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86
M
w
=
w
i
M
i
i
∑
w
i
i
∑
Equivalent expressions are:
M
w
=
w
i
M
i
i
∑
w
i
i
∑
=
c
i
M
i
i
∑
c
i
i
∑
=
N
i
M
i
2
i
∑
N
i
M
i
i
∑
It appears from the last expression that M
w
mathematically speaking is a
higher moment of the parent distribution, compared to M
n
.
By further extending into an even higher moment we can define the z-average
molecular weight:
M
z
=
N
i
M
i
3
i
∑
N
i
M
i
2
i
∑
=
m
i
M
i
2
i
∑
m
i
M
i
i
∑
There are several good reasons for using different types of molecular weight
averages. First, a single parameter such as M
n
says nothing about the
distribution, whereas knowing additional averages provides simply more
information. As will be discussed later (Section 5.3) common types of
distributions have characteristic distributions. For example, randomly
degraded polymers approach a M
w
/M
n
ratio of 2.0 upon prolonged
degradation.
The ratio between weight and number average molecular weights is often
referred to as the polydispersity index (PI):
PI
=
M
w
M
n
Another reason for introducing different averages is that different experimental
methods for determining molecular weights in case of polydispersity provides
different values:
•
Osmometry (Section 3.1): Provides M
n
•
Light scattering (Section 6.2): Provides M
w
•
Intrinsic viscosity (Section 6.1): May provide M
n
, M
w
or M
v
(viscosity
average), depending on the standards used (see 6.1 for details)