Revista CENIC. Ciencias Químicas ISSN: 1015-8553 [email protected] Centro Nacional de Investigaciones Científicas Cuba

Wisniak, Jaime Josef Stefan. Radiation, conductivity, diffusion, and other phenomena Revista CENIC. Ciencias Químicas, vol. 37, núm. 3, 2006, pp. 188-195 Centro Nacional de Investigaciones Científicas La Habana, Cuba

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Josef Stefan. Radiation, conductivity, diffusion, and other phenomena

Jaime Wisniak.

Department of Chemical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel 84105, Email: [email protected].

Recibido: 17 de julio de 2005. Aceptado: 5 de noviembre de 2005.

Palabras clave: conductividad térmica de un gas, radiación, cuerpo negro, ley de la cuarta potencia, fusión de la capa de hielo polar, coeficiente de difusividad, tono, sonido, ley de radiación. Key words: thermal conductivity of gases, radiation, black body, fourth-power radiation law, melting of the polar ice, diffusivity coefficient, pitch of sound, radiation law.

RESUMEN. A Josef Stefan (1835-1893) le debemos el desarrollo de un método his parents are said to have been il- preciso para medir la conductividad de un gas, la determinación empírica de la literate. ley que describe la radiación de un cuerpo negro (la ley de la cuarta potencia), el Josef attended primary school in análisis del movimiento de la interfase en la fusión de la capa de hielo polar, un his hometown where his teachers método para medir el coeficiente de difusividad y un análisis del tono de un observed his talents and recom- sonido. Aun cuando la derivación de Stefan de la ley de radiación se basó en mended further schooling. In 1845 datos experimentales de otros que más tarde se probó que estaban errados, he entered the Klagenfurt gymna- Boltzmann demostró posteriormente su validez usando argumentos termodi- sium and graduated eight years later. námicos. In school he showed a particular interest in learning the Slovenian ABSTRACT. To Josef Stefan (1835-1893) we owe the development of a precise method for measuring the heat conductivity of gases, the empirical determina- language and literature. After gradu- tion of the law describing the radiation from a black body (fourth-power radia- ation he moved to Vienna to study tion law), analysis of the moving interface in the melting of the polar ice, a method Mathematics and Physics at the for measuring the diffusivity coefficient, and analysis of the pitch of sound. Philosophical Faculty of the Univer- Although Stefan’s derivation of the heat radiation law was based on experimen- sity of Vienna. In 1857, while in his tal data of others, which were later proven to be wrong, Stefan’s equation was fourth year of studies, he passed the later proven to be correct by Boltzmann based on theoretical thermodynamic teacher’s examination. At that time reasoning. he was already giving physics lec- tures for pharmacy students. After passing the examination he also taught at a private secondary school LIFE AND CAREER (the oberrealeschule in the inner Josef Stefan (Figures 1 and 2) is State) and even took part in admin- well known for his discovery of the istering the school as deputy school- fourth-power heat radiation law that master.3 carries his name. Most of the publi- On his own initiative he began cations related to him describe this research in theoretical physics, pre- particular contribution, but almost pared two papers and sent the third none give personal details about his to the Academy of Science. Its read- life or his seminal contributions to ing at the Academy was well re- other physical and thermodynamic ceived and attracted the interest of subjects.1-3 Only Strand’s pamphlet Carl Ludwig (1816-1895), a well- goes into some details and his work known professor of Physiology, who will be used here. invited him to collaborate in experi- Josef Stefan was born on March mental work at the Institute of 24, 1835, at St. Peter, now a part of Physiology. Stephan accepted his in- Celovec (Klagenfurt). He was the il- Fig. 1. Josef Stefan (1835-1893). vitation and worked with Ludwig on legitimate child of Marija Startinik the flow of water through tubes. who worked as a maid. Josef was In 1858 Stefan passed his final eleven years old when his mother ing and bakery products; the family examination at the university, the married his father and went to live was poor but could afford to provide philosophical rigorosum, and was with him. At that time his father had the child with basic education, an granted his doctorate. The following just opened a shop for selling mill- important attitude considering that year he became Privatdozent in 188 Revista CENIC Ciencias Químicas, Vol. 37, No. 3, 2006.

