Part 14

41. _Relation between Radiation and Temperature._--Assuming, in accordance with the reasoning of Balfour Stewart and Kirchhoff, that the radiation stream inside an impervious enclosure at a uniform temperature is independent of the nature of the walls of the enclosure, and is the same for all substances at the same temperature, it follows that the full stream of radiation in such an enclosure, or the radiation emitted by an ideal black body or full radiator, is a function of the temperature only. The form of this function may be determined experimentally by observing the radiation between two black bodies at different temperatures, which will be proportional to the difference of the full radiation streams corresponding to their several temperatures. The law now generally accepted was first proposed by Stefan as an empirical relation. Tyndall had found that the radiation from a white hot platinum wire at 1200 deg. C. was 11.7 times its radiation when dull red at 525 deg. C. Stefan (_Wien. Akad. Ber._, 1879, 79, p. 421) noticed that the ratio 11.7 is nearly that of the fourth power of the absolute temperatures as estimated by Tyndall. On making the somewhat different assumption that the radiation between two bodies varied as the difference of the fourth powers of their absolute temperatures, he found that it satisfied approximately the experiments of Dulong and Petit and other observers. According to this law the radiation between a black body at a temperature [theta] and a black enclosure or a black radiometer at a temperature [theta]0 should be proportional to ([theta]^4 - [theta]0^4). The law was very simple and convenient in form, but it rested so far on very insecure foundations. The temperatures given by Tyndall were merely estimated from the colour of the light emitted, and might have been some hundred degrees in error. We now know that the radiation from polished platinum is of a highly selective character, and varies more nearly as the fifth power of the absolute temperature. The agreement of the fourth power law with Tyndall's experiment appears therefore to be due to a purely accidental error in estimating the temperatures of the wire. Stefan also found a very fair agreement with Draper's observations of the intensity of radiation from a platinum wire, in which the temperature of the wire was deduced from the expansion. Here again the apparent agreement was largely due to errors in estimating the temperature, arising from the fact that the coefficient of expansion of platinum increases considerably with rise of temperature. So far as the experimental results available at that time were concerned, Stefan's law could be regarded only as an empirical expression of doubtful significance. But it received a much greater importance from theoretical investigations which were even then in progress. James Clerk Maxwell (_Electricity and Magnetism_, 1873) had shown that a directed beam of electromagnetic radiation or light incident normally on an absorbing surface should produce a mechanical pressure equal to the energy of the radiation per unit volume. A. G. Bartoli (1875) took up this idea and made it the basis of a thermodynamic treatment of radiation. P. N. Lebedew in 1900, and E. F. Nichols and G. F. Hull in 1901, proved the existence of this pressure by direct experiments. L. Boltzmann (1884) employing radiation as the working substance in a Carnot cycle, showed that the energy of full radiation at any temperature per unit volume should be proportional to the fourth power of the absolute temperature. This law was first verified in a satisfactory manner by Heinrich Schneebeli (_Wied. Ann._, 1884, 22, p. 30). He observed the radiation from the bulb of an air thermometer heated to known temperatures through a small aperture in the walls of the furnace. With this arrangement the radiation was very nearly that of a black body. Measurements by J. T. Bottomley, August Schleiermacher, L. C. H. F. Paschen and others of the radiation from electrically heated platinum, failed to give concordant results on account of differences in the quality of the radiation, the importance of which was not fully realized at first. Later researches by Paschen with improved methods verified the law, and greatly extended our knowledge of radiation in other directions. One of the most complete series of experiments on the relation between full radiation and temperature is that of O. R. Lummer and Ernst Pringsheim (_Ann. Phys._, 1897, 63, p. 395). They employed an aperture in the side of an enclosure at uniform temperature as the source of radiation, and compared the intensities at different temperatures by means of a bolometer. The fourth power law was well satisfied throughout the whole range of their experiments from -190 deg. C. to 2300 deg. C. According to this law, the rate of loss of heat by radiation R from a body of emissive power E and surface S at a temperature [theta] in an enclosure at [theta]0 is given by the formula

R = [sigma]ES([theta]^4 - [theta]0^4),

where [sigma] is the radiation constant. The absolute value of [sigma] was determined by F. Kurlbaum using an electric compensation method (_Wied. Ann._, 1898, 65, p. 746), in which the radiation received by a bolometer from a black body at a known temperature was measured by finding the electric current required to produce the same rise of temperature in the bolometer. K. Angstrom employed a similar method for solar radiation. Kurlbaum gives the value [sigma] = 5.32 X 10^(-5) ergs per sq. cm. per sec. C. Christiansen (_Wied. Ann._, 1883, 19, p. 267) had previously found a value about 5% smaller, by observing the rate of cooling of a copper plate of known thermal capacity, which is probably a less accurate method.

