Chemistry Reference
In-Depth Information
Table 2.1
Moments about Zero and Molecular Weight Averages
(a) Number distribution
Not normalized
Normalized
Averages
X
X
n
U
0
0
5
n
i
5
x
i
51
i
i
X
X
n
U
0
1
5
M
n
5
n
U
0
1
=
n
U
0
0
M
i
n
i
5
M
i
x
i
i
i
X
X
n
U
0
2
5
M
w
5
n
U
0
2
=
n
U
0
1
M
i
n
i
5
M
w
M
n
n
U
0
0
5
M
i
x
i
5
M
w
M
n
i
i
X
5
P
M
i
x
i
n
U
0
3
5
M
i
n
i
5
M
z
M
w
M
n
U
n
U
0
0
5
n
U
0
3
=
n
U
0
2
5
M
z
M
w
M
n
M
z
i
1=a
M
v
5
n
U
0
a
0
11
=
n
U
0
1
P
M
i
n
i
5
P
M
i
x
i
n
U
0
j
5
(b) Weight distribution
Not normalized
Normalized
Averages
X
X
w
U
0
21
5
c
i
M
21
w
i
M
21
i
5
i
i
X
X
w
U
0
0
5
M
n
5
w
U
0
0
=
w
U
0
21
c
i
5
w
i
51
i
i
X
5
P
w
i
M
i
5
w
U
0
1
5
M
w
5
w
U
0
1
=
w
U
0
0
c
i
M
i
5M
w
U
w
U
0
0
M
w
i
X
X
w
U
0
2
5
M
z
5
w
U
0
2
=
w
U
0
1
c
i
M
i
5
M
z
M
w
U
w
U
0
0
5
w
i
M
i
5
M
z
M
w
i
i
M
a
v
5 ð
w
U
0
a
Þ
1=a
5
P
w
i
M
i
a
M
v
is derived from solution viscosity measurements through the Mark
P
c
i
M
i
w
U
0
j
5
KM
a
Houwink equation
½
n
5
v
;
where
[
n
]
is the limiting viscosity number and K and
1
are constants which depend on the polymer,
solvent, and experimental conditions, but not on M
(Section 3.3.1)
.
The reader may notice that any moment about zero of a normalized
distribution
X
x
i
ð
M
i
Þ
X
w
i
ð
M
i
Þ
j
j
n
U
0
j
5
w
U
0
j
5
or
corresponds to the arithmeti
c m
ean of
th
e number or weight distribution of (M
i
)
j
,
respectively. Respectively, M
n
and M
w
are arithmetic means of th
e
n
um
ber
and weight distributions and the source of their names is obvious. The M
z
, M
z
1
1
,
and so on, are arithmetic means of the z, z
1
1, etc., distributions. Operational
models of these distributions would be too complicated to be useful in polymer
science.
