Chapter VI: Part 6

35. _Heating by Condensation of Steam._--It is often stated that the rate at which steam will condense on a metal surface at a temperature below that corresponding to the saturation pressure of the steam is practically infinite (e.g. Osborne Reynolds, _Proc. Roy. Soc. Ed._, 1873, p. 275), and conversely that the rate at which water will abstract heat from a metal surface by the formation of steam (if the metal is above the temperature of saturation of the steam) is limited only by the rate at which the metal can supply heat by conduction to its surface layer. The rate at which heat can be supplied by condensation of steam appears to be much greater than that at which heat can be supplied by a flame under ordinary conditions, but there is no reason to suppose that it is infinite, or that any discontinuity exists. Experiments by H. L. Callendar and J. T. Nicolson by three independent methods (_Proc. Inst. Civ. Eng._, 1898, 131, p. 147; _Brit. Assoc. Rep._ p. 418) appear to show that the rate of abstraction of heat by evaporation, or that of communication of heat by condensation, depends chiefly on the difference of temperature between the metal surface and the saturated steam, and is nearly proportional to the temperature difference (not to the pressure difference, as suggested by Reynolds) for such ranges of pressure as are common in practice. The rate of heat transmission they observed was equivalent to about 8 calories per sq. cm. per sec., for a difference of 20 deg. C. between the temperature of the metal surface and the saturation temperature of the steam. This would correspond to a condensation of 530 kilogrammes of steam at 100 deg. C. per sq. metre per hour, or 109 lb. per sq. ft. per hour for the same difference of temperature, values which are many times greater than those actually obtained in ordinary surface condensers. The reason for this is that there is generally some air mixed with the steam in a surface condenser, which greatly retards the condensation. It is also difficult to keep the temperature of the metal as much as 20 deg. C. below the temperature of the steam unless a very free and copious circulation of cold water is available. For the same difference of temperature, steam can supply heat by condensation about a thousand times faster than hot air. This rate is not often approached in practice, but the facility of generation and transmission of steam, combined with its high latent heat and the accuracy of control and regulation of temperature afforded, render it one of the most convenient agents for the distribution of large quantities of heat in all kinds of manufacturing processes.

36. _Spheroidal State._--An interesting contrast to the extreme rapidity with which heat is abstracted by the evaporation of a liquid in contact with a metal plate, is the so-called spheroidal state. A small drop of liquid thrown on a red-hot metal plate assumes a spheroidal form, and continues swimming about for some time, while it slowly evaporates at a temperature somewhat below its boiling-point. The explanation is simply that the liquid itself cannot come in actual contact with the metal plate (especially if the latter is above the critical temperature), but is separated from it by a badly conducting film of vapour, through which, as we have seen, the heat is comparatively slowly transmitted even if the difference of temperature is several hundred degrees. If the metal plate is allowed to cool gradually, the drop remains suspended on its cushion of vapour, until, in the case of water, a temperature of about 200 deg. C. is reached, at which the liquid comes in contact with the plate and boils explosively, reducing the temperature of the plate, if thin, almost instantaneously to 100 deg. C. The temperature of the metal is readily observed by a thermo-electric method, employing a platinum dish with a platinum-rhodium wire soldered with gold to its under side. The absence of contact between the liquid and the dish in the spheroidal state may also be shown by connecting one terminal of a galvanometer to the drop and the other through a battery to the dish, and observing that no current passes until the drop boils.

