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A Brief History of the T4 Radiation Law

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Since the 1700s, natural philosophers understood that heat exchange between two bodies was not precisely linearly dependent on the temperature difference, and that at high temperatures the discrepancy became greater. Over the years, many models were developed with varying degrees of success. The lack of success was due to the difficulty obtaining accurate experimental data, and a lack of knowledge of the fundamental mechanisms underlying radiation heat exchange. Josef Stefan, of the University of Vienna, compiled data taken by a number of researchers who used various methods to obtain their data, and in 1879 proposed a unique relation to model the dependence of radiative heat exchange on the temperature: the T4 law. Stefan’s model was met with some skepticism and was not widely accepted by his colleagues. His former student, Ludwig Boltzmann, who by then had taken a position at the University of Graz in Austria, felt that there was some truth to the empirical model proposed by his mentor. Boltzmann proceeded to show in 1884, treating electromagnetic radiation as the working fluid in a Carnot cycle, that in fact the T4 law was correct. By the time that Boltzmann published his thermodynamic derivation of the radiation law, physicists became interested in the fundamental nature of electromagnetic radiation and its relation to energy, specifically determining the frequency distribution of blackbody radiation. Among this group of investigators was Wilhelm Wien, working at Physikalisch-Technische Reichsanstalt in Charlottenburg, Berlin. He proposed a relation stating that the wavelength at which the maximum amount of radiation was emitted occurred when the product of the wavelength and the temperature was equal to a constant. This became known as Wien’s Displacement Law, which he deduced this by imagining an expanding and contracting cavity, filled with radiation. Later, he combined his Displacement Law with the T4 law to give a blackbody spectrum which was accurate over some ranges, but diverged in the far infrared. Max Planck, at the University of Berlin, built on Wien’s model but, as Planck himself stated, “the energy of radiation is distributed in a completely irregular manner among the individual partial vibrations...” This “irregular” or discrete treatment of the radiation became the basis for quantum mechanics and a revolution in physics. This paper will present brief biographies of the four pillars of the T4 radiation law, Stefan, Boltzmann, Wien and Planck, and outline the methodologies used to obtain their results.
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1 Copyright © 2009 by ASME
Proceedings HT 2009
2009 ASME Summer Heat Transfer Conference
July 19-23, 2009, San Francisco, California USA
HT2009-88060
A BRIEF HISTORY OF THE T4 RADIATION LAW
John Crepeau
University of Idaho
Department of Mechanical Engineering
PO Box 440902
Moscow, ID 83844-0902 USA
ABSTRACT
Since the 1700s, natural philosophers understood that heat
exchange between two bodies was not precisely linearly
dependent on the temperature difference, and that at high
temperatures the discrepancy became greater. Over the years,
many models were developed with varying degrees of success.
The lack of success was due to the difficulty obtaining accurate
experimental data, and a lack of knowledge of the fundamental
mechanisms underlying radiation heat exchange. Josef Stefan,
of the University of Vienna, compiled data taken by a number
of researchers who used various methods to obtain their data,
and in 1879 proposed a unique relation to model the
dependence of radiative heat exchange on the temperature: the
T4 law.
Stefan’s model was met with some skepticism and was
not widely accepted by his colleagues. His former student,
Ludwig Boltzmann, who by then had taken a position at the
University of Graz in Austria, felt that there was some truth to
the empirical model proposed by his mentor. Boltzmann
proceeded to show in 1884, treating electromagnetic radiation
as the working fluid in a Carnot cycle, that in fact the T4 law
was correct.
By the time that Boltzmann published his
thermodynamic derivation of the radiation law, physicists
became interested in the fundamental nature of electromagnetic
radiation and its relation to energy, specifically determining the
frequency distribution of blackbody radiation. Among this
group of investigators was Wilhelm Wien, working at
Physikalisch-Technische Reichsanstalt in Charlottenburg,
Berlin. He proposed a relation stating that the wavelength at
which the maximum amount of radiation was emitted occurred
when the product of the wavelength and the temperature was
equal to a constant. This became known as Wien’s
Displacement Law, which he deduced this by imagining an
expanding and contracting cavity, filled with radiation. Later,
he combined his Displacement Law with the T4 law to give a
blackbody spectrum which was accurate over some ranges, but
diverged in the far infrared.
