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354 having a wave-length of the order of 15 microns would be compe- tent to produce the observed effect in the case of COz, contributing, when fully excited, a term R to the specific heat. An attempt was accordingly made to investigate the relation between the variation of the specific heat of gases and the absorption and emission bands in their infra-red spectra. Some qualitative agreement was found, but it was very difficult to make quantitative measurements of the kind required, or to frame a consistent theory. For instance, there is a strong band at 4-4-4-5 microns both in the emission and absorption spectra of steam. This band corresponds to the maxi- mum ordinate of the wave-length spectrum of full radiation at a temperature T = 647C., which is the critical point of water, and appears to be closely related to other properties of steam, such as the latent heat and the cooling-effect, and the variation of the spe- cific heat with pressure. There is no doubt that the properties of any substance must be intimately related to the natural frequen- cies of the molecules, but the form of the relation cannot be pre- dicted with certainty; the experiments were interrupted by the war, and the quantitative measurements, up to 1921, had not been sufficiently exact to distinguish between many possible hypotheses.

The experiments of A. Eucken (Sitz. Akad., Berlin, 33.1, p. 141, 1912) on the specific heat of hydrogen at low temperatures were very instructive in this connexion. The gas was electrically heated at various temperatures in a thin steel vessel under considerable pressure at constant volume. The specific heat was found to dimin- ish from nearly 5R/2 at ordinary temperatures to nearly 3R/2 at T = 6o, after which it remained practically constant down to T = 35. The fall could be approximately represented by a formula of the Einstein type, R/(z), as explained in the next section, with a value /3c = 43O, corresponding to a wave-length of about 32 microns. The experiments were undoubtedly of considerable diffi- culty, owing to the large thermal capacity of the steel vessel, and its rapid diminution with fall of temperature, but there seems no reason to doubt the substantial accuracy of the result. In the kinetic theory of the specific heat of a diatomic gas, the term 3R/2 is attributed to the three degrees of freedom of translation, and the term R to the two of rotation, the energy of which must be exactly proportional to that of translation, if the effect is produced by colli- sions. The late Lord Rayleigh was never satisfied with this explana- tion, which evidently must be revised if Eucken is correct.

Specific Heat of Solids at Low Temperatures. The early experi- ments of Sir J. Dewar, Sir W. A. Tilden, and others, had shown that solids at low temperatures deviated from Dulong and Petit's law of the constancy of atomic heat in the same way as carbon, boron, and silicon, at ordinary temperatures, but they failed to show the full extent of the deviation, or to indicate a probable explanation. A great impetus to research in this direction was given by the sug- gestion of A. Einstein (Ann. Phys., 22, p. 180, 1907) that the atom of a solid might be regarded as an electric resonator with three degrees of freedom possessing a particular frequency, independent of the temperature, and capable of responding to the same frequency of radiation. Adopting Planck's theory and radiation formula, he showed that the specific heat at constant volume should approach the limit 3R = 5'94 calories per gram-atom at high temperatures, as required by Dulong and Petit's law, but that the variation at low temperatures should be given by the expression

... (2)

where z=/Sv/T=/SA/T, as in Planck's formula. The symbol v de- notes the natural frequency of the atoms, and X the correspond- ing wave-length in cm. such that yX = A = 3Xio 10, the velocity of light. The constant, /3A = 1-460, is Wien's constant of radiation. Taking H. F. Weber's observations on the variation of the specific heat of the diamond, extending from T = 222 to 1258, Einstein showed that they agreed qualitatively with this formula, if we could assume the diamond atoms to possess a single frequency cor- responding to the wave-length II microns. Taking the substances, CaFl, NaCl, KC1, CaCO 3, and SiO 2 , for which the optical fre- quencies in the infra-red were known, he showed that the frequen- cies agreed in order of magnitude with those required by his formula, but that the observed wave-lengths were somewhat shorter than those calculated from the specific heats. This could be attributed to the fact that most of the substances showed more than one fre- quency, and that the frequencies were not strictly monochromatic, as indicated by the width of the corresponding absorption bands. In any case there were other effects, such as work of expansion, included in the specific heats as ordinarily measured, and it might be doubted whether the optical frequencies corresponded exactly with the thermal vibrations of the atoms.

