Chemistry Reference
In-Depth Information
Table 5.1
Solution Behavior
a
Solution type
Δ
H
Δ
S
m
m
Ideal
Zero
Ideal
Regular
Nonzero
Ideal
Athermal
Zero
Nonideal
Irregular
Nonzero
Nonideal
a
The ideal entropy of mixing
Δ
S
m
is
5 2
R
X
N
i
ln x
i
(5-3)
where x
i
is the mole fraction of component i in the mixture and N
i
is the number of moles of species i.
Equation (5-3)
represents the entropy change in a completely random mixing of all species. The
components of the mixture must have similar sizes and shapes for this equation to be true.
S
ideal
m
Δ
formulated in terms of relative numbers of intermolecular contacts between like
and unlike molecules. Nonzero
Δ
H
m
values are assumed to be caused by the net
results of breaking solvent (1-1) contacts and polymer (2-2) contacts and making
polymer
solvent (1-2) contacts
[1,2]
.
Consider a mixture containing N
1
molecules of species 1, each of which has
molecular volume v
1
and can make c
1
contacts with other molecules. The corre-
sponding values for species 2 are N
2
, v
2
, and c
2
, respectively. Each (1-1) contact
contributes an interaction energy w
11
, and the corresponding energies for (2-2)
and (1-2) contacts are w
22
and w
12
. Assume that only first-neighbor contacts need
to be taken into consideration and that the mixing is random. If a molecule of spe-
cies i is selected at random, one assumes further that the probability that it makes
contact with a molecule of species j is proportional to the volume fraction of that
species (where i may equal j). If this randomly selected molecule were of species
1 its energy of interaction with its neighbors would be c
1
w
11
N
1
v
1
/V
c
1
w
12
N
2
v
2
/
1
V, where the total volume of the system V is equal to N
1
v
1
1
N
2
v
2
. The energy of
interaction of N
1
molecules of species 1 with the rest of the system is N
1
/2 times
the first term in the previous sum and N
1
times the second term [it takes two spe-
cies 1 molecules to make a (1-1) contact]; i.e., c
1
w
11
N
1
v
1
=
V.
Similarly, the interaction energy of N
2
species 2 molecules with the rest of the
system is c
2
w
22
N
2
v
2
=
2V
c
1
w
12
N
1
N
2
v
2
=
1
V. The total contact energy of the sys-
tem E is the sum of the expressions for (1-1) and (2-2) contacts plus half the sum
of the expressions for (1-2) contacts (because we have counted the latter once in
connection with N
1
species 1 molecules and again with reference to the N
2
spe-
cies 2 molecules):
2V
1
c
2
w
12
N
1
N
2
v
1
=
c
1
w
11
N
1
v
1
1
c
2
w
22
N
2
v
2
w
12
N
1
N
2
ð
c
1
v
2
1
c
2
v
1
Þ
1
E
5
(5-4)
2
ð
N
1
V
1
1
N
2
V
2
Þ
