Chemistry Reference
In-Depth Information
Equation (5-4)
can be manipulated to
0
@
1
A
1
0
@
1
A
1
1
2
c
1
w
11
1
2
c
2
w
22
1
2
N
1
N
2
N
1
v
1
1
E
N
1
N
2
5
N
2
v
2
(5-5)
3
½
w
12
ð
c
1
v
2
1
c
2
v
1
Þ
2
ω
11
c
1
v
2
2
ω
22
c
2
v
1
To eliminate w
12
it is assumed that
1
=
2
1
2
w
12
c
1
v
1
1
c
2
v
2
c
1
w
11
v
1
c
2
w
22
v
2
5
(5-6)
In effect, this takes w
12
to be equal to the geometric mean of w
11
and w
22
.
Here, it is worth noting that the geometric mean assumption is only valid when
the two species have comparable size and shape and interact with each other
through dispersion forces. Then
"
#
2
1
=
2
1
=
2
N
1
c
1
w
11
N
2
c
2
w
22
N
1
N
2
v
1
v
2
N
1
v
1
1
c
1
w
11
2v
1
c
2
w
22
2v
2
E
(5-7)
5
2
1
2
2
2
N
2
v
2
The first two terms on the right-hand side of
Eq. (5-7)
represent the interaction
energies of the isolated components, and the last term is the change in internal
energy
U
m
of the system when the species are mixed. If the contact energies
can be assumed to be independent of temperature, the enthalpy change on mixing,
Δ
Δ
H
m
, is then
"
#
2
1
=
2
1
=
2
N
1
N
2
v
1
v
2
N
1
v
1
1
c
1
w
11
2v
1
c
2
w
22
2v
2
Δ
H
m
5
Δ
U
m
5
2
(5-8)
N
2
v
2
The terms in (c
i
w
ii
/2v
i
)
1/2
are solubility parameters and are given the symbol
δ
i
. It is convenient to recast
Eq. (5-8)
in the form
2
H
m
5
½
N
1
N
2
v
1
v
2
=ð
N
1
v
1
1
N
2
v
2
Þ½δ
1
2
δ
2
(5-9)
2
V
2
5
ð
N
1
v
1
=
V
Þð
N
2
v
2
=
V
Þ½δ
1
2
δ
2
V
φ
1
φ
2
½δ
1
2
δ
2
5
where the
φ
i
are volume fractions. Hence the heat of mixing per unit volume of
mixture is
2
Δ
H
m
=
V
5
φ
1
φ
2
½δ
1
5
δ
2
(5-10)
where V is the total volume of the mixture. For solutions, subscript 1 refers to the
solvent and subscript 2 to the polymeric solute.
Miscibility occurs only if
S
m
in
Eq. (5-3)
is
always positive (the ln of a fraction is negative), the components of a mixture are
assumed to be compatible only if
Δ
G
m
#
0in
Eq. (5-3)
. Since
Δ
S
m
. Thus solution depends in this
analysis on the existence of a zero or small value of
Δ
H
m
#
T
Δ
H
m
. Note that this theory
allows only positive (endothermic) heats of mixing, as in
Eq. (5-10)
. In general,
Δ
