Chemistry Reference
In-Depth Information
If the solute molecules in a solution of a whole polymer are independent
agents, we may regard the viscosity of the solution as the sum of the contributions
of the
i
monodisperse species that make up the whole polymer. That is,
whole
5
5K
X
i
X
η
η
0
2
η
i
n
i
M
a1
1
i
1
η
0
2
1
(3-75)
i
From
Eq. (3-64)
,
whole
5
c
X
n
i
M
a1
1
1
c
η
η
0
2
K
½5
lim
c-
0
1
lim
c-
0
(3-76)
i
However,
X
c
i
5
X
n
i
M
i
c
5
(3-77)
and so
K
X
n
i
M
a1
i
X
n
i
M
i
½η 5
lim
c-
0
(3-78)
with
Eq. (3-71)
,
!
K
X
i
n
i
M
a1
i
X
i
KM
v
5
½η 5
n
i
M
i
lim
c-
0
(3-79)
Then, in the limit of infinite dilution,
h
i
1
=a
K
X
n
i
M
a1
i
X
n
i
M
i
M
v
5
(3-80)
Alternative definitions follow from simple arithmetic:
h
i
1
=a
X
w
i
M
i
M
v
5
(3-81)
In terms of moments,
1
=a
1
=a
U
0
a1
1
U
0
a
M
v
5 η
5 ω
(3-82)
Note that
M
v
is a function of the solvent (through the exponent
a
) as well as
of the molecular weight distribution of the polym
er.
Thu
s,
a given polymer sam-
ple ca
n b
e characterized only by a single value of
M
n
or
M
w
, but it may have dif-
ferent
M
v
's depending on the
so
lve
nt
in
whi
ch [
η
] is measured. Of course, if the
sample were monodisperse,
M
v
5
M
w
5
M
n
5 ...
. In general, the broader the
molecular weight distribution, the
m
ore
M
v
may vary in
d
ifferent solvents.
Note that
Eq. (3-80)
defines
M
n
with
a
1
. F
or poly-
mers that assum
e
random c
oil
shapes in solution, 0.5
#a#
0.8, and
M
v
w
ill
be
much closer to
M
w
than to
M
n
be
cause
a
is closer to 1 than to
52
1 and
M
w
with
a
5
1. Also,
M
v
is
much easier to measure than
M
w
once
K
and
a
are known, and it
2
is often
