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 2 7 9
. (10)
[6]
From this initially results for the mean number of molecules per cell
. (11)
The equations (6) and (7) therefore assume the form
(6a)
,, (7a)
which equations, together with
determine the constants A and B. Thus the law for the macroscopic distribution of
states for thermodynamic equilibrium is completely determined.
By entering the results of this section into (5), the expression for the equilibrium
entropy results:
. (12)
We now have to calculate the temperature of the system. For this purpose we apply
the definition equation for the entropy to an infinitely small isopycnic warming and
obtain
,[7]
which, taking into account (9), (6), and (7), yields
or
. (13)
Thus the temperature is also indirectly expressed by the energy and the rest of the
given quantities. From (12) and (13) it also follows that the free energy F of the sys-
tem is given by
βs
1
e–αs
–=
ns r ¦rps
βs¦re–α2r
r
dαs
d
¦e–αs
( ) –βs
dαs©
d 1
1
e–αs
–
------------------¹
§ ·
–βs
1
eαs
1 –
--------------- - = = = = =
n
1
eαs
1 –
-
s
¦---------------
=
E
c¦---------------
s2 /3
eαs
1 –
- =
αs
A
Bs2 /3
+=
[p. 264]
S κ® lg 1
e–αs
– ( )] An –
B
c
---E –
s
¦[
¯ ¿
¾
­ ½
–=
dE TdS
dαs
1
e–αs
–
bdA –
E
c
---dB –
Bd§
E
c¹
---·
©
–
s
¦------------------
¯ ¿
¾
­ ½
–κT® = =
dE κTBd©
E·
c
---¹
§
κT---dE
B
c
= =
1
κT
------
B
c
--- =