Chemistry Reference
In-Depth Information
information into parameters descriptive of various aspects of the distribution.
Such parameters evidently must contain less information than the original distri-
bution, but they present a concise picture of the distribution and are indispensable
for comparing different distributions.
One such summarizing parameter expresses the central tendency of the distri-
bution. A number of choices are available for this measure, including the median,
mode, and various averages, such as the arithmetic, geometric, and harmonic
means. Each may be most appropriate for different distributions. The arithmetic
mean is usually used with synthetic polymers. This is because it was very much
easier, until recently, to measure the arithmetic mean directly than to characterize
the whole distribution and then compute its central tendency. The distribution
must be known to derive the mode or any simple average except the arithmetic
mean. (Some methods like those based on measurement of sedimentation
and diffusion coefficients measure more complicated averages directly. They are
not used much with synthetic polymers, however, and will not be discussed in
this text.)
Various molecular weight averages are current in polymer science. We show
here that these are simply arithmetic means of molecular weight distributions.
It may be mentioned in passing that the concepts of small particle statistics that
are discussed here apply also to other systems, such as soils, emulsions, and car-
bon black, in which any sample contains a distribution of elements with different
sizes.
To define any arithmetic mean A, let us assume unit volume of a sample of
N polymer molecules comprising n
1
molecules with molecular weight M
1
, n
2
mole-
cules with molecular weight M
2
,
, n
j
molecules with molecular weight M
j
.
n
1
1
n
2
1
?
1
n
j
5
N
...
(2-1)
n
1
M
1
1
n
2
M
2
1
?
1
n
j
M
j
n
1
1
n
2
1
?
1
n
j
n
1
M
1
1
n
2
M
2
1
?
1
n
j
M
j
N
A
5
5
(2-2)
n
1
N
M
1
1
n
2
N
M
2
1
?
1
n
j
N
M
j
A
5
(2-3)
The arithmetic mean molecular weight A is given as usual by the total mea-
sured quantity (M) divided by the total number of elements. That is, the ratio n/N
is the proportion of the sample with molecular weight M
i
. If we call this propor-
tion f
i
, the arithmetic mean molecular weight is given by
X
A
5
f
1
M
1
1
f
2
M
2
1
?
1
f
j
M
j
5
f
i
M
i
(2-4)
i
Equation (2-4)
defines the arithmetic mean of the distribution of molecular
weights. Almost all molecular weight averages can be defined from this
equation.
