Entropy of the ideal gas - How to derive the $S(U,V,N)?$ I'd like to derive the $S(U,V,N)$ function. In the lecture, we were using the $S=k_b \log(\Omega)$, some combinatorics, approximation, $6N$ dimensional spheres, etc. But I'd like to avoid that if it's possible. So I think I should do it from the first and second law of thermodynamics:
$$\mathrm{d}U=T\mathrm{d}S-p\mathrm{d}V+\mu\mathrm{d}N$$
$$\mathrm{d}S=\frac{1}{T}\mathrm{d}U+\frac{p}{T}\mathrm{d}V-\frac{\mu}{T}\mathrm{d}N$$
And with $pV=Nk_bT$ and $U=\frac{f}{2}Nk_b T$, I could get that
$$\mathrm{d}S=\frac{f}{2}k_b \frac{N}{U}\mathrm{d}U+k_b\frac{N}{V}\mathrm{d}V-\frac{\mu}{T}\mathrm{d}N$$
But I can't get rid of the $\frac{\mu}{T}$. How could I do it?
 A: From the definition of chemical potential:
\begin{equation}\mu = \left(\frac{\partial U}{\partial N}\right) = cRT \end{equation}
where c = 3/2 for a monatomic ideal gas, c = 5/2 for a diatomic ideal gas.
To verify the consistency of the derivation:
From the “fundamental equation” of an ideal gas:
\begin{equation} PV = N RT \end{equation}
\begin{equation} U=c N RT\end{equation}
if you rewrite the equations of state as
\begin{equation} \frac{1}{T}=\frac{cN R}{U}=(∂S/∂U)_{V,N}\end{equation}
\begin{equation} PT=\frac{NR}{V}=(∂S/∂V)_{U,N} \end{equation}
By integrating both, the first equation gives:
\begin{equation} S(U, V, N) =c N R \ln(U) +f(V, N)\end{equation}
and the second 
\begin{equation} S(U, V, N) =N R \ln(V) +g(U, N) \end{equation}
Which are consistent only if we can write S in the form:
\begin{equation}S(U, V, N) =cN R \ln(U) +N R \ln(V) +f(N) \end{equation}
where $f(N)$ is a function of N alone namely:
\begin{equation}f(n) = c RN \end{equation}
which accounts also for the fact that $S$ must be extensive in $U, V$, and $N$.
the arguments of the logarithms must be dimensionless (otherwise, the logarithm doesn’t make sense)
\begin{equation}S(U, V, N) =cN R \ln \frac{U}{Nk_1} +N R \ln \frac{V}{Nk_k2} +f(N) \end{equation}
Here $k_1$ and $k_2$, are constants with the dimensions of energy per mole, volume per mole
