The Kirchoff-Clausius Law

In the course of laying out his formal derivation of the equation describing the spectral density of energy in blackbody radiation (in AOn the Law of Distribution of Energy in the Normal Spectrum@, published in Annalen der Physik, Vol 4, Pg 553ff, 1901), Max Planck referred his readers to Athe well-known Kirchoff-Clausius law@. Well, what was well-known in 1901 is not well-known today; indeed, it seems to be not known at all. The only reference to the Kirchoff-Clausius law that I can find on the Internet (as of 2007 Aug 30) is the one in Planck=s paper. And my search of books, as technical as I can find, on thermodynamics (and radiation thermodynamics in particular) I found no reference to the law. This raises the question of how a science-dominated society can lose a law of physics, though I don=t intend to pursue that question in this essay.

It also presents a challenge. Using the few clues available I must now try to reconstruct this law in a way that I can apply to Planck=s derivation. The biggest clue is Planck=s own statement in his paper, which is

t t t

Now according to the well-known Kirchoff-Clausius law, the
energy emitted per unit time at the frequency
v
and temperature θ
from a black surface in a diathermic medium is inversely proportional to the
square of the velocity of propagation c^{2}; hence the energy density u
is inversely proportional to c^{3} and we have:

where the constants associated with the function f are independent of c.

t t t

We see from comparing those two equations that Planck converted the function ψ containing the speed of light into the function f not containing the speed of light by factoring out the inverse fourth power of lightspeed. That fact implies a derivation similar to the one we used to gain the Stefan-Boltzmann law.

Planck=s statement that the rate of energy emission from a black surface is related to the radiation=s speed of propagation tells me that I need to conceive an imaginary experiment that has a blackbody emitting its heat radiation into a transparent medium that has a variable index of refraction. Because Planck attributed the law to Gustav Robert Kirchoff and Rudolf Clausius, I can guess that this will not be a problem involving electromagnetic theory: rather, I will have to work this as a problem in pure classical thermodynamics, one in which I must contrive a way to introduce the speed of light as a variable.

Rudolf Clausius did for the second law of thermodynamics pretty much what Einstein did for Relativity, so I will begin with the master equation of classical thermodynamics,

(Eq=n 0)

and look at the emission of blackbody radiation in terms of changes in entropy. Because we=re dealing with light in its various manifestations, one obvious way to introduce the speed of light into our derivation leads us to look at a situation in which we must invoke the Doppler shift.

So we begin with radiation pressure. As early as 1619 Johannes Kepler guessed that sunlight exerts pressure: he made the guess to explain the observed fact that the tails of comets always point away from the sun. Only in 1873 did a solid theory of radiation pressure come into being when James Clerk Maxwell applied his still-new electromagnetic theory to the question. But few people gave electromagnetism the attention it deserved until 1886, when Heinrich Hertz began generating and observing the effects of electromagnetic waves in his laboratory. More likely, the Italian physicist Adolfo Bartoli (Firenze 1851 Mar 19 - Pavia 1896 Jul 18) got the attention of Kirchoff and Clausius with his 1876 deduction of the existence of radiation pressure on thermodynamic grounds. Bartoli, a professor of physics at the University of Pavia, was well-known at the time and influenced Ludwig Boltzmann (in devising the Stefan-Boltzmann law) and Wilhelm Wien (in devising his 1893 displacement law) and may also have had some influence on Einstein, whose family lived in Pavia in the 1890's. As with the Kirchoff-Clausius law, I am unable to find any detailed description of how Bartoli deduced the existence of radiation pressure beyond the fact that he conceived an imaginary experiment that used a moving mirror to impose a Doppler shift on heat radiation and showed that such an imposition would lead to a violation of the second law of thermodynamics unless the mirror did work against a pressure resisting its movement. In part our modern ignorance of Bartoli=s deduction may come from the fact that Bartoli himself disliked his conclusion and disowned it. Bartoli=s imaginary experiment survives, in modified form, in the imaginary experiment that Ludwig Boltzmann devised as a means of working out his version of the Stefan-Boltzmann law. Even though we don=t know precisely the logical path that Bartoli followed to his conclusion, we can devise our own path confident that it will come close enough to Bartoli=s to cross the same intellectual terrain.

Imagine a pulse of heat radiation propagating
parallel to the x-axis of some coordinate system. The pulse carries an amount of
energy Q so distributed among the frequencies of its component electromagnetic
waves that it appears to have emanated from a body at a temperature T_{1}.
The pulse thus carries a quantity of entropy S_{1}=Q/T_{1}. The
pulse then strikes a mirror moving at the speed v parallel to the x-axis in the
direction whence the pulse came, so we have a head-on collision between the
pulse and the mirror. In that collision the radiation comes under the action of
a Doppler shift that raises its frequencies in accordance with

(Eq=n 1)

(Note that I am using the non-relativistic Doppler shift because Bartoli and his contemporaries did not know about Relativity).