Max Planck, at the University of Berlin, built on
Wien’s model but, as Planck himself stated, “the energy of
radiation is distributed in a completely irregular manner among
the individual partial vibrations...” This “irregular” or discrete
treatment of the radiation became the basis for quantum
mechanics and a revolution in physics.
This paper will present brief biographies of the four
pillars of the T4 radiation law, Stefan, Boltzmann, Wien and
Planck, and outline the methodologies used to obtain their
results.
EARLY RADIATION STUDIES
Heat transfer, as a serious branch of study in physics,
began with Sir Isaac Newton, although his contributions to this
subject are but a small part of his complete body of work
including calculus, mechanics, optics, and of course, gravity.
Natural philosophers of the day deduced that the heat transfer
rate was some function of the temperature, although they had
difficulty distinguishing between the now commonly known
modes of heat transfer: conduction, convection and radiation.
Early research into geological history motivated Joseph Fourier
to study heat conduction in solids, and then extend his work to
incompressible fluids, but postponed any kind of work studying
heat transfer in atmospheric gases. Early visual studies of
convection heat transfer in gases were performed by Rinaldini
in 1657. Euler began the first mathematical theories of
convection, and Rumford later discovered convection currents
in liquids in 1797. At the time, scientists had difficulty
separating conduction and convection in a fluid, and it was
widely held that fluids did not conduct heat.
Early researchers had a rudimentary knowledge of
radiation heat transfer, and believed that heat could be
2 Copyright © 2009 by ASME
transferred by radiation through a vacuum. To separate
conduction and convection effects from radiation, it would be
necessary to draw a vacuum between two surfaces. It was not
fully understood, however, that a sufficiently low pressure had
to be achieved and that residual gases could affect the radiation
transfer by absorption and reradiation. One observation that
became quite clear was that at high temperatures, the heat
transfer was no longer simply a function of the temperature
difference. What the dependence was, however, was another
issue. For a more detailed early history of heat transfer, see
Brush [1,2].
After some issues in trying to determine a proper
temperature scale, and some fits and starts, Dulong and Petit
[3] performed a series of experiments to determine the rate of
radiation heat transfer between two bodies. Their paper, which
was awarded a prize from the French Académie des Sciences in
1818 proposed a relation for the emissive power of radiation,
T
Ea
μ
= (1)
Where a =1.077 and
μ
was a constant dependent on the
material and size of the body. The dependence on the
temperature in the exponent helped explain the higher heat
transfer rates at higher temperatures, but had a significant
disadvantage. There would be heat transfer even at T = 0.
Nevertheless, even decades after the T4 law was discovered, a
textbook stated that the Dulong and Petit law, “seems to apply
with considerable accuracy through a much wider range of
temperature differences than that of Newton,” and is based on
“one of the most elaborate series of experiments ever
conducted.”[4] Further experiments were performed by Tyndall
[5], Provostaye and Desains [6], Draper [7] and Ericsson [8],
all making slight variations to the constants determined by
Dulong and Petit, but keeping the basic structure of the model
intact. This was the state-of-the-art when Josef Stefan,
Professor and Director of the Institute of Physics at the
University of Vienna began his own investigation of the
problem.
JOSEF STEFAN
Josef Stefan (Fig. 1) was born on March 24th, 1835 in
the small village of St. Peter, on the outskirts of Klagenfurt, in
modern-day Austria. He was the illegitimate son of ethnic
Slovenes. His father Aleš was a miller and baker, and his
mother Marija Startinik was a maidservant. He began his
studies at the University of Vienna in 1853, studying physics
and mathematics, graduating in 1857. The year after his
graduation, he passed his doctoral examination at the
University of Vienna and accepted a position at the University’s
Physiological Institute. He was more interested in physics-
related work, and was eventually offered a full professorship in
mathematics and physics in 1863, becoming the youngest
person in the Austro-Hungarian empire to hold that rank. Two
years later in 1865, he was appointed the Director of the
Institute of Physics at the University of Vienna, a position he
held until his death in 1893.[9] Among his students at the
Institute were Ludwig Boltzmann and the psychoanalyst
Sigmund Freud, who took physics courses as part of his
medical studies.
Figure 1. Portrait of Josef Stefan, taken around 1885.
Stefan did not feel comfortable with the Dulong and
Petit law, so he began to delve into their work more closely.
