Josef Stefan

How to use this course

Josef Stefan is known to every physics student for one law: a black body radiates power in proportion to the fourth power of its absolute temperature. The law gave the first sensible temperature for the Sun, and through his student Ludwig Boltzmann it became one of the first results of the thermodynamics of radiation, a step on the road to Max Planck’s quantum. But radiation was only one of Stefan’s subjects. He made the first measurements of heat conduction in gases, wrote the equations for diffusion in multicomponent gas mixtures, explained evaporation from a liquid surface, and solved the problem of how fast ice grows on a freezing sea. Engineers still use his name for many of these results.

This course follows his life from a small village in Carinthia to the vice-presidency of the Imperial Academy in Vienna, and his physics through seven lessons on the transport of heat and matter.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Stefan wrote mostly in the Sitzungsberichte of the Vienna Academy, about eighty papers in all, combining careful experiments with compact mathematical analysis. The lessons try to keep that balance.

Timeline

Carinthia, 1835–1853

Josef Stefan, in Slovene Jožef Stefan, was born on 24 March 1835 in St. Peter, a village on the outskirts of Klagenfurt, the capital of the Austrian duchy of Carinthia. His parents were Slovenes. His father, Aleš Stefan, was a miller and baker who sold flour; his mother, Marija Startinik, was a maidservant. Neither could read or write, and they were not married when he was born. He was their only child. The family was poor.

His parents married in 1846, when he was eleven, which made it possible for him to enter the Gymnasium in Klagenfurt that year. He was an outstanding pupil. His physics teacher, Karel Robida, a Benedictine and a Slovene, encouraged his interest in science. The revolutionary year 1848, when he was thirteen, stirred a generation of young Slovenes to write in their own language, and Stefan began to publish poems in Slovene. For a time he thought of entering the Benedictine order, but he gave up the idea. He finished school in 1853 at the head of his class.

A Slovene poet

Stefan kept writing poetry and prose in Slovene through his student years in Vienna, and some of it appeared in Slovene periodicals. The work was patriotic and romantic. After sharp criticism from Slovene literary authorities he stopped publishing literature around the time he graduated, and devoted himself to physics. In Slovenia he is remembered both as a scientist and as a minor poet of the national revival, and his name is borne by the country’s leading research institute.

Vienna, 1853–1866

In 1853 Stefan went to the University of Vienna to study mathematics and physics. Vienna’s physics was then being rebuilt. Christian Doppler had founded the Physical Institute in 1850, and after Doppler’s death in 1853 it was directed by Andreas von Ettingshausen, a mathematician and physicist who had built one of the first electromagnetic generators in the 1830s. Ettingshausen became Stefan’s mentor.

Stefan finished his studies in 1857. To earn a living he taught physics, first to pharmacy students and at a Realschule in Vienna. He also worked in the laboratory of the physiologist Carl Ludwig, measuring the flow of water through tubes, an early introduction to transport problems. In 1858 he received his doctorate and his habilitation in mathematical physics, and began to lecture at the university.

A rapid rise

His early papers ranged widely: the motion of fluids, the theory of sound, optics, electricity. In 1860, at twenty-five, he was elected a corresponding member of the Imperial Academy of Sciences in Vienna. In 1863 he was appointed full professor of higher mathematics and physics, and in 1865 he became a full member of the Academy and received the Lieben Prize for work in optics. In 1866 Ettingshausen retired and Stefan succeeded him as director of the Physical Institute, at the age of thirty-one. He declined a call to the Vienna Polytechnic in the same year. He directed the Institute until his death.

The Physical Institute

The Physical Institute was small, with modest apparatus, and its members worked closely together. His close colleague at the university was Josef Loschmidt, who had made the first estimate of the size of air molecules in 1865. The young Ludwig Boltzmann took his doctorate under Stefan in 1866. On Boltzmann’s first visit, Stefan handed him James Clerk Maxwell’s papers on electromagnetism and an English grammar, since Boltzmann did not yet read English. The story, told by Boltzmann himself, shows both the Institute’s orientation and Stefan’s style.

