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