Drawing on his past experience in conduction heat transfer, he
reconstructed their apparatus, shown in Fig. 2. Based on the
design, he estimated that a significant portion of the heat was
lost by conduction and not by radiation, as had been presumed
by Dulong and Petit. By subtracting out the heat lost by
conduction and reanalyzing the data, Stefan saw that the heat
transfer was proportional to the temperature to the fourth
power. Figure 3 below shows the form of the T4 law as it first
appeared in print in 1879.[10] Stefan didn’t realize the
importance of his proportionality constant, A, but noted that it,
“depend[ed] on the size and the surface of the body.” He also
noted that the temperature had to be given in absolute values.
The unusual form of his model forced Stefan to test it
on previously published data, and he wanted to see how well it
worked at temperatures higher than those used by Dulong and
Petit. He first looked at Tyndall’s work, which gave heat
transfer data on a platinum wire over a wide temperature range.
After studying Tyndall’s data, Stefan wrote, “From weak red
heat (about 525°C) to complete white heat (about 1200°C) the
intensity of the radiation increases from 10.4 to 122, thus
nearly 12-fold (more precisely 11.7). The ratio of the absolute
temperature 273+1200 and 273+525 raised to the fourth power
3 Copyright © 2009 by ASME
gives 11.6.” [11] This result gave Stefan additional confidence
in his model, and he proceeded to test it on the data of
Provostaye and Desains, Draper, and Ericsson, and found that
the T4 model fit their data better than the Dulong and Petit
model.
Figure 2. Original apparatus which Stefan used to evaluate the
amount of heat lost in an experiment by Dulong and Petit.
Although Stefan was an outstanding experimentalist,
neither he nor any of his students performed (or at least
published) radiation experiments themselves. It is not known
why. For this reason, perhaps, Stefan was not completely
confident in his model. He wrote that his analysis had a,
“hypothetical nature and reasoned support for [it] was
impossible, so long as measurements are not made of radiation
to surroundings at absolute zero, or at least a very low
temperature.”(translation from Dougal [11]) Interestingly,
Stefan never computed a value of his proportionality constant,
A, but from a straightforward analysis from Stefan’s paper, it
can be easily determined to be 5.056 x 10-8 W/m2K4, which is
about 11% lower than the currently accepted value of what we
now know as the Stefan-Boltzmann constant.
Figure 3. The first appearance in print of the T4 law.[10]
LUDWIG BOLTZMANN
Ludwig Boltzmann theoretically tackled the radiation
problem which his mentor Stefan studied experimentally, long
after Boltzmann left Vienna during his second professorship at
the University of Graz.
Boltzmann (Fig. 4) was born on February 20, 1844,
just outside the medieval protective walls of Vienna, to a
middle class family. His father, Ludwig Georg, was a civil
servant of the Habsburg empire, working in the taxation office.
His mother was Maria Pauernfeind, the daughter of a well-to-
do Salzburg merchant.[12] He was a precocious boy who early
in life was taught by a private tutor, and then as the family
moved to various locations throughout Austria, attended the
local Gymnasium. He took piano lessons from the famed
composer Anton Bruckner. Although the lessons ended
abruptly, Boltzmann continued to play the piano for the rest of
his life.[13] In 1863, Boltzmann entered the University of
Vienna to study mathematics and physics, and after publishing
two papers, received his Ph.D. just three years after
matriculating. He arrived at the Institute of Physics not long
after Stefan assumed the Directorship. Stefan was just nine
years older than his student. Also at the Institute was Josef
Loschmidt, an older scientist who took Boltzmann under his
wing. Loschmidt was the discoverer of the ring structure of the
benzene molecule and was the first to accurately determine the
size of an air molecule. Loschmidt and Boltzmann hashed out
ideas that Boltzmann had on the kinetic theory of gases, which
he was developing separately but simultaneously with James
Clerk Maxwell of Scotland. After receiving his doctorate and
working as an assistant at the Institute, Boltzmann left Vienna
in 1869 to become a professor of physics at the University of
Graz. He left Graz for Vienna in 1873, then returned to Graz in
1876, where he stayed until 1890. During this second period,
Boltzmann set out to prove from thermodynamic principles the
experimental model published by Stefan.
Figure 4. Ludwig Boltzmann in Graz, 1887, three years after he
derived the T4 law.
