Page 501 - Advanced thermodynamics for engineers

20.7 DIFFUSION AND HEAT TRANSFER 493

and the net energy passing through the wall is given from Eqn (20.115) as
_
0
E E _ 2k 1=2h 0 0 1=2 1=2 i
¼ p T pT (20.119)
A pm
It is possible to substitute for the pressure of a monatomic gas by the expression
a 3 3=2 5=2
p ¼ e h ð2pmÞ b (20.120)
and the temperature by
T ¼ 1=kb: (20.121)
Then Eqns (20.118) and (20.119) become:

E _ 3 h a 3 a 0 0 3 i
¼ 4pmh e b e b (20.122)
A
_ n c 3 h a 2 a 0 0 2 i
¼ 2pmh e b e b (20.123)
A
If the state denoted by the primed symbols is constant, then differentiating gives

_

E
d ¼ 2pmh 3 b 3 a 6b 1 db 2da (20.124)
e
A

_ n c 3 3 a
d ¼ 2pmh b e ½ 2db b daŠ (20.125)
A
Equations (20.124) and (20.125) may be written in the form
_

E
d ¼ L _ E;b db þ L _ E;a da (20.126)
A
and

_ n c
d ¼ L _ n c;b db þ L _ n c;a da (20.127)
A
where
¼ 12pmb 4 3 a
h
e
L _ E;b
3 3 a )
¼ 4pmb h e
L _ E;a
e
L ¼ 4pmb 3 3 a
h
_ n c;b
h
L ¼ 2pmb 2 3 a :
e
_ n c;a
¼ L _ n c ;b , which is equivalent to the result given by Onsager, that L 12 ¼ L 21 , and has
Hence L _ E;a
been proved from statistical thermodynamics.