2 7 8 D O C . 2 8 3 Q U A N T U M T H E O R Y O F I D E A L G A S
(4)
extended over all r’s and s’s. If one extends the product over all values of s, from 1
to
∞,
then (4) clearly represents the total number of complexions, that is, the prob-
ability in Planck’s sense of a (macroscopic) state of the gas defined by . For the
entropy S of this state, Boltzmann’s principle yields the expression
. (5) [4]
§3. Thermodynamic Equilibrium
At thermodynamic equilibrium, S is a maximum, where apart from (3) the addi-
tional conditions must be satisfied that the total number n of the atoms as well as
their total energy E have given values. These conditions evidently express them-
selves in the two
equations1)
[5] (6)
, (7)
where means the energy of a molecule belonging to the
sth
phase cell. From (1a)
it easily follows that
. (8)
By executing the variation according to the ’s as variables, one finds that with a
suitable choice of constants
,
A, and B,
(9)
have to hold. Here, according pursuant to (3), must be
1)
For, is the number of molecules belonging on average to the
sth
cell.
1
sprs
rs
∏pr
---------------- -
pr
s
S
κlg¦(
pr s lg pr s)
sr
–=
[p. 263]
ns
N
p
---rps -
r
=
n s
sr
¦rpr
=
E
s
sr
¦Esrpr
=
Es
Es cs3 /2
=
c
2m)–1h2§
(
4
3
--πV·
-
© ¹
–2
3
----- -
=
pr s
βs
pr
s
βse–αsr
=
αs A Bs2 /3 +=