Chemistry Reference
In-Depth Information
As one of the three composition variables becomes zero, this relation simplifies
to the expression for binary mixtures (
57
). The extension of (
60
) to multicomponent
systems is unproblematic and enables the calculation of phase diagrams for such
mixtures of great practical importance if one calculates the composition of the
coexisting phases by a direct minimization of the Gibbs energy [
19
]. In this manner,
it is possible to evade the laborious and sometimes even impossible calculation of
the chemical potential for each component.
The implementation of ternary interactions by simply adding the last term of
(
56
)to(
60
) does not suffice. The reason lies in the fact that the three options to form
a contact between all three components out of binary contacts (1/2
þ
þ
3, 1/3
2, and
þ
2/3
1) might differ in their contribution to the Gibbs energy of the mixture. This
supposition results in the necessity to introduce three different ternary interaction
parameters. Furthermore, it requires a weighting of these contribution to account
for the fact that they must be largest in the limit of the first addition of the third
component 3 (highest fraction of 1/2 contacts) and die out as component 3 becomes
domi
na
nt (vanishing fraction of 1/2 contacts). The simplest possibility to account
for
G
res
t
, the extra contributions of ternary contacts to the residual Gibbs energy, is
formulated in (
61
), where the negative sign was chosen by analogy to the second
term of (
57
):
D
G
res
t
RT
¼
D
½
t
1
1
ð
'
1
Þþ
t
2
1
ð
'
2
Þþ
t
3
1
ð
'
3
Þ
'
1
'
2
'
3
(61)
t
1
quantifies the changes associated with the formation of a ternary contact 1/2/3 out
of a binary contact 2/3 by adding a segment of component 1. The meaning of
t
2
and
t
3
is analogous.
Equation (
61
) makes allowance for differences in the genesis of ternary contacts
but it does not yet consider that the number of segments of the third component in
the coordination sphere of a certain binary contact might deviate from that expected
from the average composition due to very favorable or unfavorable interactions
(quasi chemical equilibria). One way to model such effects consists of the intro-
duction of composition-dependent ternary interaction parameters, as formulated in
the following equation:
G
res
t
ðÞ
RT
D
¼
½
ð
t
1
þ
t
11
'
1
Þ
ð
'
1
Þ
þ
ð
t
2
þ
t
22
'
2
Þ
ð
'
2
Þ
1
1
(62)
þð
t
3
þ
t
33
'
3
Þð
1
'
3
Þ'
1
'
2
'
3
The relations presented for ternary mixtures open the possibility for investigation of
the extent to which their thermodynamic behavior can be forecast (neglecting
possible contributions of ternary interaction parameters) if the binary interaction
parameters of the three subsystems are known as a function of composition from
independent experiments. For such calculations, it is important to make sure that
the size of a segment is identical for all subsystems. The fact that most of the
