# Weird form of entropy using grand partition function for a system

## Homework Statement

Hey guys,

Here's the question. For a distinguishable set of particles, given that the single particle partition function is $Z_{1}=f(T)$ and the N-particle partition function is related to the single particle partition function by $Z_{N}=(Z_{1})^{N}$ find the following:

(a) The grand canonical partition function
(b) The entropy
(c) Prove that the entropy is given by
$\frac{S}{k}=N[\frac{Tf'(T)}{f(T)}-\log z]-\log(1-zf(T))$ where $z=e^{\beta\mu}$ is the fugacity.

## Homework Equations

Grand particle partition function
$Z=\sum_{N=0}^{\infty}z^{N}Z_{N}$

Entropy
$S=(\frac{\partial(kT \log Z)}{\partial T})_{\beta,V}$
(i found this myself so it might not be 100% right)

## The Attempt at a Solution

So ive done everything but im struggling with part C:

(a) $Z=\frac{1}{1-zf(T)}$
(b) Using that formula I found, i get $\frac{S}{k}=\frac{Tzf'(T)}{1-zf(T)}-\log (1-zf(T))$

for part (c), i dont know how im meant to get from what I have to what's required. Basically, i dont see how

$\frac{Tzf'(T)}{1-zf(T)}=N[\frac{Tf'(T)}{f(T)}-\log z]$

Thats pretty much all i need help with...but if you guys need more info just let me know! thanks a lot!