A system of physical chemistry - Index of

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A system of physical chemistry - Index of

V

DEB YES EQUA7J0N 71

mode of vibration will be complex. Such complex modes of vibration

can be treated, as Fourier showed, as theoretically made up of a series

of true simple harmonic motions, and we have thus to integrate over a

spectrum of frequencies if we wish to calculate with greater exactness

the total energy content of a vibrating atom. The first point to be

noted is, however, that we do not deal with a spectrum extending from

= o to V = 00 . If a body consists of N atoms— treated as massive

points — the system possesses 3N degrees of freedom. The system will

therefore in general exercise 3N different periodic vibrations, i.e. 3N

different vibration frequencies. If the older view— consonant with the

principle of ^^w/partition of energy — were true, namely, that the energy

was the same for each degree of freedom over the entire spectrum, then

we could state directly that each vibration frequency corresponded to the

energy kT (where ^ = ^, N being now the number of atoms in i gram-

atom), so that the total energy of the body would be simply 3NRT per

gram-atom as Dulong and Petit's Law requires.

We have seen, however, that the whole point of the quantum theory

is the negation of the principle of equipartition throughout a spectrum,

and that instead the energy per vibration varies with the type of vibra-

tion. This lack of equipartition is expressed as we have already seen in

hv

the Planck expression ^^.^^ , which gives the true mean energy of

any single vibration v. To obtain the total energy of the vibrating

system it is necessary to sum this expression over the spectrum (of

an infinite number

absorption or emission), the spectrum not containing

of frequencies but limited to 3N, as already pointed out. The spectrum

is characterised by two factors, {a) its boundaries, {b) the density of the

lines, i.e. the number of lines in any given vibration region dv. There

is, according to Debye, a certain definite limiting frequency vm beyond

which the spectrum does not extend. Debye has reached a number of

important conclusions in the course of his investigation. The first is

this : If the temperature T be regarded as a multiple [or submultiple) of a

then the

temperature 6 {a characteristic constant for any give Ji substance),

the same

atomic heat for all monatomic substances can be represented by

T

curve. I.e. the atomic heat is a universal function of -^.

u

A second relation which is not confined to monatomic substances

states that the number of lines spread over a region dv is proportional to

vVv (a relation for black-body radiation already obtained by Jeans).

From this Debye concludes that at sufficiently low temperatures the

atomic heat of all solids is proportional to T^, that is to the third power

of the absolute temperature. This conclusion differs (theoretically) widely

from the conclusion of the Einstein and the Nemst-Lindemann equations.

It is borne out by the experimental data, but experiment at low

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