x88 A SYSTEM OF PHYSICAL CHEMISTRY vibrations constitute waves of " sound " travelling through the body with definite speeds. There are, for instance, "distortional" transverse waves travelling with a velocity v^, and " compressional " longitudinal waves travelling with a velocity v«. v^ can be calculated if one knows the density p and modulus of rigidity fi of the material ; it is in fact z\ = //f ; V.2 can be calculated if one knows in addition the bulkmodulus K— IT a. 4/* / 3. V p Now, the number of degrees of freedom for vibrations having frequen- cies between v and v + 8v is, as previously, — jv'^Sv per unit volume for the transverse waves, and ^v^Sv for the longitudinal. (N.B.—The factor Stt occurs because the transverse waves are polarisable in two rectangular planes, the longitudinal are not.) Consequently the whole modes of vibration per unit volume between v and v + 8v are ( —^-^ :, )v^Bv. We saw above that the similar formula developed foi the ether by Rayleigh and Jeans suffered from the drawback that, in in- tegrating it, no finite upper limit could be assigned on account of the assumed continuity of the ethereal medium. In this case, however, there must be a finite upper limit to v, because the number of degrees of freedom cannot exceed 3N for a gram-atom of the substance. Hence if v,„ is the — highest frequency possible .^ /Sttt 47rT\ (where t is the volume of a gram-atom) v^dv o where F is a constant calculable from the elastic constants and density of the body. /3N\i Smce v,„ = ( -p- ) ) v,n IS also calculable. For Al, Cu, Zn, Ag, Pb, Diamond it is of the order 10^- to lo^^. Now by ascribing, according to the extension of Planck's work, the energy— hv ghvlkt _ J
APPENDIX II 1S9 to every degree of freedom, we obtain for the energy-content of body due to wave-motion of frequencies v to v -l- Zv— or Hence the total energy— 3F. 9N ,,3 ' 9N ^""^ qN/^T x^ g hv\ks _ J ' hv'^lv phv\ki f' /iv^dv hv exp exp^ - I jc f'dy where x = —- and v = — • A differentiation of this expression with regard to T produces this expression for the specific heat at constant volume— U^J.cf - I ^^ - ij This is clearly a function of .v or 7=^, and therefore suitably accounts for the similarity of the specific heat curves of different solids. Further, the expression agrees remarkably well with determinations of Nernst and his co-workers, more so in fact if v,„ is calculated not from the values of elastic constants, but from a formula discovered by Lindemann,^ connecting v,,, and the melting point (taking i/„, as the frequency of vibration of an atom under the influence of its neighbours). The computation is not easily carried out, for the integral in the above expression cannot be evaluated in finite terms, and so one has to resort to the summation of series which can be obtained to suit various ranges of temperature as regards their convergency. One striking result emerges from the calculation, viz. that at very low temperatures the this would specific heat varies as the cube of the absolute temperature ; appear to be true to within i per cent, for the range of temperature 0° ^ hvm Nernst has made a very happy combination of Einstein's and Debye's formulse in the treatment of diatomic solids such as KCl,. ^ Physikalische Zeitschrift, 11, 609 (1910).