496
9 Chemical Equilibrium
or
K P 38.35 × 10
42 m
3
2
e T
5
2 e
−βI p
= 3.334 × 10
−2 T
5
2 e
−βI p .
(9.3.17)
Our final form for K P is then the expression
K P = 0.03334 T
5
2 e
−βI p .
(9.3.18)
K P for the process M → M + + e − may also be written (as in first year chemistry)
in terms of partial pressures as
K P =
(P M + /P ◦ )(P e − /P ◦ )
(P M /P ◦ )
.
(9.3.19)
Naturally, the thermal ionization process is, in general, not 100% efficient, and only
a fraction of the atoms will be ionized at a given temperature T . The expression for
the ionization process can then be written in terms of the fraction α of the M atoms
that are ionized as
(1 − α)M → αM
+
+ αe
− ,
(9.3.20)
from which we obtain P M + = αP , P e − = αP , and P M = (1 − α)P , with P the
total gas pressure. In this way, we find that K P and α are related by
K P =
α 2
1 − α
P
P ◦ α
2 P
P ◦ ,
for α 1 .
(9.3.21)
From this result, we obtain the expression
α = K
1
2
P
P ◦
P
1
2 = 0.1826
P ◦
P
1
2
T
5
4 e
−βI p /2
(9.3.22)
for the degree of ionization α of M atoms. This equation is known in plasma physics
as the Saha equation.
If we consider the specific example of Cs, we have I p = 6.237 × 10 −19 J/atom,
and βI p is given by
βI p =
6.237 × 10 −19 J
1.3807 × 10 −23 J K −1 T
4.518 × 10 4 K
T
.
(9.3.23)