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  Planck understood that Clausius was not simply stating the obvious, but something of deep significance. Heat, the transfer of energy from A to B due to a temperature difference, explained such everyday occurrences as a hot cup of coffee getting cold and an ice cube in a glass of water melting. But left undisturbed, the reverse never happened. Why not? The law of conservation of energy did not forbid a cup of coffee from getting hotter and the surrounding air colder, or the glass of water becoming warmer and the ice cooler. It did not outlaw heat flowing from a cold to a hot body spontaneously. Yet something was preventing this from happening. Clausius discovered that something and called it entropy. It lay at the heart of why some processes occur in nature and others do not.

  When a hot cup of coffee cools down, the surrounding air gets warmer as energy is dissipated and irretrievably lost, ensuring that the reverse cannot happen. If the conservation of energy was nature’s way of balancing the books in any possible physical transaction, then nature also demanded a price for every transaction that actually occurred. According to Clausius, entropy was the price for whether something happened or not. In any isolated system only those processes, transactions, in which entropy either stayed the same or increased were allowed. Any that led to a decrease of entropy were strictly forbidden.

  Clausius defined entropy as the amount of heat in or out of a body or a system divided by the temperature at which it takes place. If a hot body at 500 degrees loses 1000 units of energy to a colder body at 250 degrees, then its entropy has decreased by –1000/500 = –2. The colder body at 250 degrees has gained 1000 units of energy, +1000/250, and its entropy has increased by 4. The overall entropy of the system, the hot and cold bodies combined, has increased by 2 units of energy per degree. All real, actual processes are irreversible because they result in an increase in entropy. It is nature’s way of stopping heat from passing spontaneously, of its own accord, from something cold to something hot. Only ideal processes in which entropy remains unchanged can be reversed. They, however, never occur in practice, only in the mind of the physicist. The entropy of the universe tends towards a maximum.

  Alongside energy, Planck believed that entropy was ‘the most important property of physical systems’.26 Returning to Munich University after his year-long sojourn in Berlin, he devoted his doctoral thesis to an exploration of the concept of irreversibility. It would be his calling card. To his dismay, he ‘found no interest, let alone approval, even among the very physicists who were closely concerned with the topic’.27 Helmholtz did not read it; Kirchhoff did, but disagreed with it. Clausius, who had such a profound influence on him, did not even answer his letter. ‘The effect of my dissertation on the physicists of those days was nil’, Planck recalled with some bitterness even 70 years later. But driven by ‘an inner compulsion’, he was undeterred.28 Thermodynamics, particularly the second law, became the focus of Planck’s research as he began his academic career.29

  German universities were state institutions. Extraordinary (assistant) and ordinary (full) professors were civil servants appointed and employed by the ministry of education. In 1880 Planck became a privatdozent, an unpaid lecturer, at Munich University. Employed neither by the state nor the university, he had simply gained the right to teach in exchange for fees paid by students attending his courses. Five years passed as he waited in vain for an appointment as an extraordinary professor. As a theorist uninterested in conducting experiments, Planck’s chances for promotion were slim, as theoretical physics was not yet a firmly established distinct discipline. Even in 1900 there were only sixteen professors of theoretical physics in Germany.

  If his career was to progress, Planck knew that he had ‘to win, somehow, a reputation in the field of science’.30 His chance came when Göttingen University announced that the subject for its prestigious essay competition was ‘The Nature of Energy’. As he worked on his paper, in May 1885, ‘a message of deliverance’ arrived.31 Planck, aged 27, was offered an extraordinary professorship at the University of Kiel. He suspected it was his father’s friendship with Kiel’s head of physics that had led to the offer. Planck knew there were others, more established than he, who would have expected advancement. Nevertheless, he accepted and finished his entry for the Göttingen competition shortly after arriving in the city of his birth.

  Even though only three papers were submitted in search of the prize, an astonishing two years passed before it was announced that there would be no winner. Planck was awarded second place and denied the prize by the judges because of his support for Helmholtz in a scientific dispute with a member of the Göttingen faculty. The behaviour of the judges drew the attention of Helmholtz to Planck and his work. After a little more than three years at Kiel, in November 1888, Planck received an unexpected honour. He had not been first, or even second choice. But after others had turned it down, Planck, with Helmholtz’s backing, was asked to succeed Gustav Kirchhoff at Berlin University as professor of theoretical physics.

  In the spring of 1889, the capital was not the city Planck had left eleven years earlier. The stench that always shocked visitors had disappeared as a new sewer system replaced the old open drains, and at night the main streets were lit by modern electric lamps. Helmholtz was no longer head of the university’s physics institute but running the PTR, the majestic new research facility three miles away. August Kundt, his successor, had played no part in Planck’s appointment, but welcomed him as ‘an excellent acquisition’ and ‘a splendid man’.32

  In 1894 Helmholtz, aged 73, and Kundt, only 55, both died within months of each other. Planck, only two years after finally being promoted to the rank of ordinary professor, found himself as the senior physicist at Germany’s foremost university at just 36. He had no choice but to bear the weight of added responsibilities, including that of adviser on theoretical physics for Annalen der Physik. It was a position of immense influence that gave him the right of veto on all theoretical papers submitted to the premier German physics journal. Feeling the pressure of his newly elevated position and a deep sense of loss at the deaths of his two colleagues, Planck sought solace in his work.

