Chemical Forums

Chemistry Forums for Students => Physical Chemistry Forum => Topic started by: Vrig on September 26, 2013, 12:52:47 PM

Title: Hooke's law and some algebra
Post by: Vrig on September 26, 2013, 12:52:47 PM

I got a question at school which I can't solve for the life of me.

"A one-dimensional linear chain of N links, where each link may be folded to the left or right and of length l may be taken as a simple model of a rubber band. The length, L, of the rubber band is L = (N_l - N_r)l and of course N = N_l + N_r. For convenience we say that N_r =< N_l. The weight is then W = N!/(N_r!N_l!).
a) Write the entropy of such a system.
b) Obtain the tending force, F (quivalent of pressure), and show that it obeys Hooke's law for small extensions, L << Nl. (Note that it says N*l and not N_l)

F = -T(dS/dL)|N"

For a) I just did S = k ln[W], where W is the weight.
For b) I've tried various things to see if it gets me anywhere, such as substituting N! (from the weight), deriving and using Stirling approx. But,.. yeah.. I simply don't know!

Any help (in any form or shape) would be much appreciated!

Title: Re: Hooke's law and some algebra
Post by: Enthalpy on October 04, 2013, 09:09:35 AM

I'd try a slightly different way:

with tension F, the energy of each link is E=2*F*I higher in the folded position than extended position.
The statistics is 1/[1+exp(+E/kT)] for each link to have the higher energy.
For N big, L is just N times the mean value of extended minus folded contributions.

Well, probably not what the professor expected.

Title: Re: Hooke's law and some algebra
Post by: Yggdrasil on October 04, 2013, 11:51:35 AM

Try doing a Taylor expansion of F around zero.