Physical Chemistry Third Edition

1194 28 The Structure of Solids, Liquids, and Polymers

PROBLEMS

Section 28.6: Approximate Theories of Transport
Processes in Liquids

28.34The self-diffusion coefficient of liquid CCl 4 is given by
Rathbun and Babb:^36

tC/◦C25. 040. 050. 060. 0

D/ 10 −^9 m^2 s−^11. 30 1. 78 2. 00 2. 44

Find the value ofAdand the value ofEadin Eq. (28.6-9).

Assuming thata 4 × 10 −^10 m, find∆S‡◦and∆G‡◦.

28.35The viscosity of water at 20◦C is equal to

0.001005 kg m−^1 s−^1 , and at 50◦C it is equal to

0.0005494 kg m−^1 s−^1. Calculate the Arrhenius

activation energy for water’s viscosity.

28.36Using the data in Problem 28.34, calculate the value of

the correlation timeτin Eq. (28.6-16) for carbon

tetrachloride at 50◦C. Explain the temperature

dependence of this parameter.

28.7 Polymer Conformation

The simplest polymers have linear chain-like molecules, which can take on a large

number of possible conformations. The principal piece of information about the con-

formation of a polymer molecule is the end-to-end distance. Even if we had a monodis-

perse sample of a polymer (one in which all molecules had the same molecular mass)

there would be a distribution of end-to-end distances, because each molecule would

coil up differently from the others.

We approximate a chain-like polymer molecule by afreely jointed chain, which is a

model system consisting of a set of links of fixed lengthafastened together end-to-end

such that each joint can rotate into any orientation, even folding one link back onto the

previous link. Since singly bonded carbon atoms can form a ring of six atoms, each link

might represent a set of about three carbon atoms in a chain of singly bonded carbon

atoms. To simplify the problem even further we suppose that each link of the chain can

be directed in one of only six directions, parallel to thex,y, andzaxes of a Cartesian

coordinate system. We place one end of the chain at the origin, so that the ends of the

links can fall only on the lattice points of a simple cubic lattice with lattice spacinga,

very much like a crystal lattice.

The probability that the end of link numbern+1 is at a lattice point with Cartesian

coordinates (x,y,z) is denoted byp(n+1,x,y,z). If the end of link numbern+1is

at (x,y,z), then the end of link numberncan be at one of only six possible locations:

(x+a,y,z), (x−a,y,z), (x,y+a,z), (x,y−a,z), (x,y,z+a), or (x,y,z−a). We

assume that the probabilities of the six possible directions of a link are equal so that

p(n+1,x,y,z)

1

6

[p(n,x+a,y,z)+p(n,x−a,y,z)+p(n,x,y+a,z)

+p(n,x,y−a,z)+p(n,x,y,z+a)+p(n,x,y,z−a)]

(28.7-1)

Equation (28.7-1) is adifference equationthat can be solved.^37 We assume that the

beginning of the chain is at the origin:

p(0,x,y,z)

{

1ifx0,y0, andz 0

0 otherwise

(28.7-2)

(^36) R. E. Rathbun and A. L. Babb,J. Phys. Chem., 65 , 1072 (1961).
(^37) F. T. Wall,Chemical Thermodynamics, 2nd ed., W. H. Freeman, San Francisco, 1974, p. 341ff.