# Polymer Chemistry. The Basic Concepts - Himenz P.C.

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gives N as a function of M. This is the most general way of describing

the

36

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

function.

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,--------)

(1'4)

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 --

+M2 (1.5)

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

(1.6)

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

(1.7)

We shall make use of this relationship presently.

Molecular Weight Averages

37

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:

Ni

f= = -

(1.8)

1 Nt

v

Introducing this notation means that Eqs. (1.3) and (1.4) may be written

as

M = ZjfjMi

(1.9)

and

o= [ZjfKMj-Si)2]*

(1.10)

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

(1.11)

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

(U2>

2imi

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.

38

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

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