laboratory at the Institute, where he also lived. He did not travel and par- ticipate at scientific meetings that were organized in Europe already at that time. According to Strnad3 Stefan could well have become a Slovenian poet if he had not chosen physics. While living in Carinthia, German was spoken in towns at that time but Slovenians populated the country. In 1848 the Slovenian language be- came an obligatory subject and also a subject at the general examination for Slovenians and a non-obligatory subject for the others. When this event took place Stefan was in the fifth class and from there on he be- gan to write poems in Slovenian under the tutorship of Anton Janezic (1829-1869). In the same year (1849) he and his schoolfellows launched a manuscript literary journal called Slavija. Some of the poems written by Stefan were later published in Slovenian journals appearing in Austro-Hungary. Up to 1853 the po- ems were signed Stefan, late on ei- ther Stefan or with a pseudonym, mainly J. A. Spleteni, after the mean- Fig. 2. Stamps issued by Slovenia and Austria honoring Stefan. ing of the Greek word stephanos (in- terlaced).3 In addition, he started publishing popular scientific articles Mathematical Physics and as such he Andreas von Ettingshausen (1796- in Slovenian magazines and thus was officially entitled to lecture at 1878), retired owing to illness and contributed to the development of the University. In 1860, thanks to the the position was offered to Stefan.3 Slovenian language in natural sci- proposal of Ludwig and his colleague Stefan was a brilliant experimen- ence. Ernst Wilhelm Brücke (1819-1892) he talist and well-liked teacher. He was Stefan was known as a cheerful was elected a corresponding member dean of the Philosophical Faculty youth who loved singing, took part of the Imperial Academy of Sciences. during 1869-1870 and rector mag- in choirs, and was even involved in All these achievements did not ad- nificus of the University in 1876- organizing them. In 1891, two years vance Stefan’s chances for obtaining 1877. In 1860 he became a corre- before his death, he married the a position at the Institute of Phys- sponding member of the Imperial widow Marija Neumann. By the end ics, which would allow him to carry Academy of Sciences (Mathematics of 1892 he had an apoplectic stroke his own research. Nevertheless, the and Natural Sciences Class) and in when he was visiting a friend. For a situation improved. Both professors 1865 he was promoted to member. In couple of weeks he lay unconscious of physiology persuaded a high offi- 1875 he was appointed secretary of and could not be moved back home. cial in the ministry of education to the mathematics-science class of the Then his health improved somewhat attend one of Stefan’s lectures. The same and served as its vice-presi- and he was transferred to his apart- official was so impressed by the lec- dent from 1885 until his death in ment at the Institute where he died ture that when a new position of full 1893. on January 7, 1893, at the age of fifty- professor (Professor Ordinarius) of In the meantime, he pursued re- eight. mathematics and physics was opened search actively and published sev- Josef Stefan may be considered it was offered to Stefan. Thus, in eral papers in the proceedings of the the best-known Slovenian physi- 1863, at the age of 28 he became the Academy that attracted the interest cists; as one of the leading physicists youngest full professor in the of the scientific community. For one of the Austro-Hungarian Empire he Austro-Hungarian Empire. A series of them (on the subject of the nature took advantage of the possibilities a of fortuitous events catapulted his played by non polarized light) he prosperous capital could offer. His scientific career ahead. First, was awarded the Lieben Prize “dedi- increased work in physics separated Wilhelm Joseph Grailich (1829-?), cated to best scientific paper which him little by little from taking take who was expected to take over the was written by an Austrian citizen part in Slovenian affairs and contrib- post of Director of the Institute of during the last three years.”2 uted to his alienation from Ljubljana. Experimental Physics [founded by He dedicated much of his time He was extremely active, as a provi- Christian Doppler (1803-1853) in to lecturing and his lectures were sional list of his duties around 1883 1850], died unexpectedly and thus a carefully prepared and carried out. shows: member of the faculty board, position at the Institute opened be- Stefan was a lone wolf; he rarely Director of the Institute of Physics, came available. Three years later participated in social activities and a member of the International Com- (1865) the Director of the Institute, preferred to spend his time in his mittee on Explosions of Mining 189 Revista CENIC Ciencias Químicas, Vol. 37, No. 3, 2006.

Gases, chairman of the scientific glass plates is a hydrodynamic phe- ues calculated for the gas conductiv- committee of the World Electro- nomenon. ity came out too small or too large, technical Exhibition in Vienna Although all his contributions, according as the walls of the vessel (1883) and the first chairman of the made at the University of Vienna, conducted heat more slowly or more Austrian Society of Electrical Engi- contain important work the most quickly than the air. The first situa- neering and editor-in-chief of its important belong to thermodynam- tion occurred when the walls were publications. In 1885 he was ap- ics. The results of his many activi- made of glass or of iron, the second pointed chairman of the Interna- ties resulted in the publication of when they were made of zinc. The tional Committee on Music Pitch in 82 papers, almost without exception only clear result was that the ther- Vienna. He was a member of several in the proceedings of the Vienna mal conductivity of air was between foreign scientific academies, held Academy of Sciences. Twelve of that of iron and zinc. numerous Austrian and foreign hon- them dealt with mechanics and hy- In the second experimental ar- ors, and was both royal and privy drodynamics, seven with acoustics, rangement, an enclosed mass of air Imperial councillor. As a “councilor twenty-five with thermodynamics was uniformly warmed or cooled at court” he would be entitled to use and the kinetic theory of gases, from all sides. In the first runs, “von” with his name if only he would twelve with optics, and twenty-six Stefan employed spherical air ther- apply for this. He did not.3 with electricity and magnetism. He mometers made of copper