37. _Early Theories of Radiation._--It was at one time supposed that there were three distinct kinds of radiation--thermal, luminous and actinic, combined in the radiation from a luminous source such as the sun or a flame. The first gave rise to heat, the second to light and the third to chemical action. The three kinds were partially separated by a prism, the actinic rays being generally more refracted, and the thermal rays less refracted than the luminous. This conception arose very naturally from the observation that the feebly luminous blue and violet rays produced the greatest photographic effects, which also showed the existence of dark rays beyond the violet, whereas the brilliant yellow and red were practically without action on the photographic plate. A thermometer placed in the blue or violet showed no appreciable rise of temperature, and even in the yellow the effect was hardly discernible. The effect increased rapidly as the light faded towards the extreme red, and reached a maximum beyond the extreme limits of the spectrum (Herschel), showing that the greater part of the thermal radiation was altogether non-luminous. It is now a commonplace that chemical action, colour sensation and heat are merely different effects of one and the same kind of radiation, the particular effect produced in each case depending on the frequency and intensity of the vibration, and on the nature of the substance on which it falls. When radiation is completely absorbed by a black substance, it is converted into heat, the quantity of heat produced being equivalent to the total energy of the radiation absorbed, irrespective of the colour or frequency of the different rays. The actinic or chemical effects, on the other hand, depend essentially on some relation between the period of the vibration and the properties of the substance acted on. The rays producing such effects are generally those which are most strongly absorbed. The spectrum of chlorophyll, the green colouring matter of plants, shows two very strong absorption bands in the red. The red rays of corresponding period are found to be the most active in promoting the growth of the plant. The chemically active rays are not necessarily the shortest. Even photographic plates may be made to respond to the red rays by staining them with pinachrome or some other suitable dye.

The action of light rays on the retina is closely analogous to the action on a photographic plate. The retina, like the plate, is sensitive only to rays within certain restricted limits of frequency. The limits of sensitiveness of each colour sensation are not exactly defined, but vary slightly from one individual to another, especially in cases of partial colour-blindness, and are modified by conditions of fatigue. We are not here concerned with these important physiological and chemical effects of radiation, but rather with the question of the conversion of energy of radiation into heat, and with the laws of emission and absorption of radiation in relation to temperature. We may here also assume the identity of visible and invisible radiations from a heated body in all their physical properties. It has been abundantly proved that the invisible rays, like the visible, (1) are propagated in straight lines in homogeneous media; (2) are reflected and diffused from the surface of bodies according to the same law; (3) travel with the same velocity in free space, but with slightly different velocities in denser media, being subject to the same law of refraction; (4) exhibit all the phenomena of diffraction and interference which are characteristic of wave-motion in general; (5) are capable of polarization and double refraction; (6) exhibit similar effects of selective absorption. These properties are more easily demonstrated in the case of visible rays on account of the great sensitiveness of the eye. But with the aid of the thermopile or other sensitive radiometer, they may be shown to belong equally to all the radiations from a heated body, even such as are thirty to fifty times slower in frequency than the longest visible rays. The same physical properties have also been shown to belong to electromagnetic waves excited by an electric discharge, whatever the frequency, thus including all kinds of aetherial radiation in the same category as light.

38. _Theory of Exchanges._--The apparent concentration of cold by a concave mirror, observed by G. B. Porta and rediscovered by M. A. Pictet, led to the enunciation of the theory of exchanges by Pierre Prevost in 1791. Prevost's leading idea was that all bodies, whether cold or hot, are constantly radiating heat. Heat equilibrium, he says, consists in an equality of exchange. When equilibrium is interfered with, it is re-established by inequalities of exchange. If into a locality at uniform temperature a refracting or reflecting body is introduced, it has no effect in the way of changing the temperature at any point of that locality. A reflecting body, heated or cooled in the interior of such an enclosure, will acquire the surrounding temperature more slowly than would a non-reflector, and will less affect another body placed at a little distance, but will not affect the final equality of temperature. Apparent radiation of cold, as from a block of ice to a thermometer placed near it, is due to the fact that the thermometer being at a higher temperature sends more heat to the ice than it received back from it. Although Prevost does not make the statement in so many words, it is clear that he regards the radiation from a body as depending only on its own nature and temperature, and as independent of the nature and presence of any adjacent body. Heat equilibrium in an enclosure of constant temperature such as is here postulated by Prevost, has often been regarded as a consequence of Carnot's principle. Since difference of temperature is required for transforming heat into work, no work could be obtained from heat in such a system, and no spontaneous changes of temperature can take place, as any such changes might be utilized for the production of work. This line of reasoning does not appear quite satisfactory, because it is tacitly assumed, in the reasoning by which Carnot's principle was established, as a result of universal experience, that a number of bodies within the same impervious enclosure, which contains no source of heat, will ultimately acquire the same temperature, and that difference of temperature is required to produce flow of heat. Thus although we may regard the equilibrium in such an enclosure as being due to equal exchanges of heat in all directions, the equal and opposite streams of radiation annul and neutralize each other in such a way that no actual transfer of energy in any direction takes place. The state of the medium is everywhere the same in such an enclosure, but its energy of agitation per unit volume is a function of the temperature, and is such that it would not be in equilibrium with any body at a different temperature.

