Basic HTML Version
89
Example 2.
What is M
n
, M
w
and M
z
for a mixture of polymers that are themselves
polydisperse, i.e. having given number, weight, and z- average molecular
weights? This can be found by using the definitions:
Consider subpopulations 1, 2, … j
Population
Weight
M
n
M
w
M
z
1
w
1
M
n,1
M
w,1
M
z,1
2
w
2
M
n,2
M
w,2
M
z,2
j
w
j
M
n,j
M
w,j
M
z,j
Mixing subpopulations 1+2+…+j results in a new mixture where:
Easier for M
w
:
1
2
1
2
1
2
1
2
1
2
,1
,2
1
2
1
2
1
2
...
...
...
...
...
...
i
i
i
i
i
i
i
i
i
i
n
i
i
i
i
i
i
i
n
n
i
i
i
i
i
i
i
i
w
w
w
w
w
w w
M w
w w
w
w
w w
w w
M
M M
M M
M w
M w
⎛
⎞ ⎛
⎞
⎛
⎞ ⎛
⎞
+
+
+
+
⎜
⎟ ⎜
⎟
⎜
⎟ ⎜
⎟
+ +
⎝
⎠ ⎝
⎠
⎝
⎠ ⎝
⎠
=
=
=
=
⎛
⎞ ⎛
⎞
⎛
⎞
⎛
⎞
⎛ ⎞
⎛ ⎞
+ +
+
+
+
+
⎜
⎟ ⎜
⎟
⎜
⎟
⎜
⎟
⎜ ⎟
⎜ ⎟
⎝ ⎠
⎝ ⎠
⎝
⎠ ⎝
⎠
⎝
⎠
⎝
⎠
∑ ∑
∑ ∑
∑
∑ ∑ ∑
∑
∑
,1 1
,2 2
1
2
1
2
1
1
..
...
..
...
i
i
i
i
i
i
i
i
w
w
i
w
i
i
i
i
i
i
w M w M
w M
M w M w
M
w
w w
w
w
⎛
⎞ ⎛
⎞
+
+
⎜
⎟ ⎜
⎟
+
+
⎝
⎠ ⎝
⎠
=
=
=
+ +
⎛
⎞ ⎛
⎞
+
+
⎜
⎟ ⎜
⎟
⎝
⎠ ⎝
⎠
∑ ∑
∑
∑ ∑ ∑
2
i
i
n
i
i
i
i
i
i
w
i
i
i
i
i
z
i
i
i
w
M w
M
w M
M
w
w M
M
w M
=
=
=
∑
∑
∑
∑
∑
∑