plate, but began his research activities in the the resulting values of the conduct- SCIENTIFIC CONTRIBUTION field of mechanics. ing power were too large on account Stefan did research in all of the influence of currents. After branches of physics: mechanics, op- Heat and mass transfer these unsuccessful attempts experi- tics, thermodynamics, and electro- As mentioned above, the two ence led him to the invention of a dynamics.4-20 His contributions to main Stefan’s achievements in ther- simple apparatus, which he called a thermodynamics, particularly in modynamics are related to heat con- diathermometer, consisting of a heat transfer, heat conduction, radia- duction in gases and to the radiation double wall thermometer made of tion, and gas absorption, are prob- law. Stefan was the first to measure brass or copper plate, the space be- ably the best known. He was the first the thermal conductivity of gases tween the metallic envelopes being to measure correctly the heat con- and was led to the fourth-power ra- filled by the gas under examination. ductivity of gases,11 to determine the diation law by considering measure- A tube led from the inner cylinder correct relationship between ther- ments of others. These and other to a mercury manometer. The height mal radiation and temperature,14 and related subjects will be now de- difference in the manometer at con- to study the formation of ice in the scribed in more detail. stant volume was proportional to the Polar seas, giving a special solution Heat conductivity of gases temperature in the inner cylinder. to this non-linear conduction prob- Before Stefan, many physicists First the device was left in the air so lem with phase change.16,18 He was had tried to measure the conduction all its parts were at room tempera- interested in fluid flow and in oscil- of heat in gases but were unable to ture. Then the outer cylinder was lations and this led him to acoustics. achieve what they thought was an placed in a mixture of ice and water. In his first paper in 1857 he consid- indispensable condition: that the gas The outer surface of the outer cylin- ered general oscillatory equations4 stay at rest under a temperature gra- der thus had a constant temperature and problems related to the transver- dient. In all the experimental ar- of 0 oC . The gap was very narrow and sal and longitudinal oscillations of rangements devised the gas was the temperature difference was rods and on the velocity of sound.8.9 heated near the hot body so that its small so that appreciable convection In optics he studied the polarization density diminished and started to could not develop. Conduction of of light,5 double refraction, interfer- move upwards, that is, conductivity heat through the gas gap cooled the ence of light, especially Newton’s was always accompanied by convec- air in the inner cylinder and reduced rings, and measurement of the wave- tion making the results highly un- its pressure. The rate of change of length.7 Afterwards, he put most of reliable. pressure allowed calculating the his efforts on the experimental and To overcome these difficulties heat conductivity of the gas in the theoretical aspects of thermodynam- Stefan followed a different experi- gap. The values obtained with appa- ics and electrodynamics. In a well- mental strategy. Instead of trying to ratus built of different materials known work on the diffusion of achieve a stationary state he opted agreed very well with one another; gases.10-12 Stefan measured the for a non-stationary situation. In the conductivity of air was found to evaporation of liquids in thin long addition, instead of measuring the be 0.000 056 cm/s, which was nearly tubes and calculated the theoretical temperature with thermometers he 20 000 times smaller than that of cop- coefficients of diffusion and of fric- did it by way of the pressure of the per and 3 400 times less than that of tion and their dependence on the ab- gas, which he measured by means of iron. The calculated value compared solute temperature, showing that a manometer. Stefan conducted very well with the one calculated by the calculated values were in agree- many measurements on the subject Maxwell from the dynamical theory ment with the experimental results using different experimental tech- of gases: 0.000 055.21 Stefan noted obtained by James Clerk Maxwell’s niques.12,13 In his first experimental that “the conductivity of air about (1831-1879), Thomas Graham (1805- set up the air was enclosed in cylin- 3 400 times smaller than of iron. 1869), and Joseph Loschmidt (1821- ders and warmed either from above Maxwell states that air must conduct 1895). Another well-known work or cooled from below. The enclosed about 3 500 times worse than related to the relation between sur- air itself formed the thermometric iron”.11,12 Stefan also measured the face tension and evaporation, which substance and the mean temperature conductivity of other gases and included Stefan’s number and at each instant could be determined found, for example, that the conduc- Stefan’s law. He also demonstrated by manometric measurement. The tivity of hydrogen was seven times that the apparent adhesion of two results were disappointing; the val- greater than that of air. An important 190 Revista CENIC Ciencias Químicas, Vol. 37, No. 3, 2006. outcome of Stefan’s experiments ting heat through their surface; the tion, i.e., infrared light. In the part was that it supported Maxwell’s pre- hotter they were the more they emit- of the spectrum beyond red he put a diction that the conductivity of air ted and the intensity of the emitted thermopile and measured the deflec- should not depend on pressure.21 rays changed very considerably with tion of a galvanometer connected to Decreasing the pressure in the air the state of the surface.” it. He did not measure the tempera- gap to one half its normal value did François Marcet Delaroche26 ture of the wire but only gave the not alter the rate of cooling. Stefan (1803-1883) was aware that the heat color of its appearance. wrote “Also another law, which is losses due to radiation increased Adolph Wüllner (1835-1908) came given by this theory, namely the in- more rapidly than in proportion to across the 1865 German translation dependence of the conductivity the temperature difference, but he of Tyndall’s paper and included the from density, has proved correct in did not isolate the radiation from the quoted data into the new edition of a completely indisputable way by other heat losses, as Pierre-Louis his book.30 In pages 214-215 he re- this experiment”.1,12 