  As a leading member of Berlin’s close-knit community of physicists, he was well aware of the ongoing, industry-driven blackbody research programme of the PTR. Although thermodynamics was central to a theoretical analysis of the light and heat radiated by a blackbody, the lack of reliable experimental data had stopped Planck from trying to derive the exact form of Kirchhoff’s unknown equation. Then a breakthrough by an old friend at PTR meant that he could no longer avoid the blackbody problem.

  In February 1893, 29-year-old Wilhelm Wien discovered a simple mathematical relationship that described the effect of a change in temperature on the distribution of blackbody radiation.33 Wien found that as the temperature of a blackbody increases, the wavelength at which it emits radiation with the greatest intensity becomes ever shorter.34 It was already known that the rise in temperature would result in an increase in the total amount of energy radiated, but Wien’s ‘displacement law’ revealed something very precise: the wavelength at which the maximum amount of radiation is emitted multiplied by the temperature of a blackbody is always a constant. If the temperature is doubled, then the ‘peak’ wavelength will be half the previous length.

  Figure 2: Distribution of blackbody radiation which shows Wien’s displacement law

  Wien’s discovery meant that once the numerical constant was calculated by measuring the peak wavelength – the wavelength that radiates most strongly, at a certain temperature – then the peak wavelength could be calculated for any temperature.35 It also explained the changing colours of a hot iron poker. Starting at low temperatures, the poker emits predominantly long-wavelength radiation from the infrared part of the spectrum. As the temperature increases, more energy is radiated in each region and the peak wavelength decreases. It is ‘displaced’ towards the shorter wavelengths. Consequently the colour of the emitted light changes from red to orange, then yellow and finally a blui
sh-white as the quantity of radiation from the ultraviolet end of the spectrum increases.

  Wien had quickly established himself as a member of that endangered breed of physicist, one who was both an accomplished theorist and a skilled experimenter. He found the displacement law in his spare time and was forced to publish it as a ‘private communication’ without the imprimatur of the PTR. At the time he was working as an assistant in the PTR’s optics laboratory under the leadership of Otto Lummer. Wien’s day job was the practical work that was a prerequisite for an experimental investigation of blackbody radiation.

  Their first task was to construct a better photometer, an instrument capable of comparing the intensity of light – the amount of energy in a given wavelength range – from different sources such as gas lamps and electric bulbs. It was the autumn of 1895 before Lummer and Wien devised a new and improved hollow blackbody capable of being heated to a uniform temperature.

  While he and Lummer developed their new blackbody during the day, Wien continued to spend his evenings searching for Kirchhoff’s equation for distribution of blackbody radiation. In 1896, Wien found a formula that Friedrich Paschen, at the University of Hanover, quickly confirmed agreed with the data he had collected on the allocation of energy among the short wavelengths of blackbody radiation.

  In June that year, the very month the ‘distribution law’ appeared in print, Wien left the PTR for an extraordinary professorship at the Technische Hochschule in Aachen. He would win the Nobel Prize for physics in 1911 for his work on blackbody radiation, but left Lummer to put his distribution law through a rigorous test. To do so required measurements over a greater range and at higher temperatures than ever before. Working with Ferdinand Kurlbaum and then Ernst Pringsheim, it took Lummer two long years of refinements and modifications but in 1898 he had a state-of-the-art electrically heated blackbody. Capable of reaching temperatures as high as 1500°C, it was the culmination of more than a decade of painstaking work at the PTR.

  Plotting the intensity of radiation along the vertical axis of a graph against the wavelength of the radiation along the horizontal axis, Lummer and Pringsheim found that the intensity rose as the wavelength of radiation increased until it peaked and then began to drop. The spectral energy distribution of blackbody radiation was almost a bell-shaped curve, resembling a shark’s dorsal fin. The higher the temperature, the more pronounced the shape as the intensity of radiation emitted increased. Taking readings and plotting curves with the blackbody heated to different temperatures showed that the peak wavelength that radiated with maximum intensity was displaced towards the ultraviolet end of the spectrum with increasing temperature.

  Lummer and Pringsheim reported their results at a meeting of the German Physical Society held in Berlin on 3 February 1899.36 Lummer told the assembled physicists, among them Planck, that their findings confirmed Wien’s displacement law. However, the situation regarding the distribution law was unclear. Although the data was in broad agreement with Wien’s theoretical predictions, there were some discrepancies in the infrared region of the spectrum.37 In all likelihood these were due to experimental errors, but it was an issue, they argued, that could be settled only once ‘other experiments spread over a greater range of wavelengths and over a greater interval of temperature can be arranged’.38