Nineteenth Century physicists knew from
observations of the colors of progressively hotter bodies that the average
frequency of the radiation emanating from a body increases with the body=s
temperature. So the Doppler shifted radiation bouncing off the mirror appears to
have emanated from a body with a temperature T_{2} = T_{1}+ΔT.
The pulse thus comes off the mirror carrying entropy S_{2}=Q/T_{2}
such that we have a change in the pulse=s
entropy in accordance with

(Eq=n 2)

But that represents a net decrease of entropy in a passive system enacting a passively reversible process (we need only attach a second mirror to the first in such a way that the pulse will bounce off it and thereby have its frequencies Doppler shifted down to their original values).

That kind of thing, Rudolf Julius Emanuel Clausius (1822 Jan 02 - 1888 Aug 24) told everyone, can never happen. Clausius invoked the law of entropy (a word he made up in 1865 to denote a phenomenon he had first studied in 1851) to explain the workings of thermodynamic systems; in particular, he stated that the entropy of a system gaining or losing heat energy dQ at an absolute temperature T changes by

(Eq=n 3)

and that the net change of the entropy in a closed system can never be less than zero. I used that equation to calculate the entropy of the pulse by using it to calculate the entropy that the emitting body lost and by assuming that the radiation picked it up. Because the emission of the radiation is perfectly reversible, the radiation could not have picked up more or less entropy that the body lost in emitting it. That=s how I know the radiation has to carry entropy: the second law of thermodynamics requires it. But the interaction with the mirror seems to make the entropy of the pulse decrease. Clausius presented the second law of thermodynamics as a statement that the entropy of a system enacting a reversible process can never decrease, so we must infer that in our analysis we missed something.

That something must add an amount ΔQ of energy to the radiation such that, at least,

(Eq=n 4)

In that case the entropy of the system would not change and would, thus satisfy the law of entropy. The only opportunity for energy to be added to the radiation in the pulse comes when the moving mirror imposes a Doppler shift upon it. The mirror must do work upon the radiation, which necessitates that it exert a force upon the radiation. By Newton=s third law of motion the radiation must exert an equal and oppositely directed force upon the mirror, so now we know that light striking a surface exerts a pressure upon that surface.

Further, we know that

(Eq=n 5)

in which F represents the force exerted between the radiation and the mirror, p represents the pressure of the radiation, A represents the area of the mirror, and Δt represents the time interval over which the force acts (assuming that the force does not vary over that interval; otherwise we must use differentials and the process of integration). But we calculate the total energy in the pulse of radiation coming to the mirror as

(Eq=n 6)

in which u represents the energy density in the pulse and Δl represents the length of the pulse (Δl=cΔt for v<<c). We use that equation to calculate AΔt and substitute the result into Equation 5 to get

(Eq=n 7)

We can increase the value of Q without
changing the length of the pulse (by decreasing the albedo of the emitting body
to increase the efficiency at which it radiates heat, for example). That would
not change the temperature T_{2}=T_{1}+ΔT,
so Equation 4 necessitates that ΔQ
stand proportional directly to Q, which Equation 7 confirms. But increasing Q as
I described above also increases u in the same proportion, so what I just said
about Equation 7 necessitates that

p=αu,

(Eq=n 8)

in which α represents some numerical constant.

Now we invoke the master equation of classical thermodynamics,

TdS=dW+pdV,

(Eq=n 9)

in which W represents the energy content that the body in question has obtained from work and V represents the body= s volume. We have before our mind=s eyes a perfectly reversible process: the mirror has done work on the pulse of radiation, but if we were to reflect that radiation back to the mirror as it moves outward at the speed v, the radiation would do the same amount of work upon the mirror and return to its original state. Perfectly reversible processes do not change the entropy of the systems that carry them out, so in the above process we have dS=0 for the radiation. Equation 9 thus gives us

dW=-pdV,

(Eq=n 10)

which means that negative changes in the system=s volume at a given pressure (a compression) produce positive changes in the system=s internal energy.

But when light passes from one medium into another medium with a different index of refraction, it changes its volume without doing any work. We know that must stand true to Reality because if we have light expanding, we can send it into a third medium that recompresses it and it goes readily. That means that we must have

d(pV) = pdV+Vdp=0.

