Chemistry Reference
In-Depth Information
where the abbreviation r
i
can be calculated using:
1
r
I
M
1
r
I
M
ln
f
II
i
ln
f
i
r
i
¼
þ
i
¼
A
;
B
:
(9)
The activity coefficients in (
9
) can be derived using standard thermodynamics in
combination with (
5
). The replacement of the segment-molar chemical potential of
the copolymer species in (
7
) according (
4
) leads to:
X
II
W
II
X
I
W
I
ð
r
;
y
Þ¼
ð
r
;
y
Þ
exp
r
r
ð
ð
y
Þ
Þ;
(10)
where the abbreviation r is given by:
1
r
I
M
1
r
I
M
ln
f
II
ln
f
I
r
ð
r
;
y
Þ¼
ð
y
Þþ
ð
y
Þ:
(11)
Equation (
10
) is valid for all
r
and
y
values found in the system and permits the
calculation of an unknown distribution function,
W
II
(
r
,
y
). The activity coefficients
in (
11
) can be derived using standard thermodynamics in combination with (
5
).
Integration of (
10
) and applying the normalization condition (
1
) results in:
1
ð
1
X
II
X
I
W
I
¼
ð
r
;
y
Þ
exp
ð
r
r
ð
r
;
y
ÞÞ
d
y
d
r
:
(12)
0
0
To deal with the problem of calculation of the cloud-point curve and the
corresponding shadow curve, the temperature of a given phase I is changed at
constant pressure until the second phase II is formed. Thus, the unknowns of the
problem are the equilibrium temperature,
T
, the composition of the second phase,
X
II
and
X
I
A
, and the distribution function,
W
II
(
r
,
y
). To calculate them, the phase
equilibrium conditions (
8
) and (
12
) are used. In this system of equations, the
unknown distribution function
W
II
(
r
,
y
) and the other scalar unknowns
T
,
X
II
, and
X
I
A
are connected; however, the unknown distribution function
W
II
(
r
,
y
) occurs only
with the average values
r
I
N
and
y
I
W
. This situation allows a separation of the problem
of the unknown distribution function by considering
r
I
N
and
y
I
W
as additional scalar
unknowns and their defining equations:
1
ð
1
X
II
r
I
N
¼
X
I
W
I
ð
r
;
y
Þ
exp
ð
r
r
ð
r
;
y
ÞÞ
d
y
d
r
(13)
r
0
0
and
1
ð
1
y
I
W
X
II
yX
I
W
I
¼
ð
r
;
y
Þ
exp
ð
r
r
ð
r
;
y
ÞÞ
d
y
d
r
;
(14)
0
0
