Problem 3.34
Polymers, like rubber, are made of very long molecules, usually tangled up in a configuration that has lots of entropy. As a very crude model of a rubber band, consider a chain of 
links, each of length 
(see Figure 3.17). Imagine that each link has only two possible states, pointing either left or right. The total length 
of the rubber band is the net displacement from the beginning of the first link to the end of the last link.
(a) Find an expression for the entropy of this system in terms of and 
, the number of links pointing to the right.
(b) Write down a formula for in terms of and 
(c) For a one-dimensional system such as this, the length is analogous to the volume 
of a three-dimensional system. Similarly, the pressure 
is replaced by the tension force 
Taking 
to be positive when the rubber band is pulling inward, write down and explain the appropriate thermodynamic identity for this system.
(d) Using the thermodynamic identity, you can now express the tension force in terms of a partial derivative of the entropy. From this expression, compute the tension in terms of 
and 
(e) Show that when 
the tension force is directly proportional to (Hooke’s law).
(f) Discuss the dependence of the tension force on temperature. If you increase the temperature of a rubber band, does it tend to expand or contract? Does this behavior make sense?
(g) Suppose that you hold a relaxed rubber band in both hands and suddenly stretch it. Would you expect its temperature to increase or decrease? Explain. Test your prediction with a real rubber band (preferably a fairly heavy one with lots of stretch), using your lips or forehead as a thermometer. (Hint: The entropy you computed in part (a) is not the total entropy of the rubber band. There is additional entropy associated with the vibrational energy of the molecules; this entropy depends on 
but is approximately independent of .)
Solution:
Problem 3.34 Solution (Download)
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