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

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molecules in the class:

Substituting this result into Eq. (1.12) gives

2-N-M-2 2-f-M:2

M =

(1.14)

w 2, NiMi ZjfjM,

which shows that the weight average molecular weight may also be regarded

as the ratio of the second moment of the distribution to the first

moment, where each of these moments are taken with respect to the origin

of the distribution.

The weight average molecular weight of a distribution will always be

greater than the number average. This is true because the latter merely

counts the contribution of molecules in each class, whereas the former

weights their contribution in terms of mass. Thus those molecules with

higher molecular weights contribute relatively more to the average when

mass fraction rather than number fraction is used as the weighting

factor. For all polydisperse systems

and the amount by which this ratio deviates from unity is a measure of

the polydispersity of a sample. In the event that all of the molecules in

a sample have the same molecular weight, the summations in Eqs. (1.3) and

(1.4) would each consist of a single term and their ratio would equal

unity. Such a sample is said to be monodisperse.

In connection with Eq. (1.4), we noted that the standard deviation

measures the spread of a distribution; now we see that the ratio Mw/Mn

also measures this polydispersity. The relationship between these two

different measures of polydispersity is easily shown. Equation (1.14) may

be written as

m; = Nj M;

(1.13)

(1.15)

(1.16)

The left-hand side of this expression equals M2, so Eq. (1.16) may also

be written as

Molecular Weight Averages

39

Substituting this result into Eq. (1.7) gives

This result shows that the square root of the amount by which the ratio

Mw/Mn exceeds unity equals the standard deviation of the distribution

relative to the number average molecular weight. Thus if a distribution

is characterized by Mn = 10,000 and a = 3000, then Mw/Mn = 1.09.

Alternatively, if Mw/Mn = 1.50, then the standard deviation is 71% of the

value of Mn. This shows that reporting the mean and standard deviation of

a distribution or the values of Mn and Mw/Mn gives equivalent information

about the distribution. We shall see in a moment that the second

alternative is more easily accomplished for samples of polymers. First,

however, consider the following example in which we apply some of the

equations of this section to some numerical data.

Example 1.5

The first and second columns of Table 1.4 give the number of moles of

polymer in six different molecular weight fractions. Calculate Mn and Mw

for this polymer and evaluate î using both Eqs. (1.7) and (1.18).

Solution

Evaluate the product nf Mj for each class; this is required for the

calculation of both Mn and Mw. Values of this quantity are listed in the

third column of Table 1.4. From 2;Ï;Ì; and Mn = 734/0.049 = 15,000. The

matter of significant figures will not be strictly adhered to in this

example.

The products irij M; are mass-weighted contributions and are listed in

the fourth column of Table 1.4. From iti; and l,i m; Mj5 Mw = 113 X

10s/734 = 15,400.

The ratio Mw/Mn is found to be 15,400/15,000 = 1.026 for these data.

Using Eq. (1.18), we have a/Mn =(1.026- \)Vl =0.162 or a = 0.162(15,000)=

2430.

To evaluate î via Eq. (1.7), differences between Mj and M must be

considered. The fifth and sixth columns in Table 1.4 list (M; - Mn)2 and

Nj(Mi - Mn)2 for each class of data. From 2; Nj and 2j Nj (Mj - Mn)2, a2

=

28.1 X 104/ 0.049 = 5.73 X 106, and a = 2390.

The discrepancy between the two values of î is not meaningful in terms

of significant figures. The standard deviation is 2400.

We shall see that, as polymers go, this is a relatively narrow

molecular weight distribution.

a = (MWM"-M2)'^ = M"

\ M

\ n

Î

Table 1.4 Some Classified Molecular Weight Data for a Hypothetical

Polymer Used in Example 1.5

N; (g mol 1) mi rrij Mj X 10"s (M; - M)2 X 10"6 Nj (Mj - M)2

X 10"4

(mol) (g) (g2 mol"1) (g2 mol-2) (g2 mol-

1)

0.003 10,000 30 3.0 25

7.50

0.008 12,000 96 11.5 9

7.20

0.011 14,000 154 21.6 1

1.10

0.017 16,000 272 43.5 1

1.70

0.009 18,000 162 29.2 9

8.10

0.001 20,000 20 4.0 25

2.50

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