Stefan was remembered by his students as a clear and orderly lecturer. His habit was to choose a precise question, design a simple apparatus to measure the answer, and then derive the result from theory, often in a few pages. He was one of the very few physicists on the Continent who took Maxwell’s field theory seriously in the 1870s. Under him the Physical Institute became, with Boltzmann and Loschmidt, a centre of the kinetic theory of gases.

He served the university as dean of the Faculty of Philosophy in 1869–70 and as rector in 1876–77. Students who passed through the Institute in his last years included Marian Smoluchowski, later famous for his work on Brownian motion.

Diffusion in gas mixtures

For the general reader. Open a bottle of perfume and the scent spreads through the room because molecules jostle their way through the air. With two gases, a single number describes how fast they mix. With three or more, each gas pushes against every other one, and the simple rules fail. In 1871 Stefan wrote the equations that handle any number of gases. Chemical engineers still use them as the Maxwell–Stefan equations.

Friction between species

Maxwell, in his dynamical theory of gases of 1866–67, computed the momentum that molecules of one species transfer to another through collisions. Stefan, in 1871, turned this into a simple mechanical picture. Each species moves through the others as through a resisting medium. In a steady state the force driving species \(i\), its partial pressure gradient, is balanced by friction with every other species, proportional to the relative velocity and to the product of their amounts.

Binary case. With \(n=2\), write the molar fluxes \(\mathbf N_i=c\,x_i\mathbf v_i\) and the molar-average velocity \(\mathbf v=x_1\mathbf v_1+x_2\mathbf v_2\). Then \(x_1x_2(\mathbf v_1-\mathbf v_2)=x_1(\mathbf v_1-\mathbf v)\), and the equation becomes

which is Fick’s law for the diffusive flux relative to the mean motion. The binary coefficient is a property of the pair alone; for an ideal gas it varies roughly as \(T^{3/2}/p\). Stefan computed such coefficients and their temperature dependence and compared them with the diffusion measurements of Thomas Graham and Loschmidt.

More than two species

In a ternary mixture the flux of species 1 depends on the gradients of both independent mole fractions. Fick’s law with a single coefficient fails, and a species can even diffuse against its own gradient, dragged along by the others. The Maxwell–Stefan form keeps the physics in the binary coefficients and handles these effects automatically.

Heat conduction in gases

For the general reader. Maxwell’s kinetic theory predicted something surprising: a gas conducts heat equally well whether it is dense or rarefied. More molecules carry heat, but each travels a shorter distance before colliding, and the two effects cancel. Measuring the heat conduction of a gas is hard because convection and radiation swamp it. In 1872 Stefan found a way, and confirmed the prediction.

Derivation. Across a plane in a temperature gradient \(dT/dz\), molecules arriving from above last collided about \(\lambda\) away and carry energy \(c_v\,(T+\lambda\,dT/dz)\) each; those from below carry \(c_v(T-\lambda\,dT/dz)\). About \(\tfrac16n\bar v\) molecules per unit area per unit time cross in each direction, so the net heat flux is \(q=-\tfrac13nc_v\bar v\lambda\,dT/dz\). Rigorous theory changes the numerical factor but not the scaling.

The diathermometer

Stefan’s apparatus was an air thermometer in the form of two coaxial cylinders, the gap between them filled with the gas under test. The inner cylinder was the bulb of the thermometer; the outer one was held at a fixed temperature in a bath of ice. The rate at which the inner cylinder cooled gave the conductivity of the layer of gas. A narrow gap suppressed convection, and polished surfaces kept radiation small; corrections for both could be made by varying the geometry. Stefan found the conductivity of air to be close to the value Maxwell’s theory predicted, and independent of pressure over a wide range. In 1875 he reported the relative conductivities of several gases; hydrogen conducts several times better than air. He used the same instrument, which he called a diathermometer, to measure the insulating power of clothing materials.

Evaporation, Stefan flow and adhesion

For the general reader. Water in a narrow tube evaporates slowly, because vapour must diffuse up the tube to escape. Stefan noticed that the vapour does not simply diffuse: the evaporating liquid creates a gentle wind of gas up the tube, which speeds the process. This “Stefan flow” matters in drying, combustion and cloud physics. He also explained why two wet glass plates seem to stick together.

