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gives N as a function of M. This is the most general way of describing
The Chains and Averages of Polymers
distribution, since, in principle, all other aspects of the
polydispersity can be derived from the continuous distribution function.
2. The histogram is a graphical device which is both attainable in
practice and also an approximation to a theoretical distribution
3. The mean can be evaluated from the classified data of the histogram;
it measures the center of the distribution. The mean (whose symbol is an
overbar) is defined as
S: Nj M:
Mn " ('1'3^ This quantity is also called the number (subscript n)
average molecular weight.
4. The standard deviation can also be evaluated from the same
classified data; it measures the width of the distribution. The standard
deviation î is defined as
I Sj Nj (Mj - M)2
° ~ \ 27n,--------)
Note that o2 has the significance of being the mean value of the
square of the deviations of individual M; values from the mean M.
Accordingly, î is sometimes called the root mean square (rms) deviation.
In both Eqs. (1.3) and (1.4), the summations are carried out over all
classes of data. From a computational point of view, standard deviation
may be written in a more convenient form by carrying out the following
operations. First both sides of Eq. (1.4) are squared; then the
difference M; - M is squared to give
, SiNjM;2 _ SjNjM:
a2 = -- - 2M --
Z. N- Z- N-
"l A 1 1 1 1
Recalling the definition of the mean, we recognize the first
term on the right-
hand side of Eq. (1.5) to be the mean value of M2 and write
a2 = M2 - 2M2 +M2
It is important to realize that M2^ M2. An alternative to Eq. (1.4) as a
definition of standard deviation is, therefore,
î = (M2 - M2)1/z
We shall make use of this relationship presently.
Molecular Weight Averages
Since 2; Nj represents the total number of molecules Nt in the
population we are describing, each of the coefficients in Eqs. (1.3) and
(1.4) is the fraction fj of the total number of molecules in category i:
f= = -
Introducing this notation means that Eqs. (1.3) and (1.4) may be written
M = ZjfjMi
where the fractions f; are the weighting factors used in the definition
of the average. In the mean and standard deviation, the number fraction
is the weighting factor involved.
Finally we define a quantity known as the kth moment of the
distribution. In terms of molecular weight,
kth moment = 2jfi(M-Ms)k
The numerical value of the exponent ê determines which moment we are
defining, and we speak of these as moments about the value chosen for Ms.
Thus the mean is the first moment of the distribution about the origin
(Ms = 0) and o2 is the second moment about the mean (Ms = M). The
statistical definition of moment is analogous to the definition of this
quantity in physics. When Ms = 0, Eq. (1.11) defines the average value of
Mk; this result was already used in writing Eq. (1.6) with ê = 2.
Throughout this discussion we have used the numerical fraction of
molecules in a class as the weighting factor for that portion of the
population. This restriction is not necessary; some other weighting
factor could be used equally well. As a matter of fact, one important
type of average encountered in polymer chemistry is the case where the
mass fraction of the ith component is used as the weighting factor.
Defining the mass of material in the ith class as ò|; we write
2: m, M:
mw = -V2-1
This quantity is called the weight average molecular weight, reflecting
the chemist's customary carelessness about distinguishing between mass
and weight, and is given the symbol Mw. By contrast, the mean, where
number fractions are used, is called the number average molecular weight
and is given the symbol Mn.
The Chains and Averages of Polymers
The mass of material in a particular molecular weight class is given
by the product of the class mark molecular weight and the number of