An important series of experimental measurements, extending down to the temperature of liquid hydrogen, was made by W. Nernst, F. A. Lindemann, and their collaborators (Site. Akad., Berlin, p. 494, 1911), on a number of metals and other solids, including those for which the optical frequencies were known. They found, as already indicated, that Einstein's formula gave too low values for the specific heats at low temperatures, if the optical frequencies were assumed in calculating the value of /(z), and that much better agreement could be obtained by taking the mean of /(z) for the ( optical frequency, and a similar term, f(z/2) at half the optical frequency-'


 * = 3R[/(3)+/(z/2)]/2= 3 R/"(z) ... (3)

The same function, /"(z), of z was assumed to apply to other sub- stances, such as the metals, but the appropriate values of z were selected to fit the observations on the specific heats. Some sub- stances, such as SiO 2 (in the forms of quartz and quartz-glass) and benzine, CeHs, which gave a different type of curve, were represented by formulae with two or three different values of z, each value of / "(z) being multiplied by a fractional coefficient representing the proportion in which the corresponding molecule was supposed to be present. But such cases could not be regarded as a verification of the theory, because it would obviously be possible to represent almost any type of variation in this way. Einstein objected that even the simplest of these formulae, namely (3), was too empirical to be satisfactory from a theoretical standpoint; that a cubical crystal, such as KC1, or NaCl, could not have two different frequencies; and that there was no evidence in either case of an optical frequency with half the experimental value, since, according to Rubens, the crystals became again transparent before this frequency was reached, and had a value of the refractive index which was nearly normal. He also indicated two other objections to the " quantum " theory on which Planck's formula was based.

(1) According to the quantum theory it did not follow, as required by the classical mechanics, that the oscillator with three degrees of freedom would have three times the energy of a linear oscillator.

(2) It was very difficult to conceive the distribution of energy among the oscillators at low temperatures required by the theory. Thus for the diamond at T = 73 only one molecule in 100 millions would possess a single quantum of energy, all the rest would be absolutely quiescent. It was physically impossible to conceive such a distribu- tion of energy, which moreover would make the thermal conductiv- ity of the diamond at such temperatures entirely negligible, whereas, according to Eucken, it was nearly as great as that of copper at ordinary temperatures. For these reasons Einstein preferred to rely mainly on the expression for the energy of an electric oscillator in equilibrium with radiation as deduced from Maxwell's equa- tions, and to regard Planck's formula for the distribution of energy in full radiation simply as representing the results of experiment, without reference to the special hypothesis of quanta, which was subsequently invented to provide a theoretical explanation of the formula, but leads to serious difficulties in many directions.

Debye's Theory of Specific Heat of Solids. The theory now most commonly accepted is that of P. Debye (Ann. Phys., 39, p. 789, 1912), who attributes the heat energy to mechanical or acoustic vibrations of the solid with all possible frequencies up to a certain limit v m. According to a theorem attributed to the late Lord Ray- leigh (Sound, i., p. 129, 1877) the number of possible degrees of free- dom of a system of N discontinuous mass-points will be jN. Accord- ing to another theorem by the same author (Phil. Mag., 49, p. 539, 1900), the number of possible frequencies in a given volume of a continuous medium between the limits v and v+dv may be repre- sented by C'v^dv, where C' is a constant depending on the vol- ume and the velocity of propagation. The total number of possible frequencies from o up to a limit p m is CV OT 3 /3- If we equate this to 3N, we find C'=oN/v 3 ,,,. Adopting Planck's expression for the energy of an electric oscillator with one degree of freedom as apply- ing to each possible frequency of the N atoms in a gram-atom, we obtain the energy (RT/N)z/(e e l) for each frequency. Multi- plying this by the number of frequencies between v and v- namely (gN/v'n,)^!', and integrating from o to v m, we obtain the energy of a gram-atom at T, from which the specific heat at constant volume is obtained by differentiation with regard to T. Unfortunately the integral cannot be expressed in finite terms and is too complicated to reproduce here. It is evident, however, that it will be a function of 2, or 0? m /r, or T m /T, where T m =/3c m. Thus the form of the curve representing the variation of the specific heat (which depends on a single parameter T m or v m ) is the same for all substances on Debye's theory, if the temperature scale is altered for each in proportion to v m. This point has been very care- fully tested by E. H. Griffiths and E. Griffiths (Phil. Trans., A, 214, PP- 319-357) for the metals Al, Ag, Cd, Cu, Fe, Na, Pb, Zn. Their results indicate qualitative agreement with the theory, but show characteristic differences, greatly exceeding the limit of experi- mental error, which may possibly be attributed to other effects not included in the simple theory. Thus the curve for Fe differs from that for Cu by nearly 20% between corresponding temperatures, which may be attributed to the magnetic properties of Fe. The curve for Na shows a rapid rise towards the melting point, reach- ing an excess of 25 % above 3R, followed by a diminution of specific heat for the liquid, as in the case of water and mercury. Many simple compounds, such as NaCl, show curves of a very similar type to the metals, which has been used as an argument that the specific heat must be attributed entirely to the atoms, and that the free electrons supposed to exist in metals cannot make any appre- ciable contribution. Thus if there were two free electrons per atom, as required by some theories, the electrons alone would account for the whole specific heat according to the kinetic theory at ordi- nary temperatures; and it would be necessary to suppose that the number of free electrons diminished to zero at low temperatures, which would make it difficult to account for the enormous increase