The Stefan tube

In papers of 1873 and 1881 Stefan studied the evaporation of volatile liquids from narrow tubes. A liquid A sits at the bottom of a vertical tube of length \(L\); gas B flows across the top and carries away the vapour. Gas B is insoluble in the liquid, so in the steady state it does not move: \(\mathbf N_B=0\). Fick’s law for the diffusive flux relative to the mean velocity then gives a total flux of A that includes a convective term.

Derivation. \(N_A=x_A(N_A+N_B)-cD_{AB}\,dx_A/dz\) with \(N_B=0\) gives \(N_A=-\dfrac{cD_{AB}}{1-x_A}\dfrac{dx_A}{dz}\). In the steady state \(N_A\) is constant along the tube; integrating from 0 to \(L\) gives the formula. The factor \(1/(1-x_A)\) is the Stefan flow. When \(x_0\ll1\), \(\ln\frac{1-x_L}{1-x_0}\approx x_0-x_L\), and the plain Fick result is recovered. The Stefan tube is still a standard way to measure diffusion coefficients.

Stefan adhesion (1874)

Two flat plates pressed together with a thin film of liquid between them are hard to pull apart. Stefan showed that the resistance is not a static adhesion force but viscous: liquid must flow into the widening gap. For circular plates of radius \(R\) separated by a gap \(h\ll R\), lubrication theory gives

The force vanishes if the plates are separated slowly enough. The result, now called Stefan’s law of squeeze flow, underlies the theory of lubrication and of “tack” in adhesives.

The fourth-power law

For the general reader. A stove glowing dull red radiates a little heat; the same stove at twice the absolute temperature radiates sixteen times as much. Before 1879 the laws proposed for this relation were complicated and wrong at high temperatures. Stefan looked at the data and saw a simple pattern: radiated power grows as the fourth power of absolute temperature. He used it to estimate the temperature of the Sun’s surface, and got close to the modern value.

The state of the question

Newton’s law of cooling, in which heat loss is proportional to the temperature excess, worked only for small differences. In 1817 Pierre Louis Dulong and Alexis Thérèse Petit proposed an exponential law from careful cooling experiments, and it remained the standard. In 1864 John Tyndall measured the radiation from an electrically heated platinum wire and found that as the wire went from a faint red glow, about 525 °C, to white heat, about 1200 °C, its radiation increased by a factor of about 11.7. Adolph Wüllner quoted these numbers in his textbook in 1875, where Stefan saw them.

Stefan proposed the fourth power empirically and checked it against Tyndall’s numbers and against the cooling data of Dulong and Petit, showing that it fitted their measurements over the whole range as well as their own formula did. A black body absorbs all radiation falling on it; Gustav Kirchhoff had shown in 1859–60 that its emission depends only on temperature. Platinum is far from black, so Stefan’s agreement with Tyndall was partly lucky. The law was shown to hold for black-body cavities by Otto Lummer and Ernst Pringsheim in 1897.

The temperature of the Sun

Earlier estimates of the Sun’s surface temperature, using Dulong and Petit’s law or Newton’s, ranged from about 1500 °C to millions of degrees. Stefan applied his law to measurements comparing solar radiation with that of a heated metal strip, after correcting for absorption in the atmosphere. His estimate, between about 5400 and 6000 °C depending on the data used, was the first that is close to the modern effective temperature of 5772 K.

Boltzmann’s derivation and after

For the general reader. Stefan found the fourth power in data. Five years later his former student Ludwig Boltzmann showed that it must hold exactly, using only thermodynamics and one result of Maxwell’s electromagnetism: light exerts pressure. The derivation treats a box of light like a gas in an engine. It was one of the first applications of thermodynamics to radiation, and it led directly to Wien’s law and Planck’s quantum.

Radiation as a working substance

In a cavity at temperature \(T\) the radiation has energy density \(u(T)\) depending only on \(T\) (Kirchhoff). Maxwell’s theory gives its pressure: for isotropic radiation, \(p=u/3\). Adolfo Bartoli had argued in 1876 that radiation pressure is required by the second law. Boltzmann, in 1884, considered a cylinder of radiation as the working substance of a heat engine.

