Chemistry Reference
In-Depth Information
if the 10 mol% with molecular weight deviating from M
n
by at least
62,000
were all material with molecular weight 38,000 it would be only 3.8% of the
weight of the sample. Conversely, however, if this were all material with molecu-
lar weight 162,000, the corresponding weight fraction would be 16.2.
There are various ways of expressing the skewness of statistical distributions.
The method most directly applicable to polymers uses the third moment of the
distribution about its mean. The extreme molecular weights are emphasized
because their deviation from the mean is raised to the third power, and since this
power is an odd number, the third moment also reflects the net direction of the
deviations.
In mathematical terms,
6
X
3
n
U
3
5
x
i
ð
M
i
2
M
n
Þ
i
X
3M
i
M
n
2
M
n
Þ
x
i
ð
M
i
2
3M
i
M
n
1
n
U
3
5
(2-36)
i
X
3M
n
X
x
i
M
i
2
M
n
X
x
i
3M
n
P
x
i
M
i
1
x
i
M
i
2
n
U
3
5
i
For a normalized distribution,
3M
2
n
ð
M
n
Þ 2
M
3
n
U
3
5
M
z
M
w
M
n
2
3M
n
ð
M
w
M
n
Þ 1
(2-37)
n
2
n
M
w
1
3
n
n
U
3
5
M
z
M
w
M
n
2
3M
2M
(2-38)
where
n
U
3
is positive if the distribution is skewed toward high molecular weights,
zero if it is symmetrical about the mean, and negative if it is skewed to low
molecular weights.
Asymmetry of different distributionsismostreadilycomparedbyrelating
the skewness to the breadth of the distribution. The resulting measure
α
3
is
obtained by dividing U
3
by the cube of the standard deviation. For the number
distribution,
3M
2
3M
3
n
n
U
3
M
z
M
w
M
n
2
n
M
w
1
n
α
3
5
s
n
5
(2-39)
ð
M
w
M
n
2
M
2
3
=
2
n
Þ
2.6
Summarizing the Molecular Weight Distribution
Complete description of a molecular weight distribution implies a knowledge of
all its moments. The central tendency, breadth, and skewness may be summarized
by parameters calculated from the moments about zero: U
0
0
;
U
0
1
;
U
0
2
;
and
U
0
3
:
These moments also define the molecular weight averages M
n
;
M
w
, and M
z
:
