Physical Chemistry Third Edition

1200 28 The Structure of Solids, Liquids, and Polymers

molecular massMi, and sum the equations over all possible molecular masses. Since

Khas the same value for all molecules of a given polymer, we can write

[η]K

∑

i

WiMai (28.8-10)

where the sum is over different molecular masses. We define theviscosity-average

molecular mass:

〈M〉v

(

∑

i

WiMia

) 1 /a

(28.8-11)

Combining Eq. (28.8-10) and Eq. (28.8-11) gives

[η]K(〈M〉v)a (28.8-12)

The viscosity-average molecular mass is not equal to the mean molecular mass, which

is called thenumber-average molecular mass. There is also an average molecular

mass called themass-average molecular mass, in which each molecule is given an

importance in the average proportional to its mass. The viscosity-average molecular

mass is numerically more nearly equal to the mass-average value than to the number-

average value.

PROBLEMS

Section 28.8: Polymers in Solution

28.41Find the number-average and mass-average molecular
masses for a sample of a polyvinyl alcohol that has
molecules with the following percentages and molecular
masses:

10 .00% 5000 amu

25 .00% 15,000 amu

60 .00% 25,000 amu

5 .00% 35,000 amu

28.42Assume that the diameter of a molecule of polyvinyl
alcohol is equal to the root-mean-square end-to-end
distance. Assume that (since 6 carbon atoms can make a

ring) each set of three carbons constitutes a link in a

freely jointed chain, of length 2.57× 10 −^10 m. (Each link

constitutes one and one-half monomer units.)

a.Find the volume of a sphere containing one molecule

of polyvinyl alcohol of mass 60,000 amu.

b.Find the viscosity at 20◦C of a solution of such

molecules (all assumed identical) of 1.00 g L−^1 , using

the Einstein equation, Eq. (28.8-3).

c.Find the viscosity at 20◦C of the solution of part a

using Eq. (28.8-9) and assuming that the limiting

value of the specific viscosity can be used at this

concentration.

28.43Find the viscosity-average molecular mass of the

polyvinyl alcohol sample of Problem 28.41.

28.9 Rubber Elasticity

Rubber is a naturally occurring polymer, first used as pencil erasers, which was the

origin of the English name. It is an addition polymer of isoprene (2-methyl-

1,3-butadiene). In forming an addition polymer, a double bond “opens up” to link

each monomer to the chain without forming a second product. Since each molecule

of the monomer has two double bonds, there remains one double bond per monomer