CAPACITY FOR HEAT OF METALS AT LOW TEMPERATURES. 
343 
PART II. 
Comparison of the Experimental Results with those deduced from 
Formulae based on the “Quantum” Theory. 
(l) Theoretical Formulae for C„. 
In order to appreciate the limitations of the various theoretical formulse which 
have been given to represent the variation of atomic heat with temperature, it is 
necessary to review briefly the fundamental assumptions on which they are based. 
(a) Einstein’s Theory. —Einstein* was the first to show that if we regard the 
atoms of solid substances as charged with electricity and oscillating about fixed 
positions, we may consider them as equivalent to a number of Planck’s resonators. 
On this theory each atom is regarded as possessing three degrees of freedom, and 
consequently the total vibrational energy of an atom is three times that of a linear 
oscillator. From these considerations he deduces the expression 
C„ = 3R 
h 
e kT 
( e tT—1)2 
(7) 
for the atomic heat of any elementary substance, where. 
R is the gas constant, 1'9876, 
h is the quantum constant, 7'10x 10 -27 erg sec,t 
Jc, Boltzmann’s constant, 1’47 x 10 -16 erg, 
T, the absolute temperature, 
v, the monochromatic frequency of vibration—a constant characteristic of the 
substance. 
In the case of salts, the quantity v is the frequency found by means of the 
reststralilen , but, in the case of the metals, it cannot be obtained directly. Einstein 
showed that if the elastic forces which maintain the atoms in their equilibrium 
position are the same as those which oppose a diminution of volume when the solid is 
compressed, and if we assume that these forces are due to the mutual action of 
adjacent atoms spaced in the form of a lattice, then 
” = 2’54x 10'x .(8) 
where V is the atomic volume, 
m is the atomic weight, 
k is the coefficient of compressibility. 
* ‘Ann. d. Phys.,’ vol. 22, pp. 180, 800, 1907. 
t ‘ Ann. d. Phys.,’ vol. 38, p. 41, 1912. 
