Page 101 - Carrahers_Polymer_Chemistry,_Eighth_Edition

64 Carraher’s Polymer Chemistry

5
5
5
5
5
molecular weights of 1.00 × 10 , 2.00 × 10 , and 3.00 × 10 is (6.00 × 10 )/3 = 2.00 × 10 . Recalling
that W = Σ W = Σ M N the general solution is shown mathematically as
i i i
i
M n = total weight of sample = W = ∑ MN i (3.6)
number of molecules of N i ∑ N i ∑ N i
Most thermodynamic properties are related to the number of particles present and thus are
dependent on M .
n
Colligative properties are dependent on the number of particles present and are thus related to M .
n
M values are independent of molecular size and are highly sensitive to small molecules present in the
n
mixture. Values of M are determined by Raoult’s techniques that are dependent on colligative proper-
n
ties such as ebulliometry (boiling point elevation), cryometry (freezing point depression), osmometry,
and end-group analysis.
Weight-average molecular weight, M , is determined from experiments in which each mole-
w
cule or chain makes a contribution to the measured result relative to its size. This average is more
dependent on the number of longer chains than is the number-average molecular weight, which is
dependent simply on the total number of each chain.
The M is the second moment average and is shown mathematically as
w
∑ WM ∑ M N
2
M w = i i = i i (3.7)
∑ W i ∑ M N i
i
Thus, the M of the three chains cited above is
w
(1.00 10 ) + (4.00 10 ) + (9.00 10 ) = 2.33× 10 5
×
×
×
10
10
10
×
5
(6.00 10 )
Bulk properties associated with large deformations, such as viscosity and toughness, are most
closely associated with M . M values are most often determined by light-scattering photometry.
w
w
However, melt elasticity is more closely related to the third moment known as the z-average
molecular weight, M . The M is most often determined using either light-scattering photometry or
z
z
ultracentrifugation. It is shown mathematically as
∑ MN
3
M z = i 2 i (3.8)
∑ MN i
i
5
The M value for the three polymer chains cited above is 2.57 × 10 :
z
×
15
(1 10) + (8 10) + (27 10) = 2.57× 10 5
15
15
×
×
10
×
×
10
10
×
[(1 10) + (4 10) + (9 10)]
While Z + 1 and higher average molecular weight values can be calculated, the major interests are
in M , M , M , and M , which is the order of increasing size for a heterodisperse polymer sample
n
w
z
v
as shown in Figure 3.9. Thus, for heterogeneous molecular weigh systems M > M > M . The ratio
z
w
n
of M / M is called the polydispersity index. The most probable polydispersity index for polymers
n
w
produced by the condensation technique with respect to molecular weight is 2. As the heterogeneity
decreases, the various molecular weight values converge until M = M = M .
w
n
z
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