History of the Entropy of Radiation
Discovering the past to understand the present
(\Huge{The}) history of the entropy of radiation is the story of radiation thermodynamics. It is usually attributed to William Churchill the quote "the history is written by the winners", and in this history, the winner was the energy radiation distribution, or better said, the loser was the entropy of radiation. Nowadays, many textbooks explain the history of radiation eliminating almost completely the role of entropy, giving only a partial picture of the real story.
The study of radiation goes back to Newton, Galileo and even further. Even though the Muslim world discovered many principles of the optics, those works were not fully known nor available in the occidental world. The interest in those principles and laws reappeared with the origin of the telescope and the study of the refraction and reflexion processes. Johannes Kepler, who found the inversesquare law governing the intensity of light, also noticed something that will be mentioned later: In the "De Cometis libelli tres" [Kepler, 1619] he pointed out that the tail of comets points away from the sun ^{1}.
The next great advance in the study of radiation was by the hand of Sir Isaac Newton. He was a defender of the corpuscular theory of light, and wrote his thoughts about it in his book "Opticks" [Newton, 1704]. Theodoric of Freiberg had show that white light was composed by a mix of colours like a raibow, and using a set of prisms, Newton was able to determine that white light could be separated into its component parts and could be joined back together with a second prism. Although this experiment was a breakthrough by itself, it is interesting to mention something else: in Newton's mind, colour was infused by pressure of radiation on the eye. For him and his corpuscular vision of nature, heat was an indestructible particle that had mass. With the only intention to prove his theory, he jammed a darning needle around the side of this eye until he could poke at its backside. Unfortunately, he was not able to prove his theory, and only noticed white, black, and coloured circles, but he was convinced that light can perform pressure over a surface.
In his study of the optics, he investigated the refraction of light in an attempt to improve the quality of the telescopes, demonstrating that a prism could decompose white light into a spectrum of colours, and then a lens and a second prism could recompose the multicoloured spectrum into white light. Newton was, perhaps, the first one in understanding that light was distributed in different parts. Unluckily, his investigations were not further in the topic, and the world had to wait until the 1800, the date in which another magnificent discovery was made: the existence of infrared radiation by Sir Friedrich William Herschel $($17381822$)$.
Besides the discovery of Uranus, Herschel published a series of papers in which he described a form of heat radiation beyond the red color of the spectrum of visible light, radiation that cannot be seen on a naked eye. As an astronomer, Herschel used glass filters to see the Sun, with the intention to reduce the intensity of the light. He noticed that different filters leaded to different feelings of heat in his eyes. Using a prism and a thermometer, Herschel measured the increment of temperature in a thermometer by the different colours in radiation. He placed the thermometer below the red end of the solar spectrum, in the dark zone, and found that even though he could not see any color, the temperature of the thermometer increased quickly, which necessarily leaded him to the conclusion that some sort of infrared $($"below" red in latin words$)$ radiation existed. With this experiment, Herschel went a step further than Newton and this was, perhaps, the first attempt to measure the distribution of energy in radiation. Jonhann Wilhem Ritter $($17761810$)$ completed this task when he detected ultraviolet radiation by irradiating silver chloride.
The discovery of Herschel brought more questions than answers. He proved that radiation can heat up a thermometer, so the question to be solved was 'what is heat?'. At the time, scientist around the world were working on this question with the purpose of optimizing the efficiency of heat engines.
Particularly in England, inventors such as Thomas Savery $($16501715$)$ and Thomas Newcomen $($16631729$)$ developed stream engines which used coal to heat the boiler of the engine and produce work. The engine was originally used to pump water from mines, but other inventors rapidly recognized its importance and spread the use of the stream engine to other fields.
Although the stream engine idea was good, the efficiency was extremely low to be considered useful, and engineers in England had an incentive to work in the development of more efficient machines. England was one of the more advanced countries in patent law, and engineers could make a lot of money if their ideas improve the efficiency of the engine. James Watt $($17361819$)$ was one of those engineers that improved the efficiency of the machine up to three or four times, making himself rich in the way. Engineers showed a crucial fact, that heat could produce work, but none of them was interested in the nature of heat, only in increasing the efficiency of the transformation.