42. _Theoretical Proof of the Fourth Power Law._--The proof given by Boltzmann may be somewhat simplified if we observe that full radiation in an enclosure at constant temperature behaves exactly like a saturated vapour, and must therefore obey Carnot's or Clapeyron's equation given in section 17. The energy of radiation per unit volume, and the radiation-pressure at any temperature, are functions of the temperature only, like the pressure of a saturated vapour. If the volume of the enclosure is increased by any finite amount, the temperature remaining the same, radiation is given off from the walls so as to fill the space to the same pressure as before. The heat absorbed when the volume is increased corresponds with the latent heat of vaporization. In the case of radiation, as in the case of a vapour, the latent heat consists partly of internal energy of formation and

## partly of external work of expansion at constant pressure. Since in

the case of full or undirected radiation the pressure is one-third of the energy per unit volume, the external work for any expansion is one-third of the internal energy added. The latent heat absorbed is, therefore, four times the external work of expansion. Since the external work is the product of the pressure P and the increase of volume V, the latent heat per unit increase of volume is four times the pressure. But by Carnot's equation the latent heat of a saturated vapour per unit increase of volume is equal to the rate of increase of saturation-pressure per degree divided by Carnot's function or multiplied by the absolute temperature. Expressed in symbols we have,

[theta](dP/d[theta]) = L/V = 4P,

where (dP/d[theta]) represents the rate of increase of pressure. This equation shows that the percentage rate of increase of pressure is four times the percentage rate of increase of temperature, or that if the temperature is increased by 1%, the pressure is increased by 4%. This is equivalent to the statement that the pressure varies as the fourth power of the temperature, a result which is mathematically deduced by integrating the equation.

43. _Wien's Displacement Law._--Assuming that the fourth power law gives the quantity of full radiation at any temperature, it remains to determine how the quality of the radiation varies with the temperature, since as we have seen both quantity and quality are determinate. This question may be regarded as consisting of two parts. (1) How is the wave-length or frequency of any given kind of radiation changed when its temperature is altered? (2) What is the form of the curve expressing the distribution of energy between the various wave-lengths in the spectrum of full radiation, or what is the distribution of heat in the spectrum? The researches of Tyndall, Draper, Langley and other investigators had shown that while the energy of radiation of each frequency increased with rise of temperature, the maximum of intensity was shifted or displaced along the spectrum in the direction of shorter wave-lengths or higher frequencies. W. Wien (_Ann. Phys._, 1898, 58, p. 662), applying Doppler's principle to the adiabatic compression of radiation in a perfectly reflecting enclosure, deduced that the wave-length of each constituent of the radiation should be shortened in proportion to the rise of temperature produced by the compression, in such a manner that the product [lambda][theta] of wave-length and the absolute temperature should remain constant. According to this relation, which is known as Wien's Displacement Law, the frequency corresponding to the maximum ordinate of the energy curve of the normal spectrum of full radiation should vary directly (or the wave-length inversely) as the absolute temperature, a result previously obtained by H. F. Weber (1888). Paschen, and Lummer and Pringsheim verified this relation by observing with a bolometer the intensity at different points in the spectrum produced by a fluorite prism. The intensities were corrected and reduced to a wave-length scale with the aid of Paschen's results on the dispersion formula of fluorite (_Wied. Ann._, 1894, 53, p. 301). The curves in fig. 7 illustrate results obtained by Lummer and Pringsheim (_Ber. deut. phys. Ges._, 1899, 1, p. 34) at three different temperatures, namely 1377 deg., 1087 deg. and 836 deg. absolute, plotted on a wave-length base with a scale of microns ([mu]) or millionths of a metre. The wave-lengths Oa, Ob, Oc, corresponding to the maximum ordinates of each curve, vary inversely as the absolute temperatures given. The constant value of the product [lambda][theta] at the maximum point is found to be 2920. Thus for a temperature of 1000 deg. Abs. the maximum is at wave-length 2.92 [mu]; at 2000 deg. the maximum is at 1.46 [mu].

44. _Form of the Curve representing the Distribution of Energy in the Spectrum._--Assuming Wien's displacement law, it follows that the form of the curve representing the distribution of energy in the spectrum of full radiation should be the same for different temperatures with the maximum displaced in proportion to the absolute temperature, and with the total area increased in proportion to the fourth power of the absolute temperature. Observations taken with a bolometer along the length of a normal or wave-length spectrum, would give the form of the curve plotted on a wave-length base. The height of the ordinate at each point would represent the energy included between given limits of wave-length, depending on the width of the bolometer strip and the slit. Supposing that the bolometer strip had a width corresponding to .01 [mu], and were placed at 1.0 [mu] in the spectrum of radiation at 2000 deg. Abs., it would receive the energy corresponding to wave-lengths between 1.00 and 1.01 [mu]. At a temperature of 1000 deg. Abs. the corresponding part of the energy, by Wien's displacement law, would lie between the limits 2.00 and 2.02 [mu], and the total energy between these limits would be 16 times smaller. But the bolometer strip placed at 2.0 [mu] would now receive only half of the energy, or the energy in a band .01 [mu] wide, and the deflection would be 32 times less. Corresponding ordinates of the curves at different temperatures will therefore vary as the fifth power of the temperature, when the curves are plotted on a wave-length base. The maximum ordinates in the curves already given are found to vary as the fifth powers of the corresponding temperatures. The equation representing the distribution of energy on a wave-length base must be of the form

E = C[lambda]^(-5) F([lambda][theta]) = C[theta]^5 ([lambda][theta])^(-5) F([lambda][theta])

where F([lambda][theta]) represents some function of the product of the wave-length and temperature, which remains constant for corresponding wave-lengths when [theta] is changed. If the curves were plotted on a frequency base, owing to the change of scale, the maximum ordinates would vary as the cube of the temperature instead of the fifth power, but the form of the function F would remain unaltered. Reasoning on the analogy of the distribution of velocities among the particles of a gas on the kinetic theory, which is a very similar problem, Wien was led to assume that the function F should be of the form e^(-c/[lambda][theta]), where e is the base of Napierian logarithms, and c is a constant having the value 14,600 if the wave-length is measured in microns [mu]. This expression was found by Paschen to give a very good approximation to the form of the curve obtained experimentally for those portions of the visible and infra-red spectrum where observations could be most accurately made. The formula was tested in two ways: (1) by plotting the curves of distribution of energy in the spectrum for constant temperatures as illustrated in fig. 7; (2) by plotting the energy corresponding to a given wave-length as a function of the temperature. Both methods gave very good agreement with Wien's formula for values of the product [lambda][theta] not much exceeding 3000. A method of isolating rays of great wave-length by successive reflection was devised by H. Rubens and E. F. Nichols (_Wied. Ann._, 1897, 60, p. 418). They found that quartz and fluorite possessed the property of selective reflection for rays of wave-length 8.8 [mu] and 24 [mu] to 32 [mu] respectively, so that after four to six reflections these rays could be isolated from a source at any temperature in a state of considerable purity. The residual impurity at any stage could be estimated by interposing a thin plate of quartz or fluorite which completely reflected or absorbed the residual rays, but allowed the impurity to pass. H. Beckmann, under the direction of Rubens, investigated the variation with temperature of the residual rays reflected from fluorite employing sources from -80 deg. to 600 deg. C., and found the results could not be represented by Wien's formula unless the constant c were taken as 26,000 in place of 14,600. In their first series of observations extending to 6 [mu] O. R. Lummer and E. Pringsheim (_Deut. phys. Ges._, 1899, 1, p. 34) found systematic deviations indicating an increase in the value of the constant c for long waves and high temperatures. In a theoretical discussion of the subject, Lord Rayleigh (_Phil. Mag._, 1900, 49, p. 539) pointed out that Wien's law would lead to a limiting value C[lambda]^(-5), of the radiation corresponding to any particular wave-length when the temperature increased to infinity, whereas according to his view the radiation of great wave-length should ultimately increase in direct proportion to the temperature. Lummer and Pringsheim (_Deut. phys. Ges._, 1900, 2, p. 163) extended the range of their observations to 18 [mu] by employing a prism of sylvine in place of fluorite. They found deviations from Wien's formula increasing to nearly 50% at 18 [mu], where, however, the observations were very difficult on account of the smallness of the energy to be measured. Rubens and F. Kurlbaum (_Ann. Phys._, 1901, 4, p. 649) extended the residual reflection method to a temperature range from -190 deg. to 1500 deg. C., and employed the rays reflected from quartz 8.8 [mu], and rocksalt 51 [mu], in addition to those from fluorite. It appeared from these researches that the rays of great wave-length from a source at a high temperature tended to vary in the limit directly as the absolute temperature of the source, as suggested by Lord Rayleigh, and could not be represented by Wien's formula with any value of the constant c. The simplest type of formula satisfying the required conditions is that proposed by Max Planck (_Ann. Phys._, 1901, 4, p. 553) namely,

E = C[lambda]^(-5) (e^c/[lambda][theta] - 1)^(-1),

[Illustration: FIG. 7.--Distribution of energy in the spectrum of a black body.]

[Illustration: FIG. 8.--Distribution of energy in the spectrum of full radiation at 2000 deg. Abs. according to formulae of Planck & Wien.]

which agrees with Wien's formula when [theta] is small, where Wien's formula is known to be satisfactory, but approaches the limiting form E = C[lambda]^(-4)[theta]/c, when [theta] is large, thus satisfying the condition proposed by Lord Rayleigh. The theoretical interpretation of this formula remains to some extent a matter of future investigation, but it appears to satisfy experiment within the limits of observational error. In order to compare Planck's formula graphically with Wien's, the distribution curves corresponding to both formulae are plotted in fig. 8 for a temperature of 2000 deg. abs., taking the value of the constant c = 14,600 with a scale of wave-length in microns [mu]. The curves in fig. 9 illustrate the difference between the two formulae for the variation of the intensity of radiation corresponding to a fixed wave-length 30 [mu]. Assuming Wien's displacement law, the curves may be applied to find the energy for any other wave-length or temperature, by simply altering the wave-length scale in inverse ratio to the temperature, or vice versa. Thus to find the distribution curve for 1000 deg. abs., it is only necessary to multiply all the numbers in the wave-length scale of fig. 8 by 2; or to find the variation curve for wave-length 60 [mu], the numbers on the temperature scale of fig. 9 should be divided by 2. The ordinate scales must be increased in proportion to the fifth power of the temperature, or inversely as the fifth power of the wave-length respectively in figs. 8 and 9 if comparative results are required for different temperatures or wave-lengths. The results hitherto obtained for cases other than full radiation are not sufficiently simple and definite to admit of profitable discussion in the present article.

[Illustration: FIG. 9.--Variation of energy of radiation corresponding to wave-length 30 [mu], with temperature of source.]

BIBLIOGRAPHY.--It would not be possible, within the limits of an article like the present, to give tables of the specific thermal properties of different substances so far as they have been ascertained by experiment. To be of any use, such tables require to be extremely detailed, with very full references and explanations with regard to the value of the experimental evidence, and the limits within which the results may be relied on. The quantity of material available is so enormous and its value so varied, that the most elaborate tables still require reference to the original authorities. Much information will be found collected in Landolt and Bornstein's _Physical and Chemical Tables_ (Berlin, 1905). Shorter tables, such as Everett's _Units and Physical Constants_, are useful as illustrations of a system, but are not sufficiently complete for use in scientific investigations. Some of the larger works of reference, such as A. A. Winkelmann's _Handbuch der Physik_, contain fairly complete tables of specific properties, but these tables occupy so much space, and are so misleading if incomplete, that they are generally omitted in theoretical textbooks.

Among older textbooks on heat, Tyndall's _Heat_ may be recommended for its vivid popular interest, and Balfour Stewart's _Heat_ for early theories of radiation. Maxwell's _Theory of Heat_ and Tait's _Heat_ give a broad and philosophical survey of the subject. Among modern textbooks, Preston's _Theory of Heat_ and Poynting and Thomson's _Heat_ are the best known, and have been brought well up to date. Sections on heat are included in all the general textbooks of Physics, such as those of Deschanel (translated by Everett), Ganot (translated by Atkinson), Daniell, Watson, &c. Of the original investigations on the subject, the most important have already been cited. Others will be found in the collected papers of Joule, Kelvin and Maxwell. Treatises on special branches of the subject, such as Fourier's _Conduction of Heat_, are referred to in the separate articles in this encyclopaedia dealing with recent progress, of which the following is a list: CALORIMETRY, CONDENSATION OF GASES, CONDUCTION OF HEAT, DIFFUSION, ENERGETICS, FUSION, LIQUID GASES, RADIATION, RADIOMETER, SOLUTION, THERMODYNAMICS, THERMOELECTRICITY, THERMOMETRY, VAPORIZATION. For the practical aspects of heating see HEATING. (H. L. C.)

FOOTNOTES:

[1] _Units of Work, Energy and Power._--In English-speaking countries work is generally measured in _foot-pounds_. Elsewhere it is generally measured in _kilogrammetres_, or in terms of the work done in raising 1 kilogramme weight through the height of 1 metre. In the middle of the 19th century the terms "force" and "motive power" were commonly employed in the sense of "power of doing work." The term "energy" is now employed in this sense. A quantity of energy is measured by the work it is capable of performing. A body may possess energy in virtue of its state (gas or steam under pressure), or in virtue of its position (a raised weight), or in various other ways, when at rest. In these cases it is said to possess _potential energy_. It may also possess energy in virtue of its motion or rotation (as a fly-wheel or a cannon-ball). In this case it is said to possess _kinetic energy_, or energy of motion. In many cases the energy (as in the case of a vibrating body, like a pendulum) is

## partly kinetic and partly potential, and changes continually from one