(Eq=n 11)

Substituting from that equation into Equation 10 gives us

dp = du,

(Eq=n 12)

which necessitates that in Equation 8 we have α=1.

I note that some texts say that the radiation pressure equals one third of the energy density of the radiation. If we take the simple case of a cubic enclosure, then we can divide the radiation into three portions, each portion propagating in one of the three cardinal directions of space. Each portion exerts a pressure equal to its own energy density on one of the walls of the enclosure in accordance with Equation 12, but all three portions contribute to the overall energy density in the enclosure, so we must represent the pressure on any of the walls as one third of the overall energy density. In our version of Bartoli=s imaginary experiment we have heat radiation flying in only one direction, so we use Equation 12 without modification.

So now we know that

(Eq=n 13)

and, by Equation 4, that

(Eq=n 14)

The heat content of the pulse and its temperature both stand subject to the Doppler shift. Those equations, when combined with Equation 1, give us two new facts.

First, we have the fact that

Q=Kν,

(Eq=n 15)

in which either ν can represent the average frequency of the radiation comprising a heat pulse or Q can represent the amount of heat manifested in that part of the radiation with frequency ν. That equation looks like the equation that expresses Planck=s theorem (E=hν), but in this case K can take for its value any positive real number, while in Planck=s theory K can only take those values equal to positive integer multiples of the number that we now call Planck=s constant. In the work above, heat radiation still exists in our conception as a continuum and not as the grainy entity of the quantum theory.

The fact that light exerts a force necessitates that it carry linear momentum. Now we want to describe how much linear momentum rides in a pulse of light. If the pulse falls onto a perfectly black surface that absorbs it, the total linear momentum that it puts into the surface equals the product of the exerted force and the time over which the force acts, with the force equal to the pressure multiplied by the area on which it acts, so we have

(Eq=n 16)

But the energy content of the pulse equals the product of the energy density and the volume of the pulse, the volume being equal to the product of the area on which the pulse falls and the length of the pulse, the latter equal to the product of the speed of the pulse and the time interval over which it acts, so we have

(Eq=n 17)

the latter equality coming from Equation 15 (though I have substituted the angular frequency ω=2πν). So now we have Aδt=Q/uc and Equation 16 becomes

(Eq=n 18)

in which k represents the wave number of the waves comprising the pulse.

And second we have the fact that

T=Jν,

(Eq=n 19)

in which ν must represent either the average frequency in the radiation or some other frequency characteristic of the radiation. That looks a bit like Wien=s displacement law (and we recall that Wien was aware of Bartoli=s work) but we cannot claim it as such because we don= t know what kind of function produces the coefficient J.

How did Kirchhoff and Clausius make use of that information? Again, I don=t know the path of their reasoning, but again I can lay out a logical path that certainly comes close to theirs if it does not coincide with it, simply because theoretical physics gives us few alternatives.

According to Kirchhoff=s law, if we have a perfectly black body that has reached thermal equilibrium with the radiation in which we have immersed it, then the rate at which the body emits heat energy from a given area equals the rate at which it absorbs radiant energy falling on that area. We want to begin with a simple noncommital description of the emitted energy that we can elaborate later, so we simply multiply the energy density in the radiation by a unit of volume and divide it by the time required for that volume to emerge from the surface of the body and we get

(Eq=n 20)

if we have our radiation propagating entirely in the x-direction.

Imagine, as Clausius may have done, a cavity
inside a body held at a uniform temperature. We fill that cavity with two
perfectly transparent substances with different indices of refraction and bring
those substances together on a flat plane that crosses the middle of the cavity.
We know that the radiation travels through one or the other of those substances
at a speed inversely proportional to the substance=
s index of refraction, c_{1}=c/n_{1} for substance-one for
example, so the radiation moves slower in the high index substance and faster in
the low index substance. Now look at the heat radiation bouncing around inside
the cavity and pay particular attention to the radiation crossing the interface
between the two transparent substances.

A pulse of radiation crosses the interface, going from the high-index substance into the low-index substance. As it crosses the interface the pulse, bit by bit, speeds up its motion and thereby expands its volume. In that expansion we may conceive an analogy with Joule-Thompson (sometimes called Joule-Kelvin) throttling. If we fix a porous plug (made, let=s say, from compacted fiberglass) inside a pipe and attach the ends of the pipe to two reservoirs of gas that each reservoir keeps at a constant pressure, then the gas will pass from the high-pressure side of the pipe to the low-pressure side through the plug in a way that keeps its enthalpy unchanged.