Instead of creating new machines, a theoretical approach was adopted by a young french named Nicolas Léonard Sadi Carnot $($17961832$)$. Carnot devoted himself to the analysis of heat engines in a different way, asking himself about the limit on the efficiency, and in 1824 published his book "Réflexions sur la puissance motrice du feu et sur les machines propres à déveloper cette puissance" ^{}. Carnot did not investigate heat in relation to radiation directly, but his results proved that the efficiency of the machine is optimal when the working agent is always homogeneous in temperature. Nowadays we call Carnot efficiency to that efficiency, regarding the agents, since all provide the same work.
La puissance motrice de la chaleur est indépendente des agens mis en oeuvre pour la réaliser ; sa quantité est fixée uniquement par les temperatures entre lesquels se fait en dernier résultat le transport du calorique.^{}
The nature of heat remained still unknown, and Carnot believed in what was called the caloric theory. He thought that in the optimal engine the caloric entering came out unchanged in amount. In his studies of efficiency, Carnot was not able to determine the mathematical description of the efficiency of the transformation. He reached the conclusion that it was an universal function of the temperature, but the functional form was undetermined.
Seems that at the time of his dead $($at the age of 36 of cholera, after being interned in an asylum suffering from "mania" and "general delirium" years before$)$, Carnot was not so sure about the caloric theory. He left unfinished manuscripts, where he expressed his thoughts about the impossibility to produce work by cooling a heat bath without transmitting heat to a reservoir of lower temperature.
Suggested reading: Reflections on the motive power of fire and on machines fitted to develop that power. #Carnot
The work of Carnot was continued by Benoît Pierre Émile Clapeyron $($17991864$)$. Although in the same line of work, the methods employed by Clapeyron were very different from the discuss and reflexions of Carnot. While Carnot used words and thought experiments, Clapeyron used the methods that are currently used in theoretical physics, developing graphical representations and the mathematical formalism of classical thermodynamics.
For Clapeyron, the work of a process equals the area below its graph, and the Carnot cycle was a closed curve. In his work, Clapeyron developed the idea of a reversible process and made a definitive statement of Carnot's principle, now known as the second law of thermodynamics. In 1834 he published his best known report called "Mémoire sur la puissance de la chaleur" ^{}.
The nature of heat was starting to be a fundamental piece of knowledge in the development of thermodynamics, and scientist and engineers questioned the validity of the caloric theory. William Thomson $($18241907$)$, known as Lord Kelvin since 1892, was a strong believer in the caloric theory and a pioneer in the studies about the concept of the temperature. Among many other things, we owe him the absolute scale for its measurement and the correct determination of the absolute lower limit, the absolute zero. Recognized in living, his work was commended by scientist of the size of Maxwell.
Kelvin was a men truly interested in the nature of heat. While at the beginning the claimed that:
The conversion of heat $($or caloric$)$ into mechanical effect is probably impossible, certainly undiscovered.
Years later he supported the view that "the whole theory of the motive power of heat is founded on ... two ... propositions, due respectively to Joule, and to Carnot and Clausius".
The works of Mayer, Joule and Helmholtz on the first law of thermodynamics were very well known by Kelvin $($a partner of Joule who, at the time, was trying to discard the caloric theory of heat$)$, and he had the idea that there is a continuous degradation of energy, which is dissipated into heat, showing an excellent insight on the fundamentals of thermodynamics. He was also aware about the work of Carnot, and tried to determine the universal function proposed by Carnot based on experimental measurements while working on Paris under the supervision of Regnault. He did not succeed, and the solution came shortly after by the hand of Clausius.
One could say that Rudolf Julius Emmanuel Clausius $($18221888$)$, born Rudolf Gottlieb, made to thermodynamics what Newton made to mechanics. His PhD thesis was a mathematical approach to the refraction of light, but his most famous paper, published in 1850, dealt with the mechanical nature of heat, "Über die bewegende Kraft der Wärme" ^{}. Clausius showed that there was a contradiction between Carnot's principle and the conservation of energy, and that the total heat exchange of an infinitesimal Carnot cycle is equal to the work and not zero. In this way, heat was not a state function anymore, and he introduced the notion of internal energy, U, generally a function of temperature and volume.
