Physical Chemistry Third Edition

28.7 Polymer Conformation 1197

We transform the equation to spherical polar coordinates, in whichpis a function of

r,θ, andφ. However, all directions in space are equivalent, so thatpcannot depend on

θandφ, and we writepp(n,r). From the expression for the∇^2 operator in spherical

polar coordinates in Eq. (B-45) in Appendix B,

∂p

∂n



a^2

6 r^2

∂

∂r

(

r^2

∂p

∂r

)

(28.7-17)

The solution to Eq. (28.7-17) is^39

p(n,r)

(

3

2 πna^2

) 3 / 2

e−^3 r

(^2) / 2 na 2
(28.7-18)
where the constant (3/ 2 πηa^2 )^3 /^2 provides for normalization:
∫∞
0
p(n,r) 4 πr^2 dr 1 (28.7-19)
Exercise 28.13
a.Carry out the substitution of the Taylor series into Eq. (28.7-1) to obtain Eq. (28.7-16).
b.Verify that the function of Eq. (28.7-18) satisfies Eq. (28.7-17).
c.Verify that the function of Eq. (28.7-18) is normalized.
The freely jointed chain that we have discussed is only a crude first approximation
for real polymers. Every real polymer has some rigidity built into its bonds so that
the chain is not freely jointed. We have also ignored the problem ofexcluded volume,
which means that two parts of a polymer chain cannot occupy the same location at
the same time. We have also ignored the effects of intermolecular attractions on the
conformation. Any of the books on polymer chemistry contain more elaborate theories.
The Polymer Science Department of the University of Southern Mississippi maintains
theMacrogalleriawebsite (http://www.pslc.usm.edu/macrog), which is a good source
of information about polymers.
There are many naturally occurring polymers. Proteins, which are polymers of
amino acids, can form intramolecular hydrogen bonds that hold the chains in a helical
conformation or a pleated sheet conformation. The proper conformation is essential
to the biological function of the molecule. If the molecule is transformed into a more
random conformation, it loses its biological function and is said to bedenatured.
Nucleic acids are polymers of five-carbon sugars (either ribose or deoxyribose),
phosphoric acid residues, and certain ring-containing molecules called nitrogen bases.
Deoxyribonucleic acid (DNA) is held in a double helix of two chains by hydrogen
bonds between specific pairs of bases: cytosine (C) hydrogen-bonds to guanine (G),
and adenine (A) hydrogen-bonds to thymine (T), so that in an intact DNA molecule a
C must be opposite every G on the other chain and a T must be opposite every A on
the other chain.
(^39) F. T. Wall,op. cit., p. 341ff (note 37).