In a deceptively simple analysis, Boltzmann
considered a Carnot cycle, using radiation particles as the
working fluid. He based his ideas on an earlier paper of Adolfo
Bartoli [14] who described some ideas on radiation pressure.
Boltzmann combined thermodynamics and Maxwell’s
electromagnetic equations with the then novel idea that
electromagnetic waves exert a pressure on the walls of an
enclosure filled with radiation. In a piston-cylinder system,
4 Copyright © 2009 by ASME
when the piston moves slowly, the pressure exerted on the
piston (from [1,12]) is,
3
e
p= (2)
where e is the energy density. From conservation of energy one
can then write,
()
3
de e
TdS d eV pdV edV V dT dV
dT
=+=+ +
(3)
So,
4
3
de
TdS V dT edV
dT
⎛⎞
=+
⎜⎟
⎝⎠ (4)
The derivatives of the entropy with respect to T and V are
respectively,
4
,3
SVde S e
TTdT V T
∂∂
==
∂∂
(5)
By taking cross-derivatives of the entropy with respect to V and
T and equating, one finds,
2
14 1
3
de de
Te
TdT dT T
⎛⎞
=−
⎜⎟
⎝⎠
(6)
Rearranging gives,
4
de e
dT T
= (7)
Separating and integrating gives the familiar form of the fourth
power radiation law, or as Boltzmann originally published it
[15], shown below in Fig. 5.
Figure 5. The last step of Boltzmann’s derivation of the T4
law.[15]
Once Boltzmann published his theoretical results confirming
Stefan’s analysis of experimental data, the T4 law became more
widely accepted. But work on radiation did not cease. Towards
the end of his life, Boltzmann spent a large portion of his time
and energy defending the kinetic theory of gases, leading the
so-called atomistic school of thought, defending the existence
of atoms. The opposing school, the energeticists lead by Ernst
Mach, held that without visual observation it was impossible to
prove that atoms existed. This struggle drained Boltzmann’s
strength and physics focus. He committed suicide on
September 5th, 1906 in his hotel room in Duino, near Trieste,
Italy.
WILHELM WIEN
Wilhelm Wien (Fig. 6) was born on January 13th, 1864
in the town of Fischhausen (currently Primorsk), then in East
Prussia (now Poland). His father Carl was a well-to-do farmer
who married the former Caroline Gertz. He began his advanced
studies in 1882, attending the Universities of Göttingen and
Berlin, then from 1883-1885 settled in Hermann Helmholtz’s
laboratory in Berlin, where he earned his Ph.D. in 1886 with a
thesis on experimental light diffraction on metal section and the
influence of materials on the color of refracted light.[16] He
believed strongly that theoretical and experimental physics
should not be separated, and was considered an expert in both
fields. At the age of 21, Wien befriended Max Planck, who was
six years his senior, and they remained lifelong friends and
scientific colleagues.
Figure 6. Portrait of Wilhelm Wien
Wien was an ardent Prussian nationalist, who thought
that the firing of the Bismarck was the greatest disaster in
German history, and that Germany was completely blameless
for the first World War.[17] He held a number of academic
positions, primarily as professor of Physics, the University of
Aachen in 1896, University of Giessen in 1899, and the
University of Wurzburg in 1900 where he stayed until 1920,
when he accepted a position at the University of Munich,
where he spent the rest of his life. He wrote an autobiography,
“Aus dem Leben und Wirken eines Physikers,” (On the Life
and Work of a Physicist), published in posthumously in 1930,
which was not published outside of Germany. Wien spent a
good portion of his career studying blackbody radiation, but
this paper will focus on two of his most well-known
contributions.
5 Copyright © 2009 by ASME
The Displacement Law
Wien felt very strongly that heat radiation could be
treated thermodynamically, and began to extend Boltzmann’s
methods. However, while Boltzmann analyzed the full energy
spectrum, Wien concentrated on the energy at a given
wavelength.[18] At the time, a thermodynamic approach to
radiation was not well-accepted, and both Wien’s mentor
Helmholtz and Lord Kelvin argued against these methods.