That analogy fails in an important way. The Joule-Thompson process is not reversible: the gas does no work as it passes through the plug but we must do work upon the gas to force it back through the plug to whence it came. But the light crossing the interface in our heated cavity gives us a perfectly reversible process: we need only put a mirror in its path to send it back across the interface and thereby recompress it. That fact means that the entropy of the radiation does not change as the radiation crosses the interface (dS=0), so we have the master equation of thermodynamics as

dQ=TdS=dE+pdV=0.

(Eq= n 21)

That equation, in turn, gives us

dE=-pdV=-d(pV)+Vdp.

(Eq=n 22)

No work gets done on or by the radiation as it crosses the interface, so we must have the product pV equal to a constant. That fact then leaves us with

dE=Vdp.

(Eq=n 23)

We cannot distinguish between the expansion of the pulse due to its crossing the interface or its coming under the Doppler shift. That statement must stand true to Reality because if it didn=t, we could exploit the difference between the two kinds of expansion/compression to diminish the entropy of a closed system, an act that the second law of thermodynamics absolutely forbids. So now we know that in radiation crossing an interface the energy changes in the same way that the wave number changes; to wit,

(Eq=n 24)

Substituting that into Equation 23 gives us

(Eq=n 25)

But E=uV and the pressure p=u/3, so we have, after dividing out the volume,

(Eq=n 26)

Rearranging that equation and integrating the result gives us

3lnk=lnu+C,

(Eq=n 27)

which we solve for

u=f(v,T)k^{3},

(Eq=n 28)

in which the function f comes from the constant of integration (well, constant relative to changes in the wave number certainly). We want to express that equation in terms of a property of the radiation that does not change when the radiation crosses the interface, so in terms of the frequency we have

(Eq=n 29)

We calculate the irradiance of radiation emanating from a body, the energy emitted per unit of area per unit of time, by multiplying the energy density of the emitted radiation by the radiation=s speed of propagation. Thus we have

(Eq=n 30)

Thus we infer that the rate at which a body emits heat radiation is inversely proportional to the square of the speed at which the radiation propagates in the medium in which the body is immersed. And that gives us the once well-known Kirchhoff-Clausius law.

Appendix: A Spooky Step Further

Equation 29 offers us one last problem to solve, one that Clausius apparently did not confront. When a pulse of radiation crosses the interface between two transparent media with different indices of refraction, it becomes longer or shorter in a process analogous to the Doppler shift (though only the wave number and not the frequency changes). The wave number of the radiation also changes in inverse proportion to the change in the pulse=s length. But in accordance with Equation 29, the energy density, the total energy of the pulse divided by the pulse=s volume, changes in proportion to the cube of the changed wave number.

That proportionality necessitates that a pulse of electromagnetic radiation dilate or contract in both directions perpendicular to its direction of propagation as well as in the direction of propagation itself. But a beam of light crossing the interface between two media at normal incidence does not expand or contract laterally; it would violate Maxwell=s Equations if it did. So we must dismiss our tacit assumption that electromagnetic energy spreads smoothly and uniformly throughout any beam or pulse that we generate. Perforce, then, we must take as our inference the only alternative option, that light consists of flocks of blobs carrying electromagnetic energy (as an aid to the imagination conceive the blobs as spheres perhaps one wavelength in diameter).

In that model the blobs expand and contract as they cross the interface so that the density of the energy they contain conforms to the description in Equation 29. Meanwhile the distribution of the blobs in the pulse or the beam conforms to the propagation geometry conforming to Maxwell=s Equations. That model gives us a natural explanation of the fact, not known in Clausius' day (for obvious reasons), that light passes through a fine wire mesh with little attenuation but radio waves don= t (an observation that Heinrich Hertz could have made in the late 1880's).

Now recall to mind the fact that in our effort to recreate Bartoli=s deduction of the pressure of light, we found that the energy in a pulse of light stands proportional to the light=s frequency. That proportionality remains true to Reality for our blobs, so we can describe the energy in a blob as

E=gv,

(Eq=n 31)

in which g represents some function that converts units of inverse time into units of energy. Can we now determine the form of the function that g represents?

In accordance with Gustav Kirchhoff=s description of blackbody radiation, we know that in calculating the energy emitted by a hot body we must use only the wavelength of the radiation and the absolute temperature in our calculation. The function g cannot involve the wavelength, because that would make the energy of the blob dependent upon a power of the frequency other than the first (recall that wavelength and frequency relate to each other through the invariant speed of light). Thus, we can only have g=g(T). That function represents the proposition that either the blobs with a given frequency all have the same energy and that it varies with the temperature of the emitting body or that the blobs with a given frequency possess energies distributed over some range as a function of the emitting body=s temperature.

