Where is the mistake in this derivation of $U(T)$ for an Ideal Gas? I have tried to develop this alternative derivation of the relationship between internal energy $U$ and temperature $T$ of an ideal monoatomic gas. From kinetic theory, it is well known that $PV=2/3U$ with $U$ being the internal energy, which for a single monoatomic gas is equal to $U=uN$ with $u$ being the specific internal energy of the ideal gas, which is just the mean kinetic energy per particle.
The derivation is as follows. From the fundamental thermodynamic relation of the internal energy for a single component system:
$$
U=TS-PV+\mu N
$$
so we can substitute $PV=2/3U$, and from the definition of chemical potential:
$$
\mu=\frac{\partial U}{\partial N}=\frac{\partial (uN)}{\partial N}=u
$$
thus
$$
U=TS-\frac{2}{3}U+uN=TS-\frac{2}{3}U+U=TS+\frac{1}{3}U
$$
which yields
$$
U=\frac{3}{2}TS
$$
The last term is strikingly similar to $U=N3/2k_BT$, and that would suggest that $S=Nk_B$.
However I know from a fact that the entropy of an ideal gas follows the Sackur-Tetrode equation, which appears to be different than $Nk_B$.
Where is the mistake? I can't wrap my head around this, which I stress IS NOT HOMEWORK.
 A: Have a look at Sackur-Tetrode entropy, there is Planck constant. This intuitively tell us that we can't derive the complete form from pure thermodynamics arguments. But we can come close to it. We know that entropy can be written as a function of extensive variable $S(U, V, N)$. We can write the differential of the entropy as sum of partial derivative respect to the proper variable times the respective increment:
$$ dS = \frac{\partial S}{\partial T} dT +  \frac{\partial S}{\partial V}dV + \frac{\partial S}{\partial N}dN $$
Now it's simple integration, using Maxwell relations, the heat capacity definition and the ideal gas law. Then it is possible to write the equation in a similar form to the Sackur-Tetrode, remembering to impose the estensivity of the entropy. The structure of the equation is:
$$ \frac{S}{Nk} = \ln(\frac{f(U, V, N)}{C}) $$
The function $f(U, V, N)$ is known and can be seen doing the calculation. I have put all the constant in C, that clearly remain undetermined. Planck constant comes in the derivation of Sackur-Tetrode using Statