Chemistry Reference
In-Depth Information
where q
i
is the quantity of polymer in unit volume of the sample with molecular
weight M
i
and respective values of q
i
5
n
i
(number of molecules, or moles) for an
unnormalized number distribution,
5
x
i
(mole fraction) for a normalized number
distribution,
5
c
i
(number of grams) for an unnormalized weight distribution,
or
5
w
i
(weight fraction) for a normalized weight distribution. In addition, we
shall use the notation
n
U to refer to a moment of the number distribution and
w
U
to denote a moment of the weight distribution.
Weight distributions will usually be encountered during analyses of polymer
samples. Considerations of polymerization kinetics are often easier in terms of
number distributions.
2.4.2
Dimensions
Molecular weight itself is dimensionless. It is the sum of the atomic weights in
the formula of the molecule. Atomic weights, in turn, are expressed in terms of
dimensionless atomic mass units (amu) which are ratios (
12) of the masses of
the particular atoms to that of the most abundant carbon isotope
12
C to which a
mass of 12 is assigned. A gram molecular weight, or gram-mole, is the amount of
polymer whose weight in grams is numerically equal to the molecular weight (in
amu). It is just as correct to use pound-moles or ton-moles if the circumstances so
dictate.
The moments of normalized distributions are products of dimensionless fre-
quencies and dimensionless molecular weights or of gram-moles with dimensions
of mass. The former moments will be unitless, and the units of the latter will
depend on the moment number and on the units of the distribution. Most equa-
tions in polymer science imply use of gram-moles, but this is not universal and
the dimensions of the particular equation should be checked to determine which
units, if any, are being used for molecular weight and concentration quantities.
3
2.4.3
Arithmetic Mean as a Ratio of Moments
As a general case the ratio of the first moment to the zeroth moment of any distri-
bution defines the arithmetic mean. For an unnormalized number distribution, n
i
is the number of moles per unit volume with molecular weight M
i
and the zeroth
and first moments of the distribution about zero are given, respectively, by
X
X
n
i
n
U
0
0
5
0
ð
M
i
Þ
n
i
5
(2-18)
i
X
X
M
i
n
i
n
U
0
1
5
1
ð
M
i
Þ
n
i
5
(2-19)
i
In these symbols the subscript n shows that the moment refers to a number
distribution, the numerical subscript is the moment order, and the prime super-
script indicates that the moment is taken about the M
5
0 axis. These equations
