Polymers, like rubber, are made of very long molecules that tangle into a configuration that has lots of S. A crude model of a rubber band contains N links, all of equal length L, which can only point either left or right (2 possible states). The total length of the band is thus the net displacement from the first to the final link. Let us to solve it step by step, and derive Hooke's law and find "elastic entropy". (a) Find an expression for the entropy of the rubber band in terms of the total number of links, N, and of the number that point 'right', Nright. (The multiplicity is as straight as u hope and use Stirling.) (b)What is L in terms of N and Nright.( In other words, eliminate Nleft from this expression). For a 1- dim system like this, we can make an analogy with P and V for a 3-dim system : V becomes L, and P becomes F. Take F, or the tension force, if u will, to be positive when the rubber band is pulling 'back'(inwards).What is the thermodynamic identity of this system-in other words, what is the re-written version of the 1st law? Comment on this. Using this identity, you can now find an expression for F in terms of a derivative involving S(dU=0). Now expand this partial derivative by putting in the following term,
∂Nright./∂Nright which is obviously unity and does not change the value of the derivative. Using your result from (b), you should now have a (1/2L) term. Use part (a) and apply the derivative w.r.t Nright now and not L, to find the tension force = f(L, T, N, Nright). Show that when L<<NL, F has a very familiar form. What is the value of the spring constant, K ? Discuss the dependence of the tension force on T. If you increase the T of a rubber band, does it (tend to) expand or contract? Comment! We did not find the total S above ; we ignored the vibrational entropy, which depends on E but not L. Stretch and contract a good rubber band, and use your lips as a probe. Comment !
The last part of your posting was missing: ''Show that when L<...''. But I think that part is probably trivial once you understand how to deal with this problem.
To avoid confusion I will call the length of one link a, and use the symbol L for the length of a chain. Also, N_r = N_right, N_l = N_left. You can write the length as:
L = a|N_r - N_l|.
Since N = N_r + N_l you can rewrite this as:
L = a|2 N_r - N|.
The entropy is related to the number of ways you can arrange the links to get the same length. This is just the number of ways you choose N_r links from ...
A detailed solution is given.