In either of those cases we can contrive the following situation: Facing each other across a gap we have a square-faced body at a high temperature and an equally square-faced body at a low temperature. The heat radiation from one body passing to the other body passes through a square area in the gap. We know that through that square area more heat must pass from the hot body to the cold body than passes from the cold body to the hot body in order that this system obey the second law of thermodynamics (which Clausius stated as Aheat cannot of itself pass from a colder to a hotter body.@) And we know that no passive device that we put onto the square area can change that fact.

We know that all blobs of the same frequency must carry their energy at the same density, so blobs carrying more energy must be wider than blobs carrying less energy at the same frequency. We have three possibilities before us.

We might assume that the hot body spews out radiation so violently that it sheds relatively small blobs in vast quantities while the more sedate emission rate from the cold body allows more energy to build up in the blobs before they break away. Thus we assume that the blobs of the same frequency coming from the cold body are wider than the blobs coming from the hot body. The numbers of blobs coming off each body are such that more heat flows from the hot body to the cold body than flows in the opposite direction.

Now we place the following apparatus on the square area between the bodies. Occupying the top half of the square we have two mirrors that come together at a right angle at the top line of the square area. Those mirrors, Mirror #2 and Mirror #3, each make a forty-five degree angle with the square area. On the lower half of the square we mount a vertical mirror. From the top of that mirror we extend Mirror #1 making a forty-five degree angle with the square area on the cold body=s side and we extend a fine wire mesh making a forty-five degree angle with the square area on the hot body=s side. We have made the openings in the wire mesh just wide enough to let blobs from the hot body pass through. With that apparatus in place, blobs from the hot body come to the square area, pass through the mesh, bounce off the vertical mirror, pass back through the mesh, and return to the hot body. Meanwhile, the blobs from the cold body bounce off Mirror #1, Mirror #2, Mirror #3, and the mesh and go to the hot body. Thus we have created a passive device that makes heat move from a cold body to a hot body, in violation of the law of entropy.

We might alternatively assume that the hot body, having more energy to give, sheds larger blobs of radiation of a given frequency than the cold body does. In this case we need only lay a wire mesh on the square area. If the holes in the mesh are just large enough to let the smaller blobs pass through, then the larger blobs coming from the hot body will bounce off and return to the hot body while the smaller blobs coming from the cold body will pass through the mesh and go to the hot body. Again we have contrived a passive device that makes heat flow from cold to hot, in violation of the second law of thermodynamics.

We cannot create any passive device that violates the second law, so in the analysis above we must dismiss the premise that makes the violation possible. The only premise available for dismissal is our assumption that blobs of heat radiation at a given frequency can carry different amounts of energy. So we are left with only our third possibility, which must, of necessity, be true to Reality. Now we must assert as true to Reality the statement that all blobs of a given frequency carry the same amount of energy, regardless of how they are created. That statement necessarily includes creating a blob by applying the Doppler shift to a blob of a different frequency. That fact, in turn, necessitates that g have the same value for all frequencies and that it depend on no variables whatsoever.

So now we know that g represents an absolute constant, which we rewrite as h, so that Equation 31 becomes

E=hv.

(Eq=n 32)

We recognize h as representing Planck=s constant and our electromagnetic blobs as the quanta in Planck=s paper on blackbody radiation and as the photons of modern physics. That recognition leads to two final thoughts.

If Clausius had pursued the implications of his energy density formula, as I have done, he could have laid the foundation of the quantum theory perhaps twenty years or more (between 1876 and 1888) before Max Planck and Albert Einstein did (1900 - 1905). How might physics have developed had he done so? Instead of Einstein using Planck=s quantum hypothesis to explain the photoelectric effect in 1905 (in AOn a Heuristic Viewpoint Concerning the Production and Transformation of Light@, published in Annalen der Physik) we might have had Heinrich Hertz in 1887 using his observation of the photoelectric effect to offer experimental proof and verification of the Clausian quantum theory. That theory would have given physicists a boost in understanding the phenomenon of radioactivity when Antoine Henri Becquerel (1852 Dec 15 - 1908 Aug 25) discovered it in March 1896. I will leave to others further contemplations of how physics might have developed differently in this case.

And secondly, I note that we have deduced the foundation of the old quantum theory from purely classical thermodynamics. We didn=t even refer to the statistical thermodynamics of Maxwell and Boltzmann. This most modern of the theories of physics comes straight from the coal, steam, and iron physics of the Victorian Era. And that fact is more bizarre to me than the quantum theory that I learned in school.

habg