A system of physical chemistry - Index of

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A system of physical chemistry - Index of

THE QUANTUM THEORY 39

[In actual practice a heated body does not give out monochromatic

radiation, but a complete spectrum. We shall define %i^ in different

terms later, still though equivalent to the above.]

Planck has shown on the basis of the classical electro- magnetic

theory (which therefore introduces no unitary hypothesis) that u^ and

U are connected by the relation—

U = ^.^^. (i)

\Cf. Planck, Annalen der Fhysik, [4], 4, 560, 1901 ; Lindemann,

Brii. Ass. Rep., 191 2. If we take X = Soofx/x as a mean value for a

wave-length in the red part of the spectrum, it is easily calculated from

this formula that U = Su (approx.). Similarly for the violet end of the

spectrum, taking A. = ^oonfx, one finds U = 2U (approx).]

By combining the two expressions obtained above, the mean value

of the density of energy radiated from a single resonator in a system

consisting of a large number of similar resonators, all emitting monochromatic

light of frequency v {i.e. between the limits v and v + dv),

is given by the expression —

d,irhv^ I

This is one of the forms of Planck's Radiation Formula.

We wish now to change the shape of this expression a little, in order

to be clear about the relation of the term u^ and the term E^, which

latter we have already met with in Wien's radiation formula {c/. Chap.

XIV., Vol. II.).

The total energy (say in ergs) radiated per second from unit area of

a black body emitting a continuous spectrum covering the wave-lengths

o to 00 has been denoted by S. This is the term which appears in

Stefan's Law, viz. S = o-T*, the temperature of the source being T and

the radiation purely "temperature" radiation. Now S is the quantity

of energy which would be present in an imaginary cylinder, i square

centimetre base and length 3 x lo^*' cms. (since 3 x lo^'^ cms. is the

distance which the radiation will travel in one second). The cylinder

is supposed to be placed with its base on the radiating body and extending

out into space. The volume of this cylinder is 3 x loi** c.c, and

since this contains S ergs of energy the space density of the radiation,

i.e. the amount of energy per c.c. is

density of the energy by E, we have E =

^o ^^S- Denoting the space

3x10

S

^u orS = 3 x io"'E,

3 X 10

the term E referring, of course, to the entire range of wave-lengths

between o and 00 emitted by the body. This total density E may be

expressed thus—

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