Stefan ex- Dulong (1785-1838) and Alexis- marked that Tyndall’s experiments plained the deviations from theory Thèrese Petit (1791-1820) attempted indicated that ”the quantity of heat as resulting from the movements of to do a few years later. For radiation emitted increases considerably more atoms against each other within the in empty space Dulong and Petit quickly than does the temperature, molecules. developed a much more complicated especially at higher temperatures.” Heat radiation law, introducing an absolute tem- Moreover, he supplemented Tyndall’s Stefan’s most important work perature scale and extending results by assigning somewhat arbi- dealt with heat radiation. At his time Newton’s law. As was later seen, how- trary numerical quantities to the experimental information about the ever, their law also possessed only observed temperatures: 525 oC to phenomenon and its characteristics limited validity and did not agree faint red and 1 200 oC to full white was accumulating very fast, but still with measured results even up to red. Thus, from the weak red glow there was no equation that described 300 oC . up to the full white glow the inten- it accurately. In the period 1800-1835, experi- sity of the radiation increased almost Newton seems to have been the ments on radiant heat by William twelve-fold, from 10.4 to 122 (exactly first to consider the law of cooling Herschel (1738-1822), Leslie, Mace- 11.7 fold). Wüllner could not guess of a body subject to any constant donio Melloni (1798-1854), and oth- that some 25 years later his arbitrary cooling action, such as, for example, ers showed that radiant heat had quantities would open the stage to the influence of a uniform current most if not all of the properties of the development of the exact rela- of air. He found that during the cool- light. Melloni, in particular, demon- tion between temperature and rate ing of incandescent iron in a con- strated that radiant heat shared all of radiation. stant stream of air equal quantities the qualitative properties of light: By the middle of the nineteenth of air were heated by quantities of reflection, refraction, diffraction, century the science of spectroscopy heat proportional to those that they polarization, interference, etc. had developed enough to prove that removed from the iron.22 In other Until 1861 no experimenter had all glowing solids emitted continu- words, Newton claimed that a hot been able to detect any absorption ous spectra when heated unlike body subject to cooling by a constant of radiant heat by gaseous matter heated gases which emitted bands or temperature source, like an air and it was generally supposed that lines. Eventually, Gustav Robert stream, should lose heat proportion- matter in the gaseous state transmit- Kirchhoff31 (1824-1887) would dis- ally to the instant temperature dif- ted perfectly all kinds of radiation. cover that the power emitted was ference, and the heat losses at equal In 1861 and 1863 Tyndall27-29 con- proportional to the power absorbed, time intervals should form a de- ducted the first convincing experi- that the proportionality constant creasing geometric progression. ments on the transmission of radiant was some function of the tempera- Georg Wolfgang Kraft (1701-1754) heat and the radiative properties of ture and frequency, and to define a and Georg Wilhelm Richmann (1711- gases demonstrating that “perfectly perfectly black body as the one that 1753) found that Newton’s formula colorless and invisible gases and absorbs all the radiations, which fall was able to represent the facts fairly vapors” were able to absorb and emit upon it, of whatever wavelength well for small differences in tem- radiant heat. The elementary gases they may be. For a black body the peratures (a few degrees). For differ- were almost transparent to radiant power absorbed was one so that the ences above 40 or 50 oC they and heat while others were opaque. power emitted was a function of the other experimenters such as George Tyndall’s results indicated that air, temperature and frequency alone. Martine23 (1704-1742), John Leslie oxygen, and nitrogen showed no ab- Draper32 set out to (a) determine (1726-1832), and John Dalton (1766- sorption at all, but compound gases the point of incandescence of plati- 1844), found it to deviate seriously especially ammonia and ethylene, num and to “prove” that different from experimental evidence, and exhibited a very marked effect. The bodies become incandescent at the attempted to replace it with another absorption increased with pressure, same temperature, (b) to determine law according to which the heat but not according to any simple law. the color of the rays emitted by self- losses increases more rapidly than The influence of the temperature of luminous bodies at different tem- what Newton’s law predicted. the source on the transmission of ra- perature, and (c) to determine the Richmann24 restated Newton’s in the diant heat by vapors was very relation between the brilliancy of the form: the speed of cooling is propor- marked. light emitted by a shining body and tional to the difference in tempera- The experimental procedure its temperature. He found that the ture between the heated body and used by Tyndall consisted of heating point of incandescence of platinum the surrounding atmosphere. a platinum wire with an electric cur- was 977 oF and to his conviction, this Fourier in his famous book Ana- rent and leading the radiation was the temperature at which all lytical Theory of Heat25 stated that through a rock salt lens and a prism. solids begin to shine. The luminous “All bodies had the property of emit- He was investigating obscure radia- effects were due to a vibratory move- 191 Revista CENIC Ciencias Químicas, Vol. 37, No. 3, 2006.