  Within three months Friedrich Paschen announced that his measurements, though conducted at a lower temperature than those of Lummer and Pringsheim, were in complete harmony with the predictions of Wien’s distribution law. Planck breathed a sigh of relief and read out Paschen’s paper at a session of the Prussian Academy of Sciences. Such a law appealed deeply to him. For Planck the theoretical quest for the spectral energy distribution of blackbody radiation was nothing less than the search for the absolute, and ‘since I had always regarded the search for the absolute as the loftiest goal of all scientific activity, I eagerly set to work’.39

  Soon after Wien published his distribution law, in 1896, Planck set about trying to place the law on rock-solid foundations by deriving it from first principles. Three years later, in May 1899, he thought he had succeeded by using the power and authority of the second law of thermodynamics. Others agreed and started calling Wien’s law by a new name, Wien-Planck, despite the claims and counter-claims of the experimentalists. Planck remained confident enough to assert that ‘the limits of validity of this law, in case there are any at all, coincide with those of the second fundamental law of the theory of heat’.40 He advocated further testing of the distribution law as a matter of urgency, since for him it would be a simultaneous examination of the second law. He got his wish.

  At the beginning of November 1899, after spending nine months extending the range of their measurements as they eliminated possible sources of experimental error, Lummer and Pringsheim reported that they had found ‘discrepancies of a systematic nature between theory and experiment’.41 Although in perfect agreement for short wavelengths, they discovered that Wien’s law consistently overestimated the intensity of radiation at long wavelengths. However, within weeks Paschen contradicted Lummer and Pringsheim. He presented another set of new data and claimed that the distribution law ‘appears to be a rigorously valid law of nature’.42

  With most of the leading experts living and working in Berlin, the meetings of the German Physical Society held in the capital became the main forum for discussions concerning blackbody radiation and the status of Wien’s law. It was the subject that again dominated the proceedings of the society at its fortnightly meeting on 2 February 1900 when Lummer and Pringsheim disclosed their latest measurements. They had found systematic discrepancies between their measurements and the predictions of Wien’s law in the infrared region that could not be the result of experimental error.

  This breakdown of Wien’s law led to a scramble to find a replacement. But these makeshift alternatives proved unsatisfactory, prompting calls for further testing at even longer wavelengths to unequivocally establish the extent of any failure of Wien’s law. It did, after all, agree with the available data covering the shorter wavelengths, and all other experiments bar those of Lummer-Pringsheim had found in its favour.

  As Planck was only too well aware, any theory is at the mercy of hard experimental facts, but he strongly believed that ‘a conflict between observation and theory can only be confirmed as valid beyond all doubt if the figures of various observers substantially agree with each other’.43 Nevertheless, the disagreement between the experimentalists forced him to reconsider the soundness of his ideas. In late September 1900, as he continued to review his derivation, the failure of Wien’s law in the deep infrared was confirmed.

  The question was finally settled by Heinrich Rubens, a close friend of Planck’s, and Ferdinand Kurlbaum. Based at the Technische Hochschule on Berlinerstrasse, where at the age of 35 he had recently been promoted to ordinary professor, Rubens spent most of his time as a guest worker at the nearby PTR. It was there, with Kurlbaum, that he built a blackbody that allowed measurements of the uncharted territory deep within the infrared region of the spectrum. During the summer they tested Wien’s law between wavelengths of 0.03mm and 0.06mm at temperatures ranging from 200 to 1500°C. At these longer wavelengths, they found the difference between theory and observation was so marked that it could be evidence of only one thing, the breakdown of Wien’s law.

  Rubens and Kurlbaum wanted to announce their results in a paper to the German Physical Society. The next meeting was on Friday, 5 October. With little time to write a paper, they decided to wait until the following meeting two weeks later. In the meantime, Rubens knew that Planck would be eager to hear the latest results.

  It was among the elegant villas of bankers, lawyers, and other professors in the affluent suburb of Grunewald in west Berlin that Planck lived for 50 years in a large house with an enormous garden. On Sunday, 7 October, Rubens and his wife came for lunch. Inevitably the talk between the two friends soon turned to physics and the blackbody problem. Rubens explained that his
latest measurements left no room for doubt: Wien’s law failed at long wavelengths and high temperatures. Those measurements, Planck learnt, revealed that at such wavelengths the intensity of blackbody radiation was proportional to the temperature.

  That evening Planck decided to have a go at constructing the formula that would reproduce the energy spectrum of blackbody radiation. He now had three crucial pieces of information to help him. First, Wien’s law accounted for the intensity of radiation at short wavelengths. Second, it failed in the infrared where Rubens and Kurlbaum had found that intensity was proportional to the temperature. Third, Wien’s displacement law was correct. Planck had to find a way to assemble these three pieces of the blackbody jigsaw together to build the formula. His years of hard-won experience were quickly put into practice as he set about manipulating the various mathematical symbols of the equations at his disposal.

  After a few unsuccessful attempts, through a combination of inspired scientific guesswork and intuition, Planck had a formula. It looked promising. But was it Kirchhoff’s long-sought-after equation? Was it valid at any given temperature for the entire spectrum? Planck hurriedly penned a note to Rubens and went out in the middle of the night to post it. After a couple of days, Rubens arrived at Planck’s home with the answer. He had checked Planck’s formula against the data and found an almost perfect match.