This internal energy, being a state function, must be conserved in a Carnot cycle. It should be the sum of the free heat and the heat consumed in doing the internal work
\begin{equation} dU = dQ  pdV \end{equation}And so, the nature of heat was identified as a form of energy, and the first law of thermodynamics had a mathematical formulation. This formulation allowed him to determine the universal function proposed by Carnot, determining the maximum efficiency as a function of the absolute temperature of the cold reservoir, $T_C$, and the absolute temperature of the hot reservoir, $T_H$:
\begin{equation} \eta_{\text{max}} = 1  \frac{T_C}{T_H} \end{equation}where $\eta$ is the ratio of the work done by the engine to the heat drawn out of the hot reservoir.
Although Clausius was already famous among scientist, for him this was just the beginning of his research. He gained international respect for his most famous statement published in 1854
Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.
but in 1865 made a discovery even more important. Due to the energy conservation, the fist law of thermodynamics is unable to quantify the effects of friction and dissipation. Therefore, in his research Carnot described that if the body of the working substance, such as a body of steam, is returned to its original state at the end of a complete engine cycle, then "no change occurs in the condition of the working body". Clausius objected to that supposition, arguing that there is indeed a change, a transformation of the energy content. He coined the term entropy to quantify this change and gave a mathematical formulation of the concept. The word "entropy" was chosen because of its meaning, from Greek entropein, "content transformative" or "transformation content". Clausius "intentionally formed the word Entropy as similar as possible to the word Energy". The two laws of thermodynamics were formulated, and summarized in 1865 by Clausius as:
The energy of the universe is constant.
The entropy of the universe tends to a maximum.
When analyzing the Carnot's cycle, Clausis found that the heat transfer along the isotherms proportional to the temperature of the system. The mathematical formulation of entropy was given as increments equal to the ratio of incremental heat transfer divided by temperature. If the entropy change in the process is zero, then the process is called reversible. Entropy is then a state function defined as:
\begin{equation} dS = \frac{\delta Q_{\text{rev}}}{T} \end{equation}In reversible processes, the integral is pathindependent:
\begin{equation} \oint \frac{\delta Q_{\text{rev}}}{T} = 0 \end{equation}and, in general, the entropy is defined as:
\begin{equation} dS \geq \frac{\delta Q}{T} \end{equation}So far, we have only described the fundamentals of thermodynamics and no reference has been made to radiation.
The standard story of the development of radiation law and the beginning of quantum mechanics is usually told in terms of energy, when the reality is that the thermodynamical entropy played the main role in the analysis of radiation. The unfortunate fate of the entropy was to be buried under an unrealistic history about the failure of classic physics, profound disagreements between theory and experiments and the so called ultraviolet catastrophe. Then a prodigious Max Planck introduced  almost magically  the energy quanta and quantum mechanics was generally accepted immediately.
The truth is that Wien's law was believed to be correct at the time, only a slight disagreement was found for long wave radiation, and RayleighJeans equation and the "Ultraviolet catastrophe", named by Ehrenfest, come across years after Planck's proposition of his law. The quantum theory and the radiation law were the result of the mastering of thermodynamics by Wien, Boltzmann, Planck, Einstein and their contemporary.
According to the electromagnetic theory, the energy distribution law was determined as soon as the entropy S of a linear resonator which interacts with the radiation were known as function of the vibrational energy U. Wien was aware of the importance of the entropy [Wien, 1894], and so was Planck.
At the time, Wien's law [Wien, 1896] was proven valid for short wavelengths but not completely accurate for the whole spectra, and Rayleigh had proposed a formula valid for long wave radiation [Rayleigh, 1900]. From Rayleigh formula, the relation between entropy and energy was of the kind of:
\begin{equation} \frac{\partial^2 S}{\partial^2 U} = \frac{\text{const.}}{U} \end{equation}The expression on the righthand side of this functional equation is the change in entropy since n identical processes occur independently, the entropy changes of which must simply add up [Planck, 1900].
On the other hand, from Wien's distribution law the relation would be something of the sort:
\begin{equation} \frac{\partial^2 S}{\partial^2 U} = \frac{\text{const.}}{U^2} \end{equation}Analyzing a variety of completely arbitrary expressions, Planck proposed the simplest equation $($besides the Wien's one$)$ which yield S as a logarithmic function of U, and coincided with the Wien's law for small values of U. The logarithmic relation was a constrain from the probability theory of Mr. Boltzmann, who's works were known by Planck and were the base upon which the theory was developed afterwards. Without further justification, Planck included a new term as a series expansion, proportional to $U^2$:
\begin{equation} \frac{\partial^2 S}{\partial^2 U} = \frac{\text{const.}}{U(\beta + U)} \end{equation}Using this last expression and the relation $\partial S/\partial U = 1/T$, one gets a radiation formula with two constants:
\begin{equation} U = \frac{C\lambda^{5}}{e^{c/\lambda T }  1} \end{equation}Using the available data to fit the constants, the equation resulted in the nowadays named Planck's law.
