Physical Chemistry Third Edition

1198 28 The Structure of Solids, Liquids, and Polymers

PROBLEMS

Section 28.7: Polymer Conformation

28.37Show that Eq. (28.7-10) is compatible with the fact that a
“chain” of one link is rigid.

28.38For a freely jointed chain with links 3× 10 −^10 m
long, find the root-mean-square end-to-end distance
and the ratio of this length to the sum of the link
lengths for
a. 100 links
b. 10000 links

28.39For a freely jointed chain with links 3. 5 × 10 −^10 m long,

find the root-mean-square end-to-end distance and the

ratio of this length to the sum of the link lengths for

a.1000 links

b.100,000 links

28.40Assume that a polymer chain is constrained to lie in a

plane. Find the formula for the root-mean-square

end-to-end distance using the same approximations as

were used to derive Eq. (28.7-13).

28.8 Polymers in Solution

The conformation of polymer molecules in solution is affected by intermolecular forces

between polymer molecules and solvent molecules and between solvent molecules

and other solvent molecules. Polar polymer molecules will tend to form tight balls in

nonpolar solvents, but can attract solvent molecules and can swell in polar solvents.

A nonpolar polymer molecule will attract polar solvent molecules less strongly than the

polar solvent molecules attract each other and will tend to form a tight ball in a polar

solvent such as water. Similarly, nonpolar polymers can swell in nonpolar solvents (try

placing a rubber object in benzene or toluene).

Theexpansion coefficientαis defined by the relation

〈r^2 〉α^2 〈r^2 〉 0 (28.8-1)

where〈r^2 〉 0 is the mean-square end-to-end distance in the pure polymer and〈r^2 〉is the

mean-square end-to-end distance of the polymer in the solution. A solvent in whichα

is equal to 1 is called atheta solvent. In a poor solvent for the particular polymer,αwill

be smaller than unity, and in a solvent that causes the polymer to swell,αwill exceed

unity.

In a typical polymer solution, a polymer molecule and its associated solvent molecules

will occupy a roughly spherical region in space with a diameter approximately equal

to the end-to-end distance of the molecule, and will move through a solution in much

the same way as would a sphere of that size. If the total volume of the spheres in a

solution is denoted byvand the volume of the entire suspension is denoted byV, the

volume fraction of the spheres in the solution is denoted byφ:

φv/V (28.8-2)

Einstein solved the hydrodynamic equations for flow around hard spheres in a dilute

suspension in a viscous fluid. His result was that the viscosity of a suspension of hard

spheres is given by^40

ηη 0

(

1 +

5

2

φ

)

(28.8-3)

(^40) A. Einstein,Ann. Physik, 19 , 289 (1906). Einstein’s productivity around the year 1905, when he was
pursuing theoretical physics in his spare time, is truly astounding.