- #1
Dixanadu
- 254
- 2
Homework Statement
Hey guys,
Here's the question. For a distinguishable set of particles, given that the single particle partition function is [itex]Z_{1}=f(T)[/itex] and the N-particle partition function is related to the single particle partition function by [itex]Z_{N}=(Z_{1})^{N}[/itex] find the following:
(a) The grand canonical partition function
(b) The entropy
(c) Prove that the entropy is given by
[itex]\frac{S}{k}=N[\frac{Tf'(T)}{f(T)}-\log z]-\log(1-zf(T))[/itex] where [itex]z=e^{\beta\mu}[/itex] is the fugacity.
Homework Equations
Grand particle partition function
[itex]Z=\sum_{N=0}^{\infty}z^{N}Z_{N}[/itex]
Entropy
[itex]S=(\frac{\partial(kT \log Z)}{\partial T})_{\beta,V}[/itex]
(i found this myself so it might not be 100% right)
The Attempt at a Solution
So I've done everything but I am struggling with part C:
(a) [itex]Z=\frac{1}{1-zf(T)}[/itex]
(b) Using that formula I found, i get [itex]\frac{S}{k}=\frac{Tzf'(T)}{1-zf(T)}-\log (1-zf(T))[/itex]
for part (c), i don't know how I am meant to get from what I have to what's required. Basically, i don't see how
[itex]\frac{Tzf'(T)}{1-zf(T)}=N[\frac{Tf'(T)}{f(T)}-\log z][/itex]
Thats pretty much all i need help with...but if you guys need more info just let me know! thanks a lot!