In his 1893 paper[19], Wien argued that two separate
processes should give the same energy distribution over the
wavelengths if the final temperature of both processes were the
same. The first process was the increase in temperature as the
energy density increased, and the second the corresponding
adiabatic decrease of the volume of the enclosure containing
the radiation. By using the Doppler effect, he showed the
wavelength
λ
depended on the velocity of the source, so that
the spectral energy densities,
ψ
, were related to the
wavelengths by,
4
0
0
λ
ψ
ψ
λ
⎛⎞
=⎜⎟
⎝⎠
(8)
Wien then introduced the Stefan-Boltzmann relation to show, as
he did originally below in Fig. 7, what is now known as Wien’s
Displacement Law,
Figure 7: Wien’s Displacement law.[19] Here, the
θ
denotes
temperature.
In Wien’s description, “In the normal emission spectrum…each
wavelength is displaced (italics added) on change of
temperature, that the product…remains constant.”[20] The
equation shown in Figure 7 was first referred to as Wien’s
“Displacement Law” by Lummer and Pringsheim[21,22], who
also first calculated the constant to be 0.294 cm·K.
Interestingly, the concept of a “finite quantum of
energy,” often attributed to Planck, appears in Wien’s 1893
paper.
Wien’s Distribution Law
Three years later, Wien[23] sought to find a relation to
govern the spectral energy distribution of a blackbody. A key
hypothesis was that he assumed that blackbody radiation was
emitted by molecules which followed Maxwell’s velocity
distribution.[18] As Wien stated, he used, “Maxwell’s law of
the distribution of velocities as the basis of the radiation law,
but to lessen the number of hypotheses…by drawing upon the
results obtained by Boltzmann and myself by pure
thermodynamic treatment.”[20] This formed the basis of his
“molecular hypothesis.” By assuming that the wavelength of
radiation emitted by a molecule was a function only of its
velocity, he found (Fig. 8),
Figure 8. Wien’s Distribution Law as it originally appeared.[23]
By integrating
φ
λ
over all wavelengths and equating the result
with the Stefan-Boltzmann law, he found that the exponent of
lambda given in Fig. 8 had to satisfy,
α
= 5,
Wien’s distribution law agreed relatively well with
experimental data, except that it diverged at large values of the
product,
λ
T.
For his work in radiation, Wilhelm Wien won the 1911
Nobel Prize in Physics. In the words of his colleague Max von
Laue, “His immortal glory,” was that he “led us to the very
gates of quantum physics.” Wien died on August 30, 1928 in
Munich, Germany.
MAX PLANCK
Wien’s friend and colleague, Max Planck, studied
Wien’s results and understood that they modeled experimental
results well, but also realized its shortcomings. Planck knew
that the thermodynamic arguments were strong, but felt that
there were too many hypotheses, and set about a way to
minimize them and find a new energy distribution relation.
Max Planck (Fig. 9) was born into a family of scholars
on April 23rd, 1858 in Kiel, Germany. His grandparents were
theologians who taught at the University of Göttingen, and his
father Wilhelm Planck was a professor of jurisprudence at the
University of Kiel. His mother, Emma Patzig, descended from
a line of pastors. Planck, although a friend and colleague of
Wien, did not hold a similar nationalistic view of his homeland.
Planck considered himself more liberal politically than the rest
of his family. He was a good student who showed talent over a
range of subjects beyond science, including history, language
and especially music, and he played a number of
instruments.[17] By virtue of his amiable personality, he was
considered a favorite of his teachers and classmates. He first
matriculated at the University of Munich in 1874, then in 1877
studied for a year in Berlin, and in 1879 defended his doctoral
dissertation in Munich. In 1885, he was appointed an associate
professor of physics at the University of Kiel and in 1889, upon
Kirchhoff’s death, assumed his position at the University of
Berlin. He was held in high esteem for his technical abilities,
but was also noted for his leadership and organizational skills.
He combined a number of smaller, regional technical societies
to form the Deutsche Physikalische Gesellschaft. He became a
Dean at the University of Berlin, and after World War I became
the highest authority in German physics.
6 Copyright © 2009 by ASME
Figure 9. Max Planck in 1901, one year after he published his
paper on the quantum of action.
