4. Reaction equilibria
Reading: Shu, Vol.I, Ch.7
4.1 The Saha equation
In Local Thermodynamical Equilibrium the distribution of atoms with ionization state
i
over
the various energy states
m
is proportional to exp(

χ
i,m
/kT
), where
χ
i,m
is the excitation
energy of state
m
relative to the ground state. Then
N
i,m
N
i,
1
=
g
i,m
g
i,
1
exp(

χ
i,m
kT
)
(4
.
1)
and further after summing over all
m
N
i,m
N
i
=
g
i,m
u
i
(
T
)
exp(

χ
i,m
kT
)
(4
.
2)
where
u
i
(
T
) =
X
m
g
i,m
exp(

χ
i,m
kT
)
(4
.
3)
is called the
level partition function
.
We can extend this treatment to the continuity of states with positive energy by using diFeren
tials and state densities. We thus derive for the ratio of singly ionized atoms to neutral, both
in ground state,
dN
1
,
1
N
0
,
1
= 2
g
1
,
1
g
0
,
1
exp(

χ
0
+
p
2
2
m
e
kT
)
d
3
x d
3
p
h
3
(4
.
4)
where the factor 2 takes care of the two independent spin positions of a free electron. Integrating
over momentum we derive
dN
1
,
1
N
0
,
1
= 2
g
1
,
1
g
0
,
1
(2
π m
e
kT
)
3
/
2
exp(

χ
0
kT
)
d
3
x
h
3
(4
.
5)
The electron shares the available volume with all electrons, and thus
d
3
x
=
n

1
e
.
Hence we
±nally derive the
Saha
equation
N
1
,
1
N
0
,
1
n
e
= 2
g
1
,
1
g
0
,
1
(2
π m
e
kT
)
3
/
2
h
3
exp(

χ
0
kT
)
(4
.
6)
that, using the level partition sums, can be written as an equation for the ionization equilibrium.
N