Proof. With \(U=uV\), the thermodynamic identity \(\big(\partial U/\partial V\big)_T=T\big(\partial p/\partial T\big)_V-p\) gives

For the flux: radiation crossing a hole of unit area from within a solid angle \(d\Omega\) at angle \(\theta\) to the normal carries \(c\,u\cos\theta\,d\Omega/(4\pi)\). Integrating over the hemisphere gives \(cu/4\).

From Stefan to Planck

Thermodynamics fixes the \(T^4\) but not the constant. In 1893 Wilhelm Wien showed, by a similar argument, that the spectrum must have the form \(u_\nu=\nu^3f(\nu/T)\), which gives the displacement law. In 1900 Max Planck found the function:

Integrating over \(\nu\) with \(\int_0^\infty x^3/(e^x-1)\,dx=\pi^4/15\) gives \(a=\dfrac{8\pi^5k_B^4}{15h^3c^3}\), and \(\sigma=ac/4\) gives the formula in Lesson 07. Since the 2019 redefinition of the SI, \(h\), \(k_B\) and \(c\) are exact, and so is \(\sigma\).

The Stefan problem: freezing

For the general reader. On a cold lake ice grows quickly at first, then more and more slowly. The reason is that heat from the freezing water must pass up through the ice already formed, and thicker ice is a better blanket. Stefan worked out the law in 1889: ice thickness grows as the square root of time. The mathematical problem, in which the boundary between ice and water moves and must be found along with the temperature, is now called the Stefan problem.

Formulation

Water at its freezing point lies below ice of thickness \(h(t)\). The top surface of the ice is held at \(-\Delta T\) below freezing. In the ice, \(0\lt z\lt h(t)\), the temperature \(\theta(z,t)\) obeys the heat equation, and at the moving interface the latent heat released must be conducted away.

Quasi-steady approximation. When the Stefan number \(\mathrm{St}\) is small, the latent heat dominates and the temperature in the ice is nearly linear: \(\partial\theta/\partial z\approx\Delta T/h\). The Stefan condition becomes \(\rho L\,h\,\dot h=k\Delta T\), so

This is Stefan’s law of ice growth. Gabriel Lamé and Émile Clapeyron had posed a related problem in 1831, and Franz Neumann found the similarity solution in lectures in the 1860s, but Stefan’s papers of 1889 gave the first systematic treatment and applied it to the growth of sea ice in polar regions. The same mathematics describes solidification of metals, melting of permafrost and the casting of steel.

Exact solution. Try \(\theta=-\Delta T+B\operatorname{erf}\big(z/(2\sqrt{\alpha t})\big)\), which solves the heat equation and the top boundary condition. Requiring \(\theta=0\) at \(z=h=2\Lambda\sqrt{\alpha t}\) gives \(B=\Delta T/\operatorname{erf}\Lambda\). Substituting into the Stefan condition yields the transcendental equation for \(\Lambda\). For small \(\mathrm{St}\), \(\Lambda\approx\sqrt{\mathrm{St}/2}\), which reproduces the quasi-steady law.

Electrodynamics and the skin effect

For the general reader. On the Continent in the 1870s, most physicists described electricity by forces acting instantly at a distance between moving charges. Stefan was one of the few who adopted Maxwell’s picture of fields filling space. He used it for practical problems: the inductance of coils, and the way alternating current crowds toward the surface of a thick wire, the skin effect.

Fields and inductance

Stefan argued against the action-at-a-distance electrodynamics of André-Marie Ampère and Hermann Grassmann, and showed that the observable forces between closed circuits could be obtained from a continuous field. He calculated the self-inductance of wire coils and corrected an error in Maxwell’s calculation for a coil of square cross-section. The most transparent route to inductance is through field energy.

The skin effect

In a conductor of resistivity \(\rho\), with displacement current negligible, Maxwell’s equations give a diffusion equation for the current density, \(\nabla^2\mathbf J=\dfrac{\mu_0}{\rho}\,\partial_t\mathbf J\). For a field varying as \(e^{i\omega t}\) at a flat surface, \(J(z)=J_0e^{-z/\delta}e^{i(\omega t-z/\delta)}\).

Like Lord Rayleigh and Oliver Heaviside, Stefan studied how alternating current distributes itself in conductors. The question was practical as well as theoretical, since alternating-current power systems were then being developed.

