1}
of h,
finite and different from zero, such that for all values of
h<h_{1}
it would be true that

that is

so that Q_{A}>Q_{C’} 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
h_{2} different from zero,
such that for all values of h
inferior to
h_{2}

so that

Q_{C’}>Q_{A}

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 dS_{A} to the element
dS_{C’}.

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 Newton’s dynamic geometry. Such a connection, once worked out in appropriate detail, would take us a step closer to realizing Einstein’s 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; it’s easy to use; and I recommend it to anyone who needs to create
mathematical diagrams for their Internet files.

habg