Planck, although a firm believer in Wien’s distribution law
(Fig. 8), knew very well that it was not valid at large values of
λ
T, and recent experimental results reconfirmed this
shortcoming. He realized that at long wavelengths, the entropy
had to satisfy the relation,
2
22
dS const
dU U
=− (9)
which followed from the thermodynamic definition of the
temperature. After integrating and combining with the relation
for the energy in a resonator,
3
8c
U
ν
ν
π
ν
= (10)
Planck derived the relation [24],
Figure 9. Planck’s Distribution Law. [24]
Planck gave these results at a meeting of the Deutsche
Physikalische Gesellschaft in 1900 [25]. Days after his
presentation, Planck received a note stating that his equation
agreed very well with experimental data. It is interesting to note
that Planck derived this equation purely on thermodynamic
grounds. The entropy that he used to get this result was given
by,
01ln 1 ln
UUUU
S
α
β
νβνβνβν
⎛⎞⎛⎞
=− + + −
⎜⎟⎜⎟
′′
⎝⎠⎝⎠
(11)
However, he did not feel completely comfortable with the
theoretical underpinnings or the physical reasoning behind his
distribution law. This motivated him to dig deeper into his
results.
Although his distribution was based on an expression
for the equilibrium entropy, he understood that
thermodynamics alone could not accurately describe blackbody
radiation. Wien felt that if his distribution was not valid at all
wavelengths, then there was some qualitative difference
between short and long wavelength radiation, and therefore
there could be no unified theory of electromagnetic radiation.
Planck felt strongly that not only was there a unified theory, but
that principles of electromagnetism, thermodynamics and
physical fields could be combined.[18] He then discussed with
Boltzmann the derivation of the velocity distribution of
molecules in a gas, specifically the idea of dividing the energy
into discrete values.
Boltzmann gave a relation for the number of ways of
distributing discrete, equal energy values among a number of
molecules. The logarithm of the distribution was,
ln 1 ln 1 lnJnnnnn
λ
λλλ
⎛⎞⎛⎞
=+ +
⎜⎟⎜⎟
⎝⎠⎝⎠
(12)
Planck saw the similar structure between this and the entropy
equation (Eq. 11), and showed that they were equal when as he
showed below in Fig. 10 [26],
Figure 10. The first appearance of the equation E = hν. [26]
along with his calculation of the constant that would bear his
name,
Figure 11. Planck’s calculation of the constant shown in Fig.
10 [26].
Hence the quantum of action was born. By integrating Planck’s
distribution, Fig. 9, over all wavelengths, he got back the
Stefan-Boltzmann law.
7 Copyright © 2009 by ASME
Planck tried to explain the physical meaning of his
discrete energy relation, with little success. Others felt that
since Planck didn’t fully grasp the implications of his result, he
should not get credit for his discovery, but this judgment may
be too harsh.[27] As the field of quantum mechanics grew,
Planck felt less sure of its veracity. However, for his work,
Planck received the Nobel Prize for Physics in 1918. By virtue
of his ideas on discretizing the energy of molecules in a
velocity distribution, Boltzmann has been considered a
grandfather of quantum theory. Planck himself describes the
genesis of the T4 law in his The Theory of Heat Radiation [28].
He survived both World Wars, although his son Erwin was
killed by the Nazis for participating in a plot to kill Adolf
Hitler. Max Planck died on October 4th, 1947 in Göttingen,
West Germany.
ACKNOWLEDGMENTS
The author gratefully acknowledges the help of Mrs. Jennifer
O’Laughlin of Interlibrary Loans at the University of Idaho for
acquiring copies of the original papers.
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[2] Brush, S.G., 1976, The Kind of Motion We Call Heat, North
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[4] Preston, T., 1919, Theory of Heat, London, 3rd edition, pp.
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[5] Tyndall, J., 1865, Heat Considered as a Mode of Motion,
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[10] Stefan, J., 1879, “Über die Beziehung der Wärmestrahlung
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[12] Cercignani, C., 1998, Ludwig Boltzmann: The Man Who
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[13] Reiter, W., 2007, “Ludwig Boltzmann, A Life of Passion,”
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[14] Boltzmann, L., 1884, “Ueber eine von Hrn. Bartoli
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[15] Boltzmann, L., 1884, “Ableitung des Stefan’schen
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der Temperatur aus der electromagnetischen Lichttheorie,”
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[16] “Wilhelm Wien,” http://nobelprize.org/nobel_prizes/
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[17] Heilbron, J.L., 2000, Dilemmas of an Upright Man: Max
Planck and the Fortunes of German Science, Harvard.
[18] Mehra, J. and Rechenberg, H., 2000, The Historical
Development of Quantum Theory, Vol. 1, Springer.
