Chemistry Reference
In-Depth Information
where P
tot
is the sum of the partial pressures of the individual gases. HF and H
2
O are in
equilibrium with their constituent atoms, hence
1
2
m
H
2
ð
T
;
P
H
2
Þþ
1
2
m
F
2
ð
T
;
P
F
2
Þ¼
m
HF
ð
T
;
P
HF
Þ
(6
:
5)
m
H
2
ð
T
;
P
H
2
Þþ
1
2
m
O
2
ð
T
;
P
O
2
Þ¼
m
H
2
O
ð
T
;
P
H
2
O
Þ
(6
:
6)
Equations (6.2), (6.5) and (6.6) can now be used to eliminate m
Al
, m
F
2
and m
O
2
from
Equation (6.4) to obtain
2
3
G
slab
ð
T
;
P
tot
Þ
N
Al
G
bulk
ð
T
;
P
tot
Þ
N
F
3N
Al
4
5
ð
T
;
P
HF
;
P
H
2
O
;
P
H
2
Þ¼
1
2A
ð
Þ
m
HF
ð
T
;
P
HF
Þ
N
O
m
H
2
O
ð
T
;
P
H
2
O
Þ
(6
:
7)
1
2
ð
3N
Al
N
F
2N
O
þ
N
H
Þ
m
H
2
ð
T
;
P
H
2
Þ
Treating the gaseous species as ideal gases, their chemical potentials dependence on P
and T is
P
X
P
X
m
X
ð
T
;
P
X
Þ¼
m
X
ð
T
;
P
X
Þþ
kT ln
(6
:
8)
This chemical potential can be referred to the athermal limit and the DFT calculations by
rewriting Equation (6.8) as
þ
E
DFT
ð
T
¼
0
Þ
m
0
X
ð
T
;
P
X
Þ¼
m
X
ð
T
;
P
X
Þ
m
X
ð
0
;
P
X
Þ
(6
:
9)
The term in square brackets in Equation 6.9 can be obtained from thermodynamical
reference tables [18], as described previously [16]. In the current study the Gibbs free
energies of the slab and bulk crystal are computed at the athermal limit and their temperature
dependence is ignored as it is negligible compared to that of the gaseous species. The small
PV term due to the change in volume of the bulk phases is also neglected.
6.2.3 Molecular Adsorption
It is possible to extend the formulism discussed in Section 6.2.2 to calculate the extent to
which a gas will chemisorb to a given surface. In equilibrium, the rate of desorption of
molecules from the surface will be equal to their rate of adsorption. The ratio of the rate of
adsorption, r
ads
, to the rate of desorption, r
des
, is approximately
r
ads
r
des
Dm
E
ads
kT
¼
exp
(6
:
10)
