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molecules in the class:
Substituting this result into Eq. (1.12) gives
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;
The left-hand side of this expression equals M2, so Eq. (1.16) may also
be written as
Molecular Weight Averages
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.
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).
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
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)=
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"
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
(mol) (g) (g2 mol"1) (g2 mol-2) (g2 mol-
0.003 10,000 30 3.0 25
0.008 12,000 96 11.5 9
0.011 14,000 154 21.6 1
0.017 16,000 272 43.5 1
0.009 18,000 162 29.2 9
0.001 20,000 20 4.0 25