Chemistry Reference
In-Depth Information
The number-average segment number in the second phase is given by:
ð
ð
X
II
r
I
N
¼
K
ð
r
;
y
Þ
X
F
W
F
ð
r
;
y
Þ
d
y
d
r
:
(25)
r
f
r
y
The weight-average chemical composition in the second phase is given by:
ð
ð
X
F
W
F
yK
ð
r
;
y
Þ
ð
r
;
y
Þ
y
II
X
II
¼
d
y
d
r
:
(26)
f
r
y
In contrast to the cloud-point problem, the integrals occurring in (
24
) (
26
)
cannot be solved analytically. In order to calculate the coexisting curves, (
21
) and
(
24
) (
26
) must be solved simultaneously, where the occurring integrals must be
estimated using a numerical procedure. If the quantities of feed solution (
X
F
,
W
F
(
r
,
y
),
and
X
A
) are known, the unknowns are the temperature
T
(or the quantity f); the
polymer concentration in the second phase,
X
II
; the solvent concentration in the
second phase,
X
I
A
; the number-average segment number in the second phase,
r
I
N
;
and the weight-average chemical composition,
y
I
W
. The corresponding quantities of
the first phase can be estimated using the balance equations (
17
) (
20
). If the
selected
G
E
model is not dependent on the segment number, a simplification is
possible. Inserting the solvent equilibrium conditions [(
21
) with
i
B] the system
can be reduced to four equations by eliminating,
r
I
N
. The final equation reads:
¼
ð
ð
X
F
W
F
X
I
A
X
I
B
K
ð
r
;
y
Þ
ð
r
;
y
Þ
1
r
M
þ
0
¼
d
y
d
r
r
A
þ
r
B
ð
1
f
Þ
r
f
r
y
þ
X
I
B
X
I
B
1
r
B
ln
ln
f
I
B
ln
f
B
;
(27)
where
p
(
r
,
y
)in(
23
) is replaced by
p
(
y
) and can be calculated by:
þ
X
I
B
X
I
B
1
r
B
ln
ln
f
I
B
ln
f
B
ln
f
II
ln
f
I
r
ð
y
Þ¼
ð
y
Þþ
ð
y
Þ
(28)
and
þ
X
I
B
X
I
B
1
r
B
ln
ln
f
I
B
ln
f
B
þ
ln
f
A
þ
ln
f
I
A
:
r
A
¼
(29)
The coexisting-curve problem is now given by (
21
) with
i
¼
A, (
24
), (
26
), and
(
27
), where (
28
) and (
29
) can be used to compute r(
y
) and r
A
.
