Deriving a relation between entropy, volume and particles through the chemical potential

I'm a teaching assistant on a first year course on Introduction to Thermodynamics based on the book "Molecular Driving Forces" by Dill & Bromberg.


There's a question which asks the students to derive the following relation between S, V and N:


$S(V,N) = Nkln(V)$


A simple solution that is hinted is to use the differential form of the fundamental entropy equation and realizing the following identities (assuming an ideal gas):


$\frac{P}{T} = (\frac{\partial S}{\partial V})\_{U,N}=\frac{Nk}{V}$


$ds = Nk\int \frac{dV}{V} \rightarrow \Delta S = Nkln(\frac{V\_2}{V\_1})$


Now, I've meditated about and have been asked by several students the following question: could we arrive to the same (or equivalent) expression through a chemical potential route? After all, we need an expression that relates S *both* to N and V. Unfortunately, I have not been able to give a satisfactory answer.


Here's what I've tried, arriving to some ugly looking equation that intuitively looks incorrect to me:


Use the definition:
$\frac{\mu\_j}{T}=-(\frac{\partial S}{\partial N\_j})\_{U,V,N\_{j\neq i}}$, which also stems from the fundamental differential entropy equation.


Now I make the assumption that the chemical potential is equal to the density of molecules in the ideal gas. This is what I'm most dubious about, although I couldn't think of another way in which to have a relation to it. If that holds, I can then write the following identities:


$\mu = \frac{n}{V}$ then $P=\mu kT$


$\frac{-P}{T^2k} = \frac{\mu}{T} = (\frac{\partial S}{\partial N\_j})\_{U,V,N\_{j\neq i}}$


Now because $P = \frac{nkT}{V}$,


$\frac{-P}{T^2k} = \frac{-nkT}{VT^2k}=\frac{-n}{VT}=(\frac{\partial S}{\partial N\_j})\_{U,V,N\_{j\neq i}}$


Which I think can be solved:


$\int dS = \int \frac{-n}{VT}dn \rightarrow \Delta S = -\frac{\Delta n^2}{2VT}$


This result doesn't really make much sense to me (a negative entropy (?)). I'm a bit confused, because the first 'route' seems so simple but this second attempt looks fairly involved.


So, what I wrong in my logic? Apologies beforehand if I'm doing something egregiously wrong, it's been awhile since I've had to solve these 'pen and paper' problems. 


