ABSTRACT
The radiation entropy S can be expanded in terms of the spectral entropy density ,sω which is the entropy per unit volume in the frequency range between ω and ,dω ω+ as
= 0
V
where ( )ωωω ,uss = must depend on the spectral energy density ,uω defined in Eq.(27.8) which also reads
= 0
V
Since the conditions for equilibrium, 0, and 0S Uδ δ= = must hold simultaneously, their integrands should be proportional, that is,
or i.e., s ss C u u C u C u u ω ωω ω ω ω ω ω
δ δ δ δ∂ ∂= =∂ ∂ = (28.4) Upon differentiation of Eqs.(28.2) and (28.3) we obtain the relation
V VsdS du d C du d CdU u ω ω ω ω
ω ω ∞ ∞∂= =∂∫ ∫ = (28.5)
which might be compared with the definition (8.7) of the entropy change in an infinitesimal process. The interpretation of as reversible added heat gives dU
u s
T C ∂
∂== 1 (28.6) where we made use of Eq.(28.4). The temperature can be eliminated between Eq.(28.6) and the Wien formula (27.38 ) by writing
3 3 1 ln or lnB Bc u s c uk k T u
π π ω ωω ω
∂= − = −∂= == = so that, on integrating, we obtain the spectral entropy density as
3ln 1 Bk u c us ω ωω
π ω ω
⎛ ⎞= − −⎜⎜⎝ ⎠= = ⎟⎟ (28.7)
A similar relation should be true between ωω sS V= and ωω uU V= as
3ln 1V Bk U c US ω ωω
π ω ω
⎛ ⎞= − −⎜⎜⎝ ⎠= = ⎟⎟ (28.8)
so that the entropy change in the spectral range between ω and ωω d+ , associated with a reversible change of volume from V to , can be written as 0V
0V Vlnω
ωω ωω = dUk
dS B=∆ It has been emphasized by Einstein that this relation takes the form
0V VlnBnkS =∆
used to describe the n-particle gas behaviour, according to Eq.(28.1), provided that
ωωω ω
ω ω == ndUn
dU == or (28.9)
The similarity of this relation to Eq.(27.32), assumed in Planck's theory, has been considered to be an argument for the discrete nature of electromagnetic radiation. In other words, an energy
ωε == (28.10) can be associated with each radiation quantum, later called the photon. 28.2. THE OLD QUANTUM THEORY An analysis of the radiation spectrum emitted by hydrogen atoms shows intense lines, of definite frequencies ,ν which can be clustered into several series that fit the empirical formula
1,11 22 ≥>⎟⎠ ⎞⎜⎝
⎛ −= nm mn
Rν (28.11)
where m and n are integers, and Hz is the Rydberg constant. The significant fact is that the frequency of the observed spectral lines of each series can be represented by the difference between two quadratic terms, one of which, , is fixed and the other, , is a variable. Similar regularities have been found for the emission spectra of alkali metal atoms. Furthermore, in addition to the lines represented in the various series (28.11), there are other lines not included in this series, but it was observed that the frequencies of all spectral lines also correspond to the differences between two quadratic terms
151029.3 ×=R
nT
mT
mnmn TT −=ν (28.12)
and this is called the Ritz combination principle. These results are conflicting with the classical picture of an electron orbiting around a nucleus, which radiates electromagnetic waves, and hence, leads to an unstable atom: since the electron frequency of revolution would change smoothly, the emission spectrum should be expected to be continuous. A successful interpretation of the discrete spectrum of one-electron atoms has been proposed on the grounds of two assumptions known as Bohr's postulates: 1. Postulate of stationary states: an electron moves only in certain
permissible orbits which are stationary states, in the sense that no radiation is emitted. The condition for such states is that the orbital angular momentum of the electron equals an integer times / 2 .h π==
This postulate states that an electron in a stable orbit is exempt from the requirement that an accelerated charge must radiate energy, in spite of its macroscopic validity. However, the reality of stationary states has received direct experimental evidence.
