Hard thermo question.

[mewmew]

Well, it's hard to me at least
http://img169.imageshack.us/img169/206/questionzg0.png [Broken]
We are using Schroeders book and this problem was given to us from I assume the book by Reif. We haven't seem to have gotten far enough along in Schroeders book to have gone over this though.

I can do part A, but part B and C I am clueless on. I know that T = partial derivative of U/S but it doesn't seem to be helping me very much. Thanks in advance for any pointers.

 

[mewmew]

Ok, I found B more or less, by taking n=E/e. My equation for T looks pretty nasty because my multiplicity from part A was relatively nasty. I am still pretty much completely stuck on how to do C though.

 

[quasar987]

What do you get for A? I get

$$\Omega(N,n)=\binom{N}{N-n}=\frac{N!}{n!(N-n)!}$$

(the number of ways to pick the position of the N-n atomes that are not in an interstititiisal position.

Temperature is $$\frac{1}{kT}=\frac{\partial \Omega(N,n)}{\partial E}$$.

I guess you said, $$n=n(E)=E/\epsilon$$, so then

$$\frac{\partial \Omega(N,n)}{\partial E}=\frac{\partial \Omega(N,n)}{\partial n}\frac{\partial n}{\partial E}= \frac{\partial \Omega(N,n)}{\partial n}\frac{1}{\epsilon}$$

But do you know how to take the derivative of a factorial?

 

[mewmew]

Sorry, I Should have been more specific. I took a Sterling approximation of the factorials of the form Ln(x!)= x Ln(x) - x, expanded each term, substituted in n = E/epsilon and differentiated. This was messy though and I am not sure the desired method.