The formula was, indeed, exact, but it was obtained without any underling theory, so Planck devoted himself to the task of constructing a radiation theory on the base of Boltzmann's statistical mechanics and the logarithmic expression of the entropy:
\begin{equation} S = k \log W \end{equation}where S is the thermodynamic entropy, W is the number of possible microstates, and $k$ is Boltzmann's constant (introduced later by Planck). In a system composed by incoherent radiation beams [Planck, 1914] [Einstein, 1905], the total entropy can be expressed as the sum $S = S_1 + S_2$ which implies that:
\begin{equation} W = W_1\cdot W_2 \end{equation}where W is the number of ways in which we can distribute P energy elements over N hypothetical resonators. Using combinatory analysis, Planck obtained the expression for the microstates [Planck, 1900b], and using the Boltzmann's entropy expression, he obtained the entropy distribution [Planck, 1901]:
\begin{equation} S = k \left[ \left(N+P\right) \ln \left(N+P \right)  N \ln N  P \ln P \right] \end{equation}As the aim of Planck was to obtain the energy distribution, he made the marvelous hypothesis of discrete energy, $\epsilon = h\nu$, motivated by Boltzmann's works, obtaining the expression:
\begin{equation} S = k \left[ \left(1+ \frac{U}{h \nu} \right) \ln \left(1 + \frac{U }{h \nu} \right)  \frac{U}{h \nu} \ln \frac{U}{h \nu} \right] \end{equation}At this point, differentiating with respect to U and using the relation $\partial S/\partial U = 1/T$, Planck obtained:
\begin{equation} \frac{1}{T} = \frac{k}{h \nu} \ln \left(1 + \frac{h \nu}{U} \right) \end{equation}which directly gave him the expression for the energy distribution law that he was looking for:
\begin{equation} U = \frac{h\nu}{exp (h\nu / kT )  1} \end{equation}With this expression, going back to the entropy, it is possible to obtain the spectral entropy of radiation [Planck, 1914]:
\begin{equation} \begin{aligned} \displaystyle S_\nu = \frac{k \nu^2}{c^2} & \left\{ \left(1 + \frac{1}{e^{\frac{h \nu}{k T}}  1} \right) \log \left(1 + \frac{1}{e^{\frac{h \nu}{k T}}  1} \right) \right.\\ &  \left. \frac{1}{e^{\frac{h \nu}{k T}}  1} \log \left( \frac{1}{e^{\frac{h \nu}{k T}}  1} \right) \right\} \end{aligned} \label{eq:planckentropy} \end{equation}or equivalently in wavelength:
\begin{equation} \displaystyle L_\lambda = \frac{2hc^2}{\lambda^5}\frac{1}{e^{\frac{h c}{k \lambda T}}  1} \end{equation} \begin{equation} \displaystyle S_\lambda = \frac{2kc}{\lambda^4} \left\{ \left(1 + \frac{\lambda^5 L_\lambda }{2hc} \right) \log \left(1 + \frac{\lambda^5 L_\lambda }{2hc} \right)  \frac{\lambda^5 L_\lambda }{2hc} \log \frac{\lambda^5 L_\lambda }{2hc} \right\} \end{equation}Go to Quiz #1 to test your knowledge. Quiz #1

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Reflections on the motive power of fire and on machines fitted to develop that power.

The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.

On the Moving Force of Heat and the Laws of Heat which may be Deducted Thereform.
References
 Kepler, J. [1619], De cometis libelli tres, Augustae Vindelicorum, typis Andreae Apergeri.
 Newton, I. [1704], Opticks: or, A treatise of the reflexions, refractions, inflexions and colours of light, London: Printed for Sam. Smith, and Benj. Walford, Printers to the Royal Society, at the Prince's Arms in St. Paul's Churchyard