﻿ The Fortuitous Discovery and Derivation of the Stefan-Boltzmann Law for Thermal Energy Emission
San José State University

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The Fortuitous Discovery and
Derivation of the Stefan-Boltzmann
Law for Thermal Energy Emission

The most exciting topic of empirical and theoretical research in physics in the 19th century was radiation. At the beginning of the 19th century the works of Thomas Young in England and Augustin-Jean Fresnel in France overturned the corpuscular theory of light of Isaac Newton and replaced it with the wave theory. About 1870 Gustav Kirchhoff systematized the work on the spectra of light emitted from heated bodies and gases. Kirchhoff then derived the theoretical result that the emission of radiation of wavelength &lamba; from a any body of absolute temperature T is proportional to the absorptivity of that body at that wavelength and temperature. That implied that the empirical investigation could be limited to a black body, a body that absorbed all of the radiant energy that comes to it. Furthermore Kirchhoff established that a hole in body with a cavity at constant temperature functions as a black body radiator because any radiation coming to that hole enters the cavity and is effectively absorbed.

Kirchhoff's work thus set up a systematic empirical investigation of thermal radiation by physicists. By 1875 the results of this research was appearing in textbooks of physics, notably in the textbook of A. Wullner, Die Lehre von der Wärme vom Standpunkte der mechanischen Wärmetheorie (2nd edition). In that text book Josef Stefan read that John Tyndall had found that the total radiant energy emitted by a platinum wire at a temperature of 1200 °C is 11.7 times that radiated at 525°C. A temperature of 1200° C is an absolute temperature of 1473° K and 525°C is equivalent to 798°K. Josef Stefan noted that

If E=σTβ then

#### E1/E2 = (T1/T2)βand β = log(E1/E2)/log(T1/T2)

For the above data

#### β = log(11.7)/log(1.849) = 4.0016

Stefan conjectured on the basis of that one piece of data that the total emission E of radiant energy over all wavelengths from a body at absolute temperature T is proportional to the fourth power of T; i.e.,

#### E = σT4

Stefan published this speculation in an article in a journal of physics in 1879. Stefan's discovery was fortuitous because Tyndall's result was in error. Tyndall was not using test equipment approximating a black body. The result Tyndall should have obtained was a ratio of 18.6 rather than 11.7. It just happened that his erroneous measurement approximated what he would have gotten from a black body radiator.

Stefan's conjecture happened to be correct and in 1884 Ludwig Boltzmann published a derivation of the fourth power law based upon thermodynamics applied to a radiation gas. In the course of his analysis Boltzmann concluded that radiation exerts pressure just a gas of molecules does.

The empirical measurement of total emission as a function of absolute temperature found the value of the constant in the fourth power law to be

#### σ = 5.670373×10−8 (Watts per sq m) per (K°)4or, in cgs units σ = 5.670373×10−5 [(ergs per second) per sq cm] per (K°)4

Reference:

Max Jammer, The Conceptual Development of Quantum Mechanics, McGraw-Hill Book Co., New York, 1966.