Chemistry Reference
In-Depth Information
Table 2.2
lists various average molecular weights in terms of moments of the
number and weight distributions, where the quantity of polymer species with par-
ticular sizes are counted in terms of numbers of moles or weights, respectively.
Note that in general a given average is given by
M
z
1
k
5
n
U
0
k
1
3
n
U
0
k
1
2
5
w
U
0
k
1
2
w
U
0
k
1
1
(2-25)
The moment orders in the weight distribution ar
e o
ne less than the corres-
ponding orders in the number distributions. (Compare M
n
formulas.) This symme-
try arises because molar and weight concentrations are generally related by
Eqs. (2-9) and (2-10)
. Thus,
X
n
i
M
i
5
X
X
c
i
M
k
2
1
i
n
U
0
k
5
ð
n
i
M
i
Þ
M
k
2
1
i
5
w
U
0
k
2
1
5
(2-26)
The viscosity average molecular weight M
v
, which will be discussed later in
Section 3.3, is the only average listed in these tables that is not a simple ratio of
successive moments of the molecular weight distribution.
2.5
Breadth of the Distribution
The distribution of sizes in a polymer sample is not completely defined by its
central tendency. The breadth and shape of the distribution curve must also be
known, and this is determined most efficiently with parameters derived from the
moments of the distributi
on.
It is always true that M
z
.
M
w
.
M
n
;
with the equality occurring only if all
species in the sample have the same molecular weight. (This inequality is proven
in
S
ect
ion 2.
7.)
S
uc
h monodispersity is unknown in synthetic polymers. The ratio
M
w
=
M
n
,or
1, is commonly taken to be a measure of the polydisper-
sity of the sample. This ratio (the polydispersity index) is not a sound statistical
measure of the distribution breadth, and we sh
ow
l
ate
r that it is easy to make
unjustified inferences from the magnitude of the M
w
=
M
n
ratio if this parameter is
close to unity. However, the use of the polydispersity index is deeply imbedded
in polymer science and technology, where it is often called the breadth of the
distribution. We see later that it is actually related to the variance of the number
distribution of the polymer sample. In many cases, when different samples are
being compared, any changes in the number distribution will be paralleled by
changes in the weight distribution, and so variations in the polydispersity index
can substitute for comparisons of the breadth of the weight distributions, which
would be more relevant, in general, to the processing and mechanical properties
of the materials.
The most widely used statistical measure of distribution breadth
is
the st
an
-
dard deviation, which can be computed for the number distribution if M
n
and M
w
are known. This use of these molecular weight averages provides more informa-
tion than can be derived from their ratio.
ð
M
w
=
M
n
Þ 2
