Thermondynamics: Ideal Rubber Band

[alfredbester]

Set-up: Rubber band modeled as an Ideal chain (polymer) with Nr (links) monomers pointing right, and Nl monomers pointing left each of length l.
Find expressions for the length of the band, its statistical weight (iI think that means the multiplicity?) and its entropy, in terms of Nr and Nl.
The length L is just, L = (Nr + Nl)l
Statistical Weight Ω(S(R))
Entropy is given by S(R) = k.ln (Ω(S(R)))
where R is a vector representing the net displacement of the ends.
I'm not sure how to get the statistical weight. I know the general formula is
Ω(N, a) = (a + N -1) / a!(N-1)!, is a = l in my case?

 

[StatusX]

You want to figure out how many different values of N_r and N_l give each R. For example, if N_r=N and N_l=0 (ie, all the links are pointing right), then the net displacement is Nl. But this is the only way to get a displacement of Nl, so it has a degeneracy (multiplicity) of 1, and so an entropy of 0. The formula you mentioned at the end is close to the one you'll need. Look up the binomial distribution.

 

[alfredbester]

Had a look at it, and this is the best I could come up with.

Ω(Nr - Nl, N) = N! / (Nr!.Nl!),
using stirling's approximation I got this to be

Ω(Nr - Nl, N) = 2^N. exp(-(Nr - Nl)^2 / 2N)

 

[alfredbester]

Got that wrong, is it?

Ω(N, Nr) + Ω(N, Nl) = 2N! / {(0.5N + Nr)!(0.5N + Nl)!}

 

[StatusX]

Your first equation looks right. Stirling's approximation, as I learned it, is ln(n!) = n ln(n) - n, which would mean n! = (n/e)^n (those should both be approximately equals signs). This can be used to get an estimate for the number of states and the entropy. Your second equation may be correct, but I'm not familiar with whichever form of stirlings approximation you used (there are a few of them). The equation in your most recent post doesn't really make sense to me.

 

[crakadikt]

Greetings! I know this is an old thread but...

I would like to know how to get from

Ω(Nr - Nl, N) = N! / (Nr!.Nl!)

to

Ω(Nr - Nl, N) = 2^N. exp(-(Nr - Nl)^2 / 2N)

using Sterling's approximation or otherwise. I have been running into this wall:

S=k ln Ω(Nr - Nl, N) => S = k (N ln (N/(N-Nr)) + R ln ((N-Nr)/R))
after Sterling's approximation and subbing Nl=N-Nr.

This yields a (partials) dS/dNr = k ln ((N-Nr)/Nr).

I understand that f = -T (dNr/dL) (dS/dNr). Where the length, L = 2Nr - N.

So I expect dS/dNr to yield L under proper conditions. I don't see that happening, though. Where'd I go wrong? (Aside from choosing Physics...)

 

[crakadikt]

(Emoticons?!? How 'bout greek letters instead?)

Nevermind. I got it. It requires a more complete version of Sterling's Approximation:

N!= (N^N)*(e^-N)*sqrt(2*pi*N) <--- where this last square root term contains the L dependence (eventually).

Later.