ment executed by the molecules of Desains (1817-1885),38 Draper,32 ergy being approximately propor- platinum and the frequency of these Tyndall,26-29 Ericsson,39 and others. tional to the fifth power of the ab- vibrations increased with tempera- Stefan showed that his formula solute temperature, and (c) Wüllner’s ture. In addition, if the quantity of agreed with their results in all tem- temperatures, as remarked al- heat radiated by platinum at 980 oF perature ranges, if allowance was ready, were chosen somewhat ar- was taken as unity, it will have in- made for conduction through the bitrarily. creased at 1 440 oF to 2.5, at 1 900 oF gas. He suggested that Dulong and The theoretical deduction of to 7.8, and at 2 360 oF to about 17.8. Petit had described their data incor- Stefan’s relationship was first During the last quarter of the rectly because their extrapolation achieved in 1884 by Ludwig Boltz- nineteenth century Stefan contin- procedure to eliminate the influence mann40 (1844-1906), Stefan’s most ued to do research on heat transfer of air on the net heat flow could not distinguished student, within the phenomena, including radiation. have eliminated the effect of the context of thermodynamics by Apparently, Stefan’s attention was thermal conductivity of the gas. studying an ideal thermal engine directed to this issue by the low sur- Stefan estimated the thermal con- using radiation instead of a gas and face temperature of the sun calcu- ductivity through air at all pressures taking into account Maxwell’s result lated according to the Dulong-Petit contributed between 10 to 15 % of for the pressure of light.21 The most equation by Claude Pouillet’s (1790- the rates of cooling reported by important of Boltzmann’s results 1868) and Jules Violle’s (1841-1923), Dulong and Petit for a bare ther- was that the relation derived by and by Jonathan Homer Lane (1819- mometer, and up to 50 % for a silver- Stefan was exact only for completely 1880).33 His previous work on the coated thermometer (because of its black bodies. So the law nowadays conductivity of gases had made him low emissivity). is known as the Stefan-Boltzmann aware that heat conduction in a gas Moreover, with the aid of his new law. did not depend on pressure and to formula Stefan could calculate, on Today, both Stefan’s law and realize that the experimental proce- the basis of Pouillet’s and Violle’s Stefan’s constant may be derived dure used by Dulong and Petit had actinometric observations, that the from the radiation law proposed by eliminated convection but not con- surface temperature of the sun was Max Planck (1858-1947) in 1901, duction. Therefore, he decided to approximately 6 000 oC . To do so he which covers the entire frequency find a better empirical equation for had to use data about the rate of range π the heat transferred by radiation. emission of radiant energy from the ρ= 8hc λ λ (2) Dulong and Petit had used the Cel- sun and the emissivity of its surface, λ−5/(1)ehc kT sius scale in their equation and data that at that time were highly ρ Stefan, experienced in the kinetic untrustworthy. Pouillet had used the where λ is the energy of radiation theory, chose the absolute tempera- value of 84 888 cal/(cm2 · min) for the per unit volume per unit wavelength λ ture. rate of emission and Violle a value ( ), and h and k are Boltzmann's and In 1879 Stefan used Wüllner’s 44 % higher. Stefan found that the Planck's constants respectively. report of Tyndall’s data,28 transform- temperature of the sun was also Planck's law signals the beginning ing them to absolute temperature. strongly dependent of the value se- of quantum physics and modern He realized that by raising the ratio lected for the emissivity; Dulong and physics. of the absolute temperatures Petit’s equation yielded 1 450 oC for Evaporation and diffusion (273+1200)/ (273 + 525) = (1473/798) the minimum value (Pouillet) and In 1866, within the development = 1.846 to the fourth power, he got 2 025 oC for the maximum value (for of his molecular theory of gases, 11.6, almost the same value reported an emissivity of 0.025). With the Maxwell’s derived an equation describ- by Wüllner for the increase of the fourth-power formula the corre- ing the movement of a component by intensity of radiation of the weak red sponding range was from 5 600 to diffusion caused by a concentration glow up to the full white glow. From 11 000 oC . gradient in a mixture. Stefan clearly this result he made the bold state- Stefan’s findings may be consid- recognized that diffusion could give ment that the heat radiated was pro- ered a good example of serendipity: rise to a convective movement in the portional to the fourth power of the his initial purpose was to find an mixture and did a series of experi- absolute temperature. “This obser- empirical equation that would be ments to determine the characteris- vation” Stefan said, “caused me at better at high temperatures than tics of the phenomenon on a macro first to take the heat radiation as pro- that of Dulong and Petit. He achiev- scale.10,17 His apparatus consisted of portional to the fourth power of the ed this goal but at the same time he long and narrow open tubes, filled absolute temperature:“19 discovered a universal law of nature partially with a liquid and held at that is valid, however, for a special constant temperature. The tubes =σ 4 jT (1) body only, the black body which ab- were narrow to avoid a great lower- sorbs all incidental radiation and at ing of temperature at the evaporat- where j is the emitted energy flux a given temperature of all bodies it ing surface. density and σ proportionality con- is the optimal radiator. Not only that, The following results were ob- stant, which Stefan estimated to be he discovered the law using data tained: (a) the velocity of evapo- 4.5 · 10-8 W/(m2 . K4) [the present value which were later proved to be wrong: ration of a liquid from a tube is of Stefan's constant is 5.670 3 · 10-8 (a) Tyndall’s measurements referred inversely proportional to the dis- W/(m2 · K4)]. Equation (1) constitutes to infrared light and not to the ra- tance of the level of the liquid from the well-known Stefan radiation diation of all wavelengths, which is the open end of the tube. This law is law. contained in Stefan’s law; (b) for a very exact when the distance is Stefan then proceeded to discuss platinum wire the fourth-power law larger than 10 mm, (b) the velocity the experiments of Dulong and does not apply. Platinum remains of evaporation is independent of the Petit,34-37 Ferdinand Hervé de la shiny and its emissivity increases diameter of the tube (for tubes be- Provostaye (1812-1863) and Paul with temperature, the radiated en- tween 0.3 to 8 mm) and, (c) velocity 192 Revista CENIC Ciencias Químicas, Vol. 37, No. 3, 2006.

