Discussion:
Derive hooke's law and find the "elastic entropy"
Task:
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<