Chemistry Reference
In-Depth Information
The proportion of the sample with size M
i
is expressed in the present case as
the corresponding weight fraction. Equating w
i
and f
i
in
Eq. (2-4)
produces the
following expression for the arithmetic mean of the weight distribution:
X
A
5
w
i
M
i
5
M
w
(2-13)
i
where M
w
is the weight average molecular weight, which from
Eqs. (2-9) and
(2-10)
may also be expressed as
M
w
X
M
i
c
i
=
X
c
i
5
X
M
i
n
i
X
M
i
n
i
(2-14)
EXAMPLE 2-1
Given that a polymer sample contains two moles of
cha
ins wi
th
one mole having a molecular
weight of 5000 and the other 10,000, calculate its M
n
and M
w
. A different polymer sample
also contains two moles of chains but in this sample, o
ne
mol
e o
f chains has a molecular
weight of 2500 while the other has 12,500; what are its
M
n
and
M
w
?
For the first sample,
1
2
3
1
2
3
1
3
3
2
3
3
M
n
5
5000
1
10000
5
7500 and
M
w
5
5000
1
10000
5
8333
For the second sample,
1
2
3
1
2
3
1
6
3
5
6
3
M
n
5
2500
1
12500
5
7500 and
M
w
5
5000
1
10000
5
10833
Both samples have the same M
n
but the second sample has a higher M
w
, indicating that
the second sample has a wider distribution of molecular weight.
2.4
Molecular Weight Averages as Ratios of Moments
2.4.1
Moments in Statistics and Mechanics
We have seen that average molecular weights are arithmetic means of distribu-
tions of molecular weights. An alternative and generally more useful definition is
in terms of moments of the distribution. This facilitates generalizations beyond
the two averages we have considered to this point and clarifies the estimation of
parameters related to the breadth and symmetry of the distribution.
The concept of moments was adopted in statistics from the science of mechan-
ics where it was first used in the sense of “importance.” The moment of a force
about an axis meant the importance of the force in causing rotation about the
axis. Similarly, the moment of inertia of a body with respect to an axis expressed
the importance of the inertia of the body in resisting a change in the rate of rota-
tion of the body about the axis.
The first moment of a force or weight about an axis is defined as the product
of the force and the distance from the axis to the line of action of the force.