s0 increases with the temperature be- (P−−− P ) (P P ) only by conduction, fills the half P = (4) cause of the increase in vapor f −s PPs space x > 0. At the initial time it is 0 ln pressure of the liquid. If P is the 0 at a constant temperature T > 0. At PP− 2 maximum vapor pressure at the the surface x = 0 it is maintained at temperature of experimentation Stefan also derived an equation constant temperature T1 > 0, under and P the atmospheric pressure for the calculation of the total trans- the effect of which there arises crys- then the rate of evaporation is pro- port rate of a component caused by tallization, occurring isothermally portional to log[P/(P- P 0)]. diffusion in a mixture with a concen- for temperature T = 0, without su- Some curious observations were tration gradient.2 percooling, for which the small vol- made when the open end of the tube This problem is nowadays known ume effects are neglected; (b) the was immersed in ether: bubbles under the name the Stefan Diffusion heating material occupies the space were seen to form and disengage Problem. -∝ < X < ∝ . At the initial time the continually from the tube and, in the Change of state liquid fills the domain 0 < X < ∝ at beginning, the times in which suc- A very important problem of temperature T2 > 0, while the solid cessively equal numbers of bubbles theoretical and practical signifi- occupies the domain -∝ < X < 0 at formed were proportional to the odd cance is that of the change of phase, temperature T1 < 0. The remaining numbers When the upper part of the in which a substance (pure or mixed) conditions are the same as in the immersed tube was filled with hy- changes from one phase to another, first problem. It is then required to drogen instead of air, the same num- with release or absorption of heat. determine the temperature U1(x,t) ber of bubbles formed in one-fourth This phenomenon arises in many and U2(x,t) of the solid and liquid the time. In other words, evapora- contexts in which the most impor- phases and the position x = y(t) of tion in hydrogen proceeded four tant are melting and solidification. the boundary between them. Calcu- times as rapidly as in air. If now the The problem of ground freezing and lation of the thermal balance yields, tube was provided with a cock on its ice formation is of great importance as shown by Stefan, the condition open end, and submerged in ether, both in geophysics and in ice manu- ∂∂ the level of the liquid within the tube facture. A great deal of attention has λρ dy=− U12 U []kk12xy(t)= (5) sunk below that of outside and, at been paid to the solidification of dt∂∂ x x first, the depths to which the inte- castings. rior level sunk below the exterior in The essential feature of phase where λ is the latent heat of crystal- definite time was as the square-roots change phenomena is the existence lization per unit mass, ρ the density, of those times. of a moving surface of separation and k1 and k2 the coefficients of con- The Stefan diffusion tube has between the two phases. The way in ductivity of the solid and liquid been widely used for the determina- which this surface moves has to be phases, respectively. Both problems tion of vapor phase diffusion coeffi- determined. Heat is liberated or ab- examined by Stefan admit similar- cients. The liquid to be vaporized is sorbed on it, and the thermal prop- ity solutions. placed in the bottom of a vertical erties of the two phases on different In the same work of 1889, Stefan tube, which is maintained at a con- sides of it may be different, so that gave an analogous description of stant temperature. A gas is passed the problem is one of considerable process of neutralization for diffu- over the top of the tube at a rate suf- difficulty. sion transport of material in a reac- ficient to keep the partial pressure Apparently the first work of in- tion zone16 and evaporation and con- of the vapor there essentially corre- terest in this area was the paper42 of densation.8 Finally, in the same year sponding to the initial composition Gabriel Lamé (1795-1870) and Be- Stefan published his fourth work of the gas but low enough to prevent noit-Pierre-Emile Clapeyron (1799- related to the problem17 where he turbulence. The mass flux is deter- 1864) published in 1831. In this work examined the problem of melting of mined by weighing the tube during the problem was posed of determin- a layer of ice with initial temperature the quasi steady state evaporation ing the thickness of the solid crust equal to zero, subject to the influ- period. The vapor phase diffusion generated by the cooling of a liquid ence of the temperature f(t) at the coefficients are readily calculated filling the half space x > 0 under the boundary x = 0. from the mass flux and the concen- influence of a constant temperature at The work of Stefan attracted the tration gradient over the diffusion the plane x = 0. The temperature of attention of geophysicists and tube with the assumption of plug the liquid was initially everywhere caused led to several unsuccessful flow in the tube. Lee and Wilke41 have equal to the crystallization tempera- attempts to solve the more general presented a critical review of the ture. Lamé and Clapeyron found problems. experimental technique. that the thickness of the crust is pro- In the steady state the rate of va- portional to the square root of the Miscellaneous porization NA is given by the well- time, but did not determine the co- Acoustics known equation efficient of proportionality. The first Stefan also devoted part of his ∆ published general discussion seems time to acoustical problems and pub- = DPA P 19 NA (3) to be that by Stefan in a study of lished seven papers on the subject. RTPf x the thickness of polar ice and for this Here we summarize his most impor- reason the problem of freezing is fre- tant findings. where DA is the diffusion coefficient quently referred to as the problem of In a paper published in 1866 he of component A, P the total pressure, Stefan. reported on the effects of internal Ps is the saturation pressure corre- In 1889, Stefan in his work on the friction in the air on the motion of sponding to the surface temperature freezing of the ground15 posed and sound.8 His results indicated that T of the liquid, P0 the vapor pressure solved the following problems: (a) A friction increased the velocity of of the inlet gas, x the axial distance, material existing in two phases (liq- sound and that the increase was and uid and solid) and transmitting heat larger the higher the tone. Neverthe- 193 Revista CENIC Ciencias Químicas, Vol. 37, No. 3, 2006.

less, even for the highest tones this in- velocity: an increase in 1o reflected angle or rotation in a given sub- crease was very small, about 0.001 mm in a decrease of 40 m . At 30 oC the stance. Hence these two quantities in a second. The amplitudes de- velocity of sound in wax and air are must be related and the corre- creased in plane-progressive waves the same. For grease at 20 oC the ve- sponding relation may be disclosed in geometrical progression and, locity of sound was about one-half by a prismatic analysis of the light again, the exponent of the progres- that in wax. Increasing the tempera- as it leaves the polarizing appara- sion increased with the height of the ture the velocity decreased more tus. tone, and indeed, proportionally to rapidly than in air. In caoutchouc The rotation of the place of po- the square of the number of vibra- (natural rubber) the velocity of larization is proportional to the tions. The diminution of amplitude sound varied between 30 to 60 m/s; thickness of the plate of quartz. was perceptible only in the high the softer the rubber the smaller the When the latter is considerable the tones. His results indicated that velocity. Stefan made the observa- amount of rotation for the different with a tone of 10 000 vibrations the tion that his results recalled those of colors is equal to several complete amplitude diminished by 1/9 at a Hermann Ludwig von Helmholtz revolutions. When the polarizer and distance of 1 000 m and by 1/67 at (1821-1894) for the velocity of ner- analyser are placed parallel, the lat- 2 000 m . vous excitation, which is within the ter removes from the light those col- Standing vibrations are possible same limits as the velocity of soft ored rays coming through the quartz only if the length of the wave ex- rubber. that have undergone rotations. The ceeds a certain value. Yet this is very At the International Conference latter are odd multiples of 90o. In the small, equal to four times the mean on Musical Pitch, held at Vienna in places of these colors, dark bands molecular path, which, according to 1885, the proposals of the Austrian appear in the spectrum. The num- the kinetic theory of gases, a mol- commission of experts were gener- ber of bands is found multiplying ecule makes from one impact to the ally adopted in central and eastern the thickness of the plate in milli- other. Europe; the standard pitch was es- metres by 1/6 and 5/9; the number of In standing waves also the am- tablished at 435 cycles per second, odd integers between the two prod- plitude decreases with the time in as had already been done in France ucts is the number of bands. geometrical progression, with an in 1859 and in Austria in 1862. For When the analyser is rotated the exponent proportional to the square the production of this tone the con- bands move from the red towards of the number of vibrations. The ference prescribed, according to the violet end, or the reverse, accord- amplitudes of the tones of 1 000, Stefan's account, the standard tun- ing as the analyser moves in the 10 000, and 30 000 vibrations ing fork constructed to replicate the sense of rotation of the plane of po- dropped to one-half before the lapse tuning forks of Karl Rudolf König larization or the contrary. Thereby of 100, 1, and 0.1 s, respectively. (1832-1901). the number of bands may be altered Stefan was aware that the Optics by a difference of one. In addition, method introduced by Ernst Florens In optics, Stefan's interests cen- the relative position of the bands is Friedrich Chladini (1756-1827) for tered on the polarization of light, dependent upon the nature of the determining the speed of sounds double refraction, interference of substance forming the prism and from the longitudinal tones of bars, light, especially Newton's rings, and upon the thickness of the rotating was not applicable to bars, which measurement of the wavelength. plate. For a prism made of crown were not long enough to be made to In a paper about the dispersion glass or flint glass the following sound by friction [1]. For this reason of light owing to the rotation of the propositions may be deduced from he built an experimental device that plane of polarizarion5 Stefan ex- the experiments: (a) the dark bands overcame these limitations.9 plained that there are only two pos- of the spectrum are equidistant, (b) The body to be investigated was sible forms of dispersion; each color the distance between two contigu- constructed in the shape of a small in white light has either a particular ous bands is inversely proportional bar and fastened to a longer bar velocity of propagation or a particu- to the thickness of the quartz plate made of wood or glass, which could lar direction of vibration. The first employed and, (c) the bands move thus be readily made to sound. The kind of color dispersion occurs in regularly and correspondingly on compound bar was now made to refraction and diffraction; the sec- turning the analyser. In addition, sound by friction and the number of ond when light passes through a since the dark bands correspond to vibrations of the fundamental note substance, which turns its plane of colors for which their angles of rota- or of a higher tone determined. polarization inasmuch as the rota- tion differ by a constant quantity, Stefan deduced a rather compli- tion has a different magnitude for then the distances of the colors in cated mathematical expression, each color. The occurrence of disper- the spectrum are proportional to the which knowing the velocity of sion through refraction or through differences in their angles of rota- sound in the larger bar, allowed cal- alteration of the plane of polariza- tion. culating the same in the smaller bar. tion means that in the one case the Calculation of the refractive in- His results indicated that, for ex- refractive index and in the other dices of the individual dark bands ample, the velocity of sound in wax the angle of rotation is a function of indicates that equal differences of at 20 oC was 730 m/s, that is, about the wavelength of a color. Each color refractive index correspond to equal twice as great as in air. Increasing is determined by its wavelength, also differences of rotation. In other the temperature led to a decrease in by the refractive index, or by the words, there is a linear relation be-

