Chemistry Reference
In-Depth Information
The proportion of the sample with molecular weight between M and M
1
dM
is given by f(M)dM, where f(M) is the frequency distribution and f(M) will equal
x(M) for a number distribution [or w(M) for a weight distribution].
An arithmetic mean is defined in general as
0
Mq
ð
M
Þ
dM
ð
N
ð
N
A
5
q
ð
M
Þ
dM
(2-41)
0
or in equivalent terms as
ð
N
A
5
Mf
ð
M
Þ
dM
(2-42)
0
Equation (2-42)
is the integral equivalent of summative
equation (2-4)
. The
number fraction of the distribution with molecular weights in the interval M to
M
1
dM is dx(M)
5
w(M)dM. The following expressions are examples of integral equations that are
directly parallel to the summative expressions generally used in this chapter:
5
x(M)dM, and the corresponding weight fraction is dw(M)
ð
N
1
ð
N
0
w
ð
M
Þ
dM
M
M
n
5
Mx
ð
M
Þ
dM
5
(2-43)
0
[since w
ð
M
Þ 5 ð
M
=
M
n
Þ
x
ð
M
Þ
]
ð
N
0
Mw
ð
M
Þ
dM
M
w
5
(2-44)
ð
N
M
2
w
ð
M
Þ
dM
ð
N
0
M
z
5
Mw
ð
M
Þ
dM
(2-45)
0
. The results
are equivalent to those shown here for molecular weight distributions because nega-
tive values of the variable are physically impossible.
Some authors prefer to take the integration limits from
2N
to
1N
2.9
Typical Molecular Weight Distributions
As mentioned, different polymerization techniques yield different molecular weight
distributions. There exist three typical molecular weight distributions and they are
the Poisson distribution (anionic polymerization described in Chapter 12), exponen-
tial (condensation polymerization described in Chapter 8) distribution, and log-
normal distribution. Since the mathematical descriptions of these distributions are
known, one can calculate the corresponding molecular weight averages.
In the Poisson distribution, the number fraction of chains with i repeating units
is given by
e
2
a
a
i
i
!
n
i
n
5
(2-46)
