Chemistry Reference
In-Depth Information
where n is the total number of chains and a is a con
sta
nt. A
nd
i
5
.
Substituting
Eq. (2-46)
into the
orig
inal definitions of M
n
and M
w
(
Eqs. 2-8 and
2-14
) and af
ter
some derivation, M
n
5
M
1
a, where M
1
is the monomer molecular
weight and M
w
5
M
1
(a
1
0, 1, 2,
...
1). The polydi
spe
rsity index is 1
1
M
1
=
M
n
:
Obviously,
the polydispersity index approaches 1 as M
n
increases.
In the exponential distribution, the number fraction of chains with i repeating
units is given by
e
2
b
b
i
i
!
n
i
n
5
(2-47)
M
1
where b
,
1andi
5
1, 2,
...
. The corresponding M
n
and M
w
are
and
ð
1
2
b
Þ
M1
ð
1
1
b
Þ
M
1
M
n
, respectively. And the polydispersity index is 2
2
. Here, the polydis-
ð
2
b
Þ
1
persity index approaches 2 as M
n
increases. It is obvious that the e
xpo
nential distri-
bution has a broader distribution than the Poisson distribution when M
n
is
l
ow
.
The log-normal distribution resembles the one shown in
Fig 2.4
; M
n
, M
w
, and
the polydispersity index are given by the following equations.
M
n
5
Me
ð2σ
2
=
2
Þ
(2-48)
2
M
w
5
Me
ðσ
=
2
Þ
(2-49)
2
5
e
σ
Polydispersity index
(2-50)
2
is the variance.
where M is the peak molecular weight and
σ
Appendix 2A
Molecular Weight Averages of Blends of Broad Distribution
Polymers
When broad distribution polymers are blended, M
n
;
M
w
;
M
z
;
etc., of the blend are
given by the corresponding expressions listed in
Table 2.2
but the M
i
's in this
case are the appropriate average molecular weights of the broad distribution com-
ponents of the mixture. Thus, for such a mixture,
1
X
w
i
ð
M
n
Þ
i
ð
M
n
Þ
mixture
5
(2-11a)
X
w
i
ð
M
w
Þ
i
ð
M
w
Þ
mixture
5
(2-13a)
and so on.