Note 1. Ernst Florens Friedrich Chladini (1756-1827) was an 18th century physicist who found that playing a violin near a sand covered disk caused the sand to form geometric shapes. 194 Revista CENIC Ciencias Químicas, Vol. 37, No. 3, 2006. tween angle of rotation and the re- 5. Stefan J., Über die Dispersion des 25. Fourier J., Analytical Theory of Heat, fractive index, that is, both are simi- Lichtes Durch Drehung des Polari- in Great Books of the Western World, lar functions of the wavelengths. sationsebene im Quartz, Sitz. M.J. Adler, Associate Editor, Encyclo- paedia Britannica, Toronto, 1952. In a following publication Stefan Wien. Akad. Wissen., II, 50, 88-124, 1865. 26. Delaroche F., Observations Sur le described a new method he had de- 6. Stefan J., Über Einige Thermoële- Calorique Rayonnant, J. Phys., 75, veloped for measuring the lengths mente von Grosser Electromorischer 201-228,1812. 7 of light waves. The basis for his pro- Kraft, Sitz. Wien. Akad. Wissen. II, 27. Tyndall, J., On the Absorption and posal was the fact that when light 51, 260-262, 1865. Radiation of Heat by Gases and falls on a column of quartz with pol- 7. Stefan J., Über Eine Neue Methode Vapours, and on the Physical Con- ished faces parallel to the optical die Längen der Lichtwellen zu nexion of Radiation, Absorption, and axis, each ray is resolved into the or- Messen, Sitz. Wien. Akad. Wissen. II, Conduction, Phil. Mag., 22, 169-194 and 273-286, 1861. dinary and the extraordinary ray, if 53, 521-528,1866. 8. Stefan J., Über den Einfluss der 28. Tyndall, J., On Radiation Through the faces of entrance and of emer- Inneren Reibung in der Luft auf die the Earth's Atmosphere. Phil Mag., gence are parallel. Bringing both Schallbewegung, Sitz. Wien. Akad. 24, 200-207, 1863. rays into a common direction of vi- Wissen. II, 52, 529-537,1866. 29. Tyndall J., Heat Considered as a bration results in the extinction of 9. Stefan J., Anwendung der Schwin- Mode of Motion, Lecture 12, Long- those rays having a difference of op- gungen Zusamengesetzer Stäbe zur man, Green, London, 1865. tical path equal to an uneven num- Bestimmung der Schallgeschwin- 30. Wüllner A., Lehrbuch der Experi- ber of semi-wavelengths. For the digkeit, Sitz. Wien. Akad. Wissen. II, mentalphysik, 3rd Edition, III, B.G. Teuber, Leipzig, 1874. complete spectrum the extinction 57, 697-708, 1868. 10. Stefan J., Über das Gleichgewicht 31. Kirchhoff G.R., Über das Verhaltnifs appears as dark interference bands und die Begewung Insbesondere die Zwischen dem Emissionsverrmogen that are more numerous and thinner Diffusion von Gasmengen, Sitz. und dem Absorptionsvermogen der the thicker the quartz. The differ- Wien. Akad. Wissen. II, 58, 63-124, Korper fur Warme und Licht, Pogg. ence of phase between two rays may 1871. Ann., 109, 275-301, 1860. be calculated very accurately from 11. Stefan J., Untersuchungen Über die 32. Draper J.W., Examination of the Ra- the thickness of the quartz and from Wärmeleitubg in Gasen, Sitz. Wien. diation of Red-Hot Bodies. The Pro- the quotients of refraction since only Akad. Wissen. II, 55, 45-69, 1872. duction of Light by Heat, Scientific Memoirs, 20-52, 1878. the differences and not the absolute 12. Stefan J., Versuche Über die Verdamp- fung, Sitz. Wien. Akad. Wissen. II, 58, 33. Lane J.H., On the Theoretical Tem- values of the latter are required. 385-423, 1873. perature of the Sun, Am. J. Sci., 50, Twice the difference of phase di- 13. Stefan J., Sitz. Wien. Akad. Wissen. 1869. vided by the wavelength is an un- II, 75, 323-350, 1878. 34. Dulong P.L. and Petit A.T., Recher- even number for each dark band, 14. Stefan J., Über die Beziehung ches Sur les Lois de Dilatation des and for each succeeding one towards Zwischen der Wärmerstrahlung und Solides, des Liquides et des Fluides violet it is two units greater. Know- Temperatur, Sitzs. Wien. Akad. Élastiques et Sur la Mesure Exacte ing the wavelength and number of Wissen. II, 79, 391-438, 1879. des Températures, Ann. Chim. Phys., 2, 240-263, 1816. bands in one Fraunhofer line allows 15. Stefan J., Über Einige Probleme der Theorie der Wärmeleitung, Sitz. 35. Dulong P.L. and Petit A.T., Recher- calculating the wavelength of the Wien. Akad., Mat. Natur., 98, 473-484, ches Sur la Mesure des Températures following Fraunhofer line. 1889. et Sur les Lois de la Communication A wavelength may be deter- 16. Stefan J., Über die Diffusion von de la Chaleur, Ann. Chim. Phys., 7, mined directly by successively in- Saüren und Basen Gegën Einander, 113-154, 1818. creasing or decreasing the differ- Sitz. Wien. Akad. Mat. Natur., 98, 36. Dulong P.L. and Petit A.T., Recherches ence of phase for the place of the 616-634, 1889. Sur la Mesure des Températures et Sur spectrum in question. This leads to 17. Stefan J., Über die Verdampfung und les Lois de la Communication de la Chaleur, Ann. Chim. Phys., 7, 224- a removal of the interference band. die Auflösung als Vorgänge der Dif- fusion, Sitz. Wien. Akad. Mat. Natur., 264, 1818. The pertinent wavelength is them 98, 1418-1442, 1889. 37. Dulong P.L. and Petit A.T., Recher- calculated from the number of 18. Stefan J., Über die Theorie der ches Sur la Mesure des Températures bands, which have passed through Eisbuidung, Insbesondere Über di et Sur les Lois de la Communication the cross wires and the change of the Eisbildung im Polarmeere, Sitz. de la Chaleur, Ann. Chim. Phys., 7, difference of phase thus produced. Wien. Akad. Wissen. II, 98, 965-983, 337-367, 1818. Stefan illustrated his procedure 1890. 38. De La Provostaye F. and Desains P., by giving the calculated wave- 19. Stefan J., Über die Theorie der Mémoire sur le Rayonnement de la Chaleur, Ann. Chim. Phys., 16, 337- lengths of Fraunhofer's lines B, C, D, Eisbildung, Ann. Phys. Chem. (Wiede- mann) N.F. 42, 269-286, 1891. 425, 1846. E, F, G, and H, as 687.3, 657.8, 527.1, 20. Stefan J. and Ludwig C., Über den 39. Ericsson, J., The Temperature of the 486.9, and 395.9 nm, respectively. Druck, den das Fliessende Wasser Surface of the Sun, Nature, 5, 505, These values agreed very accurately Senkrecht zu Seiner Stromrichtung 1872. with those deduced from the diffrac- Ausübt., Sitz. Wien. Akad. Wissen., 40. Boltzmann L., Ableitung des Ste- tion phenomena of fine gratings. 32, 25-42, 1858. fanschen Gesetzes, Betreffend Die 21. Maxwell J.C., On the Dynamical Abhängigkeit der Wärmestrahlung von BIBLIOGRAPHY Theory of Gases, Phil. Trans., 157, 49- der Temperatur aus der Elektromag- 1. Anonymous, Obituary, Elektrotech. 88, 1866. netischen Lichttheorie, Wied. Ann., Zeitschr., 14, 31, 1893. 22. Newton. I. Opuscula Mathematica, 22, 291-308, 1884. 2. Lecher, Obituary, Innsbruck Nat. Med. Philosophica et Phiologica, vol. II, 41. Lee C.Y. and Wilke C.R., Measure- Ber., 21, x-xiv, 1893. 423, Apud Marcum-Mihaelem Bous- ment of Vapor Diffusion Coefficient, 3. Strnad J., Jozef Stefan - The Centenary quet & Socios, Lausanne, 1744. Ind. Eng. Chem., 48, 2381-2387, 1954. of his Death, Jozef Stefan Institute, 23. Martine G., Dissertations sur le 42. Lamé G. and Clapeyron B.P., Mé- Ljubljana, 993. Chaleur, Paris, 72, 1740. moire sur la Solidification par Re- 4. Stefan J., Allgemeine Gleichungen für 24. Richmann G. W., Inquisitio in Legem, froidissement d'Un Globe Solide, Oscillatorische Bewegungen, Pog- Novi Commmentarii, Ac. Sci. Imp. Ann. Chem. Phys., 47, 250-256, gend. Annal., CII, 365-387, 1857. Pet., I, 174-197,1747-1748. 1831. 195