[19] Wien, W., 1893, “Eine neue Beziehung der Strahlung
schwarzer Körper zum zweiten Hauptsatz der Wärmetheorie,”
Sitzungsberichte der preussischer Akademie, pp. 55-62.
[20] Kangro, H., 1976, Early History of Planck’s Radiation
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[21] Lummer, O. and Pringsheim, E., 1899, “Die Vertheilung
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Verhandlungen der Deutsche Physikalische Gesellschaft, 1, pp.
23-41.
[22] Lummer, O. and Pringsheim, E., 1899, “1. Die Vertheilung
der Energie im Spectrum des schwarzen Körpers und des
blanken Platins; 2. Temperaturbestimmung fester glühender
Körper,” Verhandlungen der Deutsche Physikalische
Gesellschaft, 1, pp. 215-235.
[23] Wien, W., 1896, “Über die Energievertheilung im
Emissionsspectrum eines schwarzen Körpers,” Annalen der
Physik und Chemie, 294, pp. 662-669. English translation,
1897, “On the Division of Energy in the Emission-Spectrum of
a Black Body,” Philosophical Magazine, Vol. 43, pp. 214-220.
[24] Planck, M., 1900, “Ueber eine Verbesserung der
Wien’schen Spektralgleichung,” Verhandlungen der deutschen
physikalischen Gesellschaft, 2, pp. 202-204.
[25] Planck, M., 1900, “Zur Theorie des Gesetzes der
Energieverteilung im Normalspectrum,” Verhandlungen der
deutschen physikalischen Gesellschaft, 2, 14 Dec., pp. 237-
245.
[26] Planck, M., 1901, “Ueber das Gesetz der
Energieverteilung im Normalspectrum,” Annalen der Physik,
309(3), pp. 553-563.
[27] Gearhart, C.A., 2002, Planck, the Quantum, and the
Historians, Physics in Perspective, 4, pp. 170-215.
[28] Planck, M., 1914, The Theory of Heat Radiation, P.
Blakiston’s Son, Philadelphia.
... Planck derived his distribution law first on thermodynamic grounds, but after a discussion with Boltzmann about the number of ways of distributing discrete equal energy quanta among a number of molecules, he saw the similar structure of the resulting expression with his distribution. By setting the energy quanta proportional to the photon frequency, ε = hν, where the ''Planck constant'' h entered as a proportionality factor, he obtained the number density of photons as a function of frequency that provided a perfect match with experiment [61,62]: ...
... where ρ (ν, T ) is the spectral energy density and c the velocity of light. Eq. (12) differs from Wien's formula only by the ''−1'' in the exponential term in the square bracket, but becomes identical for hν ≫ kT [61]. Photons are bosons and have integer spin equal to 1. ...
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Understanding the origin of irreversibility in thermodynamics has been a fundamental scientific challenge and puzzle for nearly a century. Initially, the discussions related to classical thermodynamic systems, but recently quantum systems became the main focus. Explanations have often been sought by reference to classical equations of motion, which are time-reversible. We conjecture that the origin of irreversibility lies in energy dissipation, a term that is at the core of the Second Law of thermodynamics. However, thermodynamic irreversibility is distinct from time-irreversibility. A system in thermodynamic equilibrium may have reached this state via a deterministic, integrable and therefore time-reversible process, or, alternatively, via an irreversible route, both resulting in thermodynamically indistinguishable states. The process with time-reversible history may become irreversible by a process called thermalization, which occurs when the system loses memory of its history without the necessity of energy dissipation. Quantum systems do this by losing phase coherence; for classical systems the decoherence is at zero frequency, due to loss of time correlation. More generally, not only equilibrium systems may have lost memory of their history. A common cause of memory loss is probabilistic/stochastic events, which are not deterministic and take place only with a certain probability at any given time. In contrast to thermalization, equilibration involves energy dissipation within a system or to the surroundings or by decrease of concentration of the system. Time-reversibility is not related to system size, and the fluctuation theorem is a probabilistic and not a deterministic phenomenon and therefore not suited to provide an understanding of the irreversibility of time in thermodynamic systems. There are also processes which are both dissipative and probabilistic, such as the radiative or non-radiative decay of electronically excited states. Dissipation of a given energy into multiple smaller energy quanta (heat) is by itself not fully reversible for kinetic reasons. It is kinetically a first-order probabilistic process, whereas the reverse is a second- or higher-order process. Thermodynamics provides empirical laws, developed for conventional matter as we know it on planet Earth and in our laboratories. Of relevance here is the Second Law, also called the arrow of time, stating that spontaneous processes take place for isolated systems with increasing entropy. It is assumed to hold also for the universe as a whole. However, over the distances of individual galaxies, self-gravitation leads to conditions where the kinetic energy of the system decreases while the total energy increases, pretending negative heat capacity, and it allows the formation of black holes. This requires an extension of the Second Law. This review aims at presenting an overarching tutorial clarification of the subject.
