48 
QUANTUM THEORY 
37 . We now introduce a third state with energy Xs (Xs > X 2 > Xi)- 
Then since 
n x 
n 2 
n. 
1 + - 
CL-t 9 \ ®c> 
;,T) 
/«23 V 
ni 
3 n 3 
^32 
0^32 1 ( V 23 J r J 1 ) / ® 
= ^ 1 + 
'l3 > ^)/ 
...(37-1), 
....(37-2). 
and by (36T) v 13 = v 12 + v 23 
This holds for all temperatures T, and T only occurs in (37-1) in the 
way explicitly indicated by the notation. We may perhaps fairly assume 
that, for a fixed value of v, I {v, T) increases without limit as T increases 
to infinity*, so that by taking a sufficiently high temperature the second 
term in each bracket is made as small as we please. Hence taking T 
infinite 
&21 ®32 ^31 
®12 ®23 ®13 
Substituting this in (37-1) we have 
.(37-3). 
1 + 
U 21 
(H2> T) 
1 + 
®32-^ ( V 23 J -^ 7 ) 
1 + 
uT) 
)■ 
Introducing Wien’s Law (32T) this becomes 
( 14 . - A 2 — 
V + f<nJT) 
where 
and c 12 is independent of T. 
This may be written 
1 + 
/K/n 
— ^2l/ a 2 
.(37-4), 
• (37-5), 
1 
1 
+ 
1 
1 
/ C 23 / <y»!T) c 12 f (v 12 /T + v n /T) c 12 c 2 
f ( V n/T)f {v^T) 
(37-6). 
Write this equation three times over, taking T equal to three temperatures 
T x , T 2 , T 3 successively. Eliminate l/c 12 and c 13 /c 12 c 23 between the three 
equations. We then obtain c 23 expressed as a function of six arguments 
v 12 IT 1 , v 12 /T 2 , v 12 /T 3 , v 23 /T 1 , v 23 jT 2 , v 23 /T 3 . These reduce to four independent 
arguments 
«/T 7 !, 
ilT 2 , 
JT 3 , 
2 A 
Obviously Cjjg cannot depend on the first three of these; not can it depend 
on the fourth since v 12 can be taken arbitrarily without affecting c 23 . 
Since c 23 does not depend on any of its arguments it must be a definite 
* I ( v, T) cannot decrease with T; for if it did, heat could be transferred from 
the cooler to the hotter of two enclosures by opening a window transparent only to v. 
But it does not seem possible to show by thermodynamics alone that it increases 
without limit (see § 40).