Academy, exhibitions and pitch, 1875–1890

Stefan’s public role grew steadily. In 1875 he became secretary of the mathematics and natural sciences class of the Imperial Academy of Sciences, and in 1885 its vice-president, a post he held until his death. The Bavarian Academy elected him a corresponding member in 1878.

Electricity and music

In 1883 Vienna held an International Electrical Exhibition in the Rotunda of the Prater, showing dynamos, arc and incandescent lamps and electric railways. Stefan presided over its scientific commission, which tested the exhibited machines and lamps, and the commission’s report appeared in 1886. In 1885 he presided over an international conference on musical pitch in Vienna, called to end the confusion of tuning standards across Europe. The conference adopted 435 vibrations per second for the note a′, a value France had used since 1859 and Austria since 1862. It remained the international standard until the present 440 Hz was agreed in the twentieth century.

Teacher and colleague

Stefan published almost entirely in the Sitzungsberichte of the Vienna Academy, and much of his influence passed through his lectures and his students. Boltzmann, who returned to Vienna as Stefan’s successor in the chair of theoretical physics in 1894, spoke of him with lasting gratitude. The Physical Institute under Stefan, Loschmidt and Boltzmann gave Vienna one of the strongest schools of kinetic theory and thermodynamics in Europe, and prepared the ground for Smoluchowski and the later Vienna physics of Franz Exner and his students.

Last years and legacy, 1889–1893

His papers on the freezing problem in 1889 showed him at the height of his powers. In 1891, aged fifty-six, he married Marija Neumann, a widow; they had no children. On 7 January 1893, a little over a year after the wedding, he died in Vienna after a stroke, at the age of 57. He is buried in the Zentralfriedhof, Vienna’s Central Cemetery. In 1895 a memorial relief was unveiled in the arcaded courtyard of the University of Vienna’s main building, with a speech by Boltzmann.

Legacy

• Radiation: the Stefan–Boltzmann law and constant, the basis of radiative heat transfer, stellar luminosity \(L=4\pi R^2\sigma T_{\text{eff}}^4\) and planetary energy balance.
• Transport: the Maxwell–Stefan equations of multicomponent diffusion, Stefan flow, the Stefan tube, and Stefan’s law of squeeze flow.
• Phase change: the Stefan problem and the Stefan number, the starting point of the mathematical theory of free-boundary problems.
• Physics in Vienna: as teacher of Boltzmann and director of the Physical Institute for twenty-seven years, he shaped the school that produced statistical mechanics in its Austrian form.
• Slovenia: the Jožef Stefan Institute in Ljubljana, founded in 1949, is the country’s leading research institute. A lunar crater bears his name.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Josef Stefan, MacTutor History of Mathematics Archive.
2. W. Böhm, Stefan, Josef, Dictionary of Scientific Biography.
3. Stefan, Josef, Neue Deutsche Biographie and Allgemeine Deutsche Biographie.
4. Josef Stefan, Wikipedia.
5. A. von Obermayer, Zur Erinnerung an Josef Stefan, Vienna and Leipzig, 1893.
6. J. Crepeau, Josef Stefan: his life and legacy in the thermal sciences, Experimental Thermal and Fluid Science 31 (2007) 795–803.

Physics

1. J. Stefan, Über die Beziehung zwischen der Wärmestrahlung und der Temperatur, Sitzungsberichte der Akademie der Wissenschaften in Wien 79 (1879).
2. J. Stefan, Über das Gleichgewicht und die Bewegung, insbesondere die Diffusion von Gasgemengen, Sitzungsberichte der Akademie der Wissenschaften in Wien (1871).
3. J. Stefan, Über die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere, Sitzungsberichte der Akademie der Wissenschaften in Wien (1889).
4. L. Boltzmann, Ableitung des Stefan’schen Gesetzes, betreffend die Abhängigkeit der Wärmestrahlung von der Temperatur aus der electromagnetischen Lichttheorie, Annalen der Physik 22 (1884).
5. Stefan–Boltzmann law and Stefan problem, Wikipedia.
6. A concise history of the black-body radiation problem, arXiv:2208.06470.
7. R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena, 2nd ed., Wiley, 2002 (Maxwell–Stefan equations, Stefan tube).
8. J. Crank, Free and Moving Boundary Problems, Oxford University Press, 1984.