Chemistry Reference
In-Depth Information
follow from the definition in
Eq. (2-17)
. The arithmetic mean of the number
distribution is the ratio of these moments:
X
M
i
n
i
X
n
i
5
M
n
n
U
0
1
n
U
0
0
5
A
5
(2-20)
(Compare
Eq. 2-8
.)
The arithmetic mean of a weight distribution (the count is in terms of the
weight c
i
, rather than number of molecules n
i
of each species) is likewise given
by the ratio of the first to the zeroth moment of the particular distribution about
zero. (The notation for moments of weight distributions follows that for number
distributions except that the subscript n is replaced by a w.)
In these last examples we have chosen unnormalized distributions. If the dif-
ferential number or weight distribution is normalized, the area under the curve in
Figs. 2.2 and 2.4
equals unity. That is,
n
U
0
0
5
w
U
0
0
5
ð
Þ
1
normalized distributions
(2-21)
The arithmetic mean is then numerically equal to the first moment of the nor-
malized distribution, as expressed in
Eqs. (2-6) and (2-13)
.
2.4.4
Extensio
n t
o Other Molecular Weight Averages
We have seen that M
n
;
the arithmetic mean of the number distribution, is equal to
the ratio of the first to the zeroth moment of this distribution (
Eq. 2-20
). If we
take ratios of successively higher moments of the number distribution, other aver-
age molecular weights are described:
n
U
0
2
n
U
0
1
5
X
M
2
i
n
i
X
M
i
n
i
5
M
w
(2-22)
X
M
i
n
i
X
M
i
n
i
5
M
z
n
U
0
3
n
U
0
2
5
(2-23)
X
M
4
i
n
i
X
M
3
n
U
0
4
n
U
0
3
5
i
n
i
5
M
z
1
1
(2-24)
We may defi
ne
an av
er
age in general as the ratio of successive moments of
the distribution. M
n
and M
w
are special cases of this definition. The process of
taking ratios of successive moments to compute higher averages of the distribu-
tio
n
can
c
on
tinue without limit. In fact, the averages usually
q
uoted are limited to
M
n
, M
w
, M
z
, and the viscosity average m
ole
cular
we
ight M
v
, which is defined
later in Section 3.3. We can measure M
n
, M
w
, and M
v
dire
ctly, but it is usually
necessary to measure the detailed distribution to estimate M
z
and higher averages.
Table 2.1
lists averages of the number and weight distributions in terms of
these moments.