39. _"Full" and Selective Radiation. Correspondence of Emission and Absorption._--The most obvious difficulties in the way of this theory arise from the fact that nearly all radiation is more or less selective in character, as regards the quality and frequency of the rays emitted and absorbed. It was shown by J. Leslie, M. Melloni and other experimentalists that many substances such as glass and water, which are very transparent to visible rays, are extremely opaque to much of the invisible radiation of lower frequency; and that polished metals, which are perfect reflectors, are very feeble radiators as compared with dull or black bodies at the same temperature. If two bodies emit rays of different periods in different proportions, it is not at first sight easy to see how their radiations can balance each other at the same temperature. The key to all such difficulties lies in the fundamental conception, so strongly insisted on by Balfour Stewart, of the absolute uniformity (qualitative as well as quantitative) of the full or complete radiation stream inside an impervious enclosure of uniform temperature. It follows from this conception that the proportion of the full radiation stream absorbed by any body in such an enclosure must be exactly compensated in quality as well as quantity by the proportion emitted, or that the emissive and absorptive powers of any body at a given temperature must be precisely equal. A good reflector, like a polished metal, must also be a feeble radiator and absorber. Of the incident radiation it absorbs a small fraction and reflects the remainder, which together with the radiation emitted (being precisely equal to that absorbed) makes up the full radiation stream. A partly transparent material, like glass, absorbs part of the full radiation and transmits part. But it emits rays precisely equal in quality and intensity to those which it absorbs, which together with the transmitted portion make up the full stream. The ideal black body or perfect radiator is a body which absorbs all the radiation incident on it. The rays emitted from such a body at any temperature must be equal to the full radiation stream in an isothermal enclosure at the same temperature. Lampblack, which may absorb between 98 to 99% of the incident radiation, is generally taken as the type of a black body. But a closer approximation to full radiation may be obtained by employing a hollow vessel the internal walls of which are blackened and maintained at a uniform temperature by a steam jacket or other suitable means. If a relatively small hole is made in the side of such a vessel, the radiation proceeding through the aperture will be the full radiation corresponding to the temperature. Such a vessel is also a perfect absorber. Of radiation entering through the aperture an infinitesimal fraction only could possibly emerge by successive reflection even if the sides were of polished metal internally. A thin platinum tube heated by an electric current appears feebly luminous as compared with a blackened tube at the same temperature. But if a small hole is made in the side of the polished tube, the light proceeding through the hole appears brighter than the blackened tube, as though the inside of the tube were much hotter than the outside, which is not the case to any appreciable extent if the tube is thin. The radiation proceeding through the hole is nearly that of a perfectly black body if the hole is small. If there were no hole the internal stream of radiation would be exactly that of a black body at the same temperature however perfect the reflecting power, or however low the emissive power of the walls, because the defect in emissive power would be exactly compensated by the internal reflection.

