Chemistry Reference
In-Depth Information
Note that M
n
and M
w
can be measured
di
rectly without knowing the distribution
but it has not been convenient to obtain M
z
of synthetic polymers as a direct mea-
surement of a property of the sample. Thus, some
in
formation about the breadth
of the number distribution can be obtained from M
n
and M
w
without analyzing
details of the distribution, but the latter information is necessary for the estimation
of the breadth of the weight distribution and for skewness calculations. This is
most conveniently done by means of gel permeation chromatography, which is
discussed in Section 3.4.
2.7
M
z
$M
w
$M
n
Equation (2-34)
can be rewritten as
2
n
1
M
w
=
M
n
5
s
n
=
M
1
(2-34a)
Since the first term on the right-hand side is the quotient of squared terms, it
is always positive or zero. Zero equality is obtained only when the distribution is
monodisperse, and s
n
then equals zero.
It is obvious then that
M
w
=
M
n
$
1
with the equality true only for monodisperse polymers.
Equation (2-35)
similarly leads to the conclusion that
M
z
=
M
w
$
1
and in general,
M
z
1
j
1
1
$
M
z
1
j
$
M
z
1
j
2
1
$ ...$
M
w
$
M
n
(2-40)
2.8
Integral and Summative Expressions
The relations presented so far have been in terms of summations for greater clar-
ity. The equations given are valid for a distribution in which the variable (mole-
cular weight) assumes only discrete values. However, differences between
successive molecular weights are trivial compared to macromolecular sizes and
the accuracy with which these values can be measured. Molecular weight distribu-
tions can therefore be regarded as continuous, and integral expressions are also
valid.
In the latter case q(M) is a function of the molecular weight such that the
quantity of polymer with molecular weight between M and M
1
dM is given by
q(M)dM. [If the quantity is expressed in moles, then q(M)
5
n(M); if in mass
units, q(M)
5
c(M).]