... The Stefan-Boltzmann (SB) law [1,2] refers to the radiant emittance of an ideal black-body cavity at a given temperature T under thermal equilibrium and thermodynamic limit conditions [3]. It takes the form: ...
... The Stefan-Boltzmann (SB) law refers to the radiant emittance of an ideal blackbody cavity at a given temperature T under thermal equilibrium and thermodynamic limit conditions [1]. It takes the form: ...
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The Stefan-Boltzmann (SB) law relates the emissivity q,to the fourth power of the absolute temperature T, the proportionaly factor, $\sigma$ firstly estimated by Stefan to within 11 per cent of the actual value. The law is a pillar of modern physics since its microscopic derivation implies the quantization of the energy related to the electromagnetic field. Somewhat astonishing, Boltzmann presented his derivation in 1878 making use only of electrodynamic and thermodynamic classical concepts, apparently without introducing any quantum hypothesis (here called first Boltzmann paradox). By using Planck (1901) quantization of the radiation field in terms of a gas of photons, the SB law received a microscopic interpretation providing also the value of the SB constant on the basis of a set of universal constants including the quantum action constant h. However, the successive consideration by Planck (1912) of the zero-point energy contribution was found to be responsible of another divergence of the radiation energy-density for the single photon mode at high frequencies. This divergence is of pure quantum origin and is responsible for a vacuum-catastrophe, to keep the analogy with the well-known ultraviolet catastrophe of the classical black-body radiation spectrum, given by the Rayleigh-Jeans law in 1900. As a consequence, from a rigorous quantum-mechanical derivation we expect the divergence of the SB law (here called second Boltzmann paradox). In this paper we revisit the SB law by accounting for genuine quantum effects associated with Planck energy quantization and Casimir size quantization thus resolving both Boltzmann paradoxes.
... Interestingly, it is this scaling law, and not the equation regarding only the shift in wavelength-peak position, that prompted the development of the blackbody radiation law, and one of the two main contributions that earned Wilhelm Wien the 1911 Nobel Prize. 10 Thus, it makes sense to discuss this topic in full only to a more advanced audience, perhaps one undergoing introductory courses to quantum mechanics and statistical physics. ...
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... No contexto da radiação de corpo negro, Crepeau (2009) destaca a lei de Stefan-Boltzmann, também com um enfoque histórico, centrado em quatro cientistas da época: Josef Stefan, Ludwig Boltzmann, Wilhelm Wien e Max Planck. Dougal (1976) ressalta a importância de John Rayleigh, bem como de vários físicos experimentais como Lummer e Rubens na descoberta da quantização da energia. ...
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... where q 0 0 is the heat flux due to radiation, ε is the emissivity of the surface, and σ is the Stefan-Boltzmann constant [10]. In order to implement these into a model, the process of conduction must be applied to the entirety of the material, whereas, the convection and radiation must only be applied to the skin as boundary conditions to ensure that the part cools properly based on the environment where it is located. ...
... a result based on a modification of the Dulong and Petit law (see Ref. [17]). Stefan used existing measurements of, e.g., temperature dependent irradiance from a platinum filament by Tyndall [18] to empirically determine the proportionality constant r. ...
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Boltzmann 1) hat auf der Grundlage eines von Bartoli ersonnenen Prozesses nachgewiesen, daß aus dem zweiten Hauptsatz der Wärmetheorie das Vorhandensein eines Druckes gefolgert werden kann, welcher von der Strahlung auf eine bestrahlte Oberfläche ausgeübt wird. Ein solcher Druck ist andererseits eine Folge der elektromagnetischen Theorie des Lichtes, und Boltzmann konnte aus dieser Beziehung das Stefansche Strahlungsgesetz für schwarze Körper ableiten.