Balfour Stewart gave a number of striking illustrations of the qualitative identity of emission and absorption of a substance. Pieces of coloured glass placed in a fire appear to lose their colour when at the same temperature as the coals behind them, because they compensate exactly for their selective absorption by radiating chiefly those colours which they absorb. Rocksalt is remarkably transparent to thermal radiation of nearly all kinds, but it is extremely opaque to radiation from a heated plate of rocksalt, because it emits when heated precisely those rays which it absorbs. A plate of tourmaline cut parallel to the axis absorbs almost completely light polarized in a plane parallel to the axis, but transmits freely light polarized in a perpendicular plane. When heated its radiation is polarized in the same plane as the radiation which it absorbs. In the case of incandescent vapours, the exact correspondence of emission and absorption as regards wave-length of frequency of the light emitted and absorbed forms the foundation of the science of spectrum analysis. Fraunhofer had noticed the coincidence of a pair of bright yellow lines seen in the spectrum of a candle flame with the dark D lines in the solar spectrum, a coincidence which was afterwards more exactly verified by W. A. Miller. Foucault found that the flame of the electric arc showed the same lines bright in its spectrum, and proved that they appeared as dark lines in the otherwise continuous spectrum when the light from the carbon poles was transmitted through the arc. Stokes gave a dynamical explanation of the phenomenon and illustrated it by the analogous case of resonance in sound. Kirchhoff completed the explanation (_Phil. Mag._, 1860) of the dark lines in the solar spectrum by showing that the reversal of the spectral lines depended on the fact that the body of the sun giving the continuous spectrum was at a higher temperature than the absorbing layer of gases surrounding it. Whatever be the nature of the selective radiation from a body, the radiation of light of any particular wave-length cannot be greater than a certain fraction E of the radiation R of the same wave-length from a black body at the same temperature. The fraction E measures the emissive power of the body for that particular wave-length, and cannot be greater than unity. The same fraction, by the principle of equality of emissive and absorptive powers, will measure the proportion absorbed of incident radiation R'. If the black body emitting the radiation R' is at the same temperature as the absorbing layer, R = R', the emission balances the absorption, and the line will appear neither bright nor dark. If the source and the absorbing layer are at different temperatures, the radiation absorbed will be ER', and that transmitted will be R' - ER'. To this must be added the radiation emitted by the absorbing layer, namely ER, giving R' - E(R' - R). The lines will appear darker than the background R' if R' is greater than R, but bright if the reverse is the case. The D lines are dark in the sun because the photosphere is much hotter than the reversing layer. They appear bright in the candle-flame because the outside mantle of the flame, in which the sodium burns and combustion is complete, is hotter than the inner reducing flame containing the incandescent particles of carbon which give rise to the continuous spectrum. This qualitative identity of emission and absorption as regards wave-length can be most exactly and easily verified for luminous rays, and we are justified in assuming that the relation holds with the same exactitude for non-luminous rays, although in many cases the experimental proof is less complete and exact.

40. _Diathermancy._--A great array of data with regard to the transmissive power or diathermancy of transparent substances for the heat radiated from various sources at different temperatures were collected by Melloni, Tyndall, Magnus and other experimentalists. The measurements were chiefly of a qualitative character, and were made by interposing between the source and a thermopile a layer or plate of the substance to be examined. This method lacked quantitative precision, but led to a number of striking and interesting results, which are admirably set forth in Tyndall's _Heat_. It also gave rise to many curious discrepancies, some of which were recognized as being due to selective absorption, while others are probably to be explained by imperfections in the methods of experiment adopted. The general result of such researches was to show that substances, like water, alum and glass, which are practically opaque to radiation from a source at low temperature, such as a vessel filled with boiling water, transmit an increasing percentage of the radiation when the temperature of the source is increased. This is what would be expected, as these substances are very transparent to visible rays. That the proportion transmitted is not merely a question of the temperature of the source, but also of the quality of the radiation, was shown by a number of experiments. For instance, K. H. Knoblauch (_Pogg. Ann._, 1847) found that a plate of glass interposed between a spirit lamp and a thermopile intercepts a larger proportion of the radiation from the flame itself than of the radiation from a platinum spiral heated in the flame, although the spiral is undoubtedly at a lower temperature than the flame. The explanation is that the spiral is a fairly good radiator of the visible rays to which the glass is transparent, but a bad radiator of the invisible rays absorbed by the glass which constitute the greater portion of the heat-radiation from the feebly luminous flame.

Assuming that the radiation from the source under investigation is qualitatively determinate, like that of a black body at a given temperature, the proportion transmitted by plates of various substances may easily be measured and tabulated for given plates and sources. But owing to the highly selective character of the radiation and absorption, it is impossible to give any general relation between the thickness of the absorbing plate or layer and the proportion of the total energy absorbed. For these reasons the relative diathermancies of different materials do not admit of any simple numerical statement as physical constants, though many of the qualitative results obtained are very striking. Among the most interesting experiments were those of Tyndall, on the absorptive powers of gases and vapours, which led to a good deal of controversy at the time, owing to the difficulty of the experiments, and the contradictory results obtained by other observers. The arrangement employed by Tyndall for these measurements is shown in Fig. 6. A brass tube AB, polished inside, and closed with plates of highly diathermanous rocksalt at either end, was fitted with stopcocks C and D for exhausting and admitting air or other gases or vapours. The source of heat S was usually a plate of copper heated by a Bunsen burner, or a Leslie cube containing boiling water as shown at E. To obtain greater sensitiveness for differential measurements, the radiation through the tube AB incident on one face of the pile P was balanced against the radiation from a Leslie cube on the other face of the pile by means of an adjustable screen H. The radiation on the two faces of the pile being thus balanced with the tube exhausted, Tyndall found that the admission of dry air into the tube produced practically no absorption of the radiation, whereas compound gases such as carbonic acid, ethylene or ammonia absorbed 20 to 90%, and a trace of aqueous vapour in the air increased its absorption 50 to 100 times. H. G. Magnus, on the other hand, employing a thermopile and a source of heat, both of which were enclosed in the same exhausted receiver, in order to avoid interposing any rocksalt or other plates between the source and the pile, found an absorption of 11% on admitting dry air, but could not detect any difference whether the air were dry or moist. Tyndall suggested that the apparent absorption observed by Magnus may have been due to the cooling of his radiating surface by convection, which is a very probable source of error in this method of experiment. Magnus considered that the remarkable effect of aqueous vapour observed by Tyndall might have been caused by condensation on the polished internal walls of his experimental tube, or on the rocksalt plates at either end.[7] The question of the relative diathermancy of air and aqueous vapour for radiation from the sun to the earth and from the earth into space is one of great interest and importance in meteorology. Assuming with Magnus that at least 10% of the heat from a source at 100 deg. C. is absorbed in passing through a single foot of air, a very moderate thickness of atmosphere should suffice to absorb practically all the heat radiated from the earth into space. This could not be reconciled with well-known facts in regard to terrestrial radiation, and it was generally recognized that the result found by Magnus must be erroneous. Tyndall's experiment on the great diathermancy of dry air agreed much better with meteorological phenomena, but he appears to have exaggerated the effect of aqueous vapour. He concluded from his experiments that the water vapour present in the air absorbs at least 10% of the heat radiated from the earth within 10 ft. of its surface, and that the absorptive power of the vapour is about 17,000 times that of air at the same pressure. If the absorption of aqueous vapour were really of this order of magnitude, it would exert a far greater effect in modifying climate than is actually observed to be the case. Radiation is observed to take place freely through the atmosphere at times when the proportion of aqueous vapour is such as would practically stop all radiation if Tyndall's results were correct. The very careful experiments of E. Lecher and J. Pernter (_Phil. Mag._, Jan. 1881) confirmed Tyndall's observations on the absorptive powers of gases and vapours satisfactorily in nearly all cases with the single exception of aqueous vapour. They found that there was no appreciable absorption of heat from a source at 100 deg. C. in passing through 1 ft. of air (whether dry or moist), but that CO and CO2 at atmospheric pressure absorbed about 8%, and ethylene (olefiant gas) about 50% in the same distance; the vapours of alcohol and ether showed absorptive powers of the same order as that of ethylene. They confirmed Tyndall's important result that the absorption does not diminish in proportion to the pressure, being much greater in proportion for smaller pressures in consequence of the selective character of the effect. They also supported his conclusion that absorptive power increases with the complexity of the molecule. But they could not detect any absorption by water vapour at a pressure of 7 mm., though alcohol at the same pressure absorbed 3% and acetic acid 10%. Later researches, especially those of S. P. Langley with the spectro-bolometer on the infra-red spectrum of sunlight, demonstrated the existence of marked absorption bands, some of which are due to water vapour. From the character of these bands and the manner in which they vary with the state of the air and the thickness traversed, it may be inferred that absorption by water vapour plays an important part in meteorology, but that it is too small to be readily detected by laboratory experiments in a 4 ft. tube, without the aid of spectrum analysis.

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),

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.

, 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
to the other throughout the motion. For instance, the energy of a
pendulum is wholly potential when it is momentarily at rest at the
top of its swing, but is wholly kinetic when the pendulum is moving
with its maximum velocity at the lowest point of its swing. The whole
energy at any moment is the sum of the potential and kinetic energy,
and this sum remains constant so long as the amplitude of the
vibration remains the same. The potential energy of a weight W lb.
raised to a height h ft. above the earth, is Wh foot-pounds. If
allowed to fall freely, without doing work, its kinetic energy on
reaching the earth would be Wh foot-pounds, and its velocity of
motion would be such that if projected upwards with the same velocity
it would rise to the height h from which it fell. We have here a
simple and familiar case of the conversion of one kind of energy into
a different kind. But the two kinds of energy are mechanically
equivalent, and they can both be measured in terms of the same units.
The units already considered, namely foot-pounds or kilogrammetres,
are gravitational units, depending on the force of gravity. This is
the most obvious and natural method of measuring the potential energy
of a raised weight, but it has the disadvantage of varying with the
force of gravity at different places. The natural measure of the
kinetic energy of a moving body is the product of its mass by half
the square of its velocity, which gives a measure in kinetic or
absolute units independent of the force of gravity. Kinetic and
gravitational units are merely different ways of measuring the same
thing. Just as foot-pounds may be reduced to kilogrammetres by
dividing by the number of foot-pounds in one kilogrammetre, so
kinetic may be reduced to gravitational units by dividing by the
kinetic measure of the intensity of gravity, namely, the work in
kinetic units done by the weight of unit mass acting through unit
distance. For scientific purposes, it is necessary to take account of
the variation of gravity. The scientific unit of energy is called the
_erg_. The erg is the kinetic energy of a mass of 2 gm. moving with a
velocity of 1 cm. per sec. The work in ergs done by a force acting
through a distance of 1 cm. is the absolute measure of the force. A
force equal to the weight of 1 gm. (in England) acting through a
distance of 1 cm. does 981 ergs of work. A force equal to the weight
of 1000 gm. (1 kilogramme) acting through a distance of 1 metre (100
cm.) does 98.1 million ergs of work. As the erg is a very small unit,
for many purposes, a unit equal to 10 million ergs, called a _joule_,
is employed. In England, where the weight of 1 gm. is 981 ergs per
cm., a foot-pound is equal to 1.356 joules, and a kilogrammetre is
equal to 9.81 joules.

The term _power_ is now generally restricted to mean "rate of
working." Watt estimated that an average horse was capable of raising
550 lb. 1 ft. in each second, or doing work at the rate of 550
foot-pounds per second, or 33,000 foot-pounds per minute. This
conventional horse-power is the unit commonly employed for estimating
the power of engines. The _horse-power-hour_, or the work done by one
horse-power in one hour, is nearly 2 million foot-pounds. For
electrical and scientific purposes the unit of power employed is
called the _watt_. The watt is the work per second done by an
electromotive force of 1 volt in driving a current of 1 ampere, and
is equal to 10 million ergs or 1 joule per second. One horse-power is
746 watts or nearly 3/4 of a kilowatt. The _kilowatt-hour_, which is
the unit by which electrical energy is sold, is 3.6 million joules or
2.65 million foot-pounds, or 366,000 kilogrammetres, and is capable
of raising nearly 19 lb. of water from the freezing to the boiling
point.

[2] In an essay on "Heat, Light, and Combinations of Light,"
republished in Sir H. Davy's _Collected Works_, ii. (London, 1836).

[3] For instance a mass of compressed air, if allowed to expand in a
cylinder at the ordinary temperature, will do work, and will at the
same time absorb a quantity of heat which, as we now know, is the
thermal equivalent of the work done. But this work cannot be said to
have been produced solely from the heat absorbed in the process,
because the air at the end of the process is in a changed condition,
and could not be restored to its original state at the same
temperature without having work done upon it precisely equal to that
obtained by its expansion. The process could not be repeated
indefinitely without a continual supply of compressed air. The source
of the work in this case is work previously done in compressing the
air, and no part of the work is really generated at the expense of
heat alone, unless the compression is effected at a lower temperature
than the expansion.

[4] Clausius (_Pogg. Ann._ 79, p. 369) and others have misinterpreted
this assumption, and have taken it to mean that the quantity of heat
required to produce any given change of state is independent of the
manner in which the change is effected, which Carnot does not here
assume.

[5] Carnot's description of his cycle and statement of his principle
have been given as nearly as possible in his own words, because some
injustice has been done him by erroneous descriptions and statements.

[6] It was for this reason that Professor W. Thomson (Lord Kelvin)
stated (_Phil. Mag._, 1852, 4) that "Carnot's original demonstration
utterly fails," and that he introduced the "corrections" attributed
to James Thomson and Clerk Maxwell respectively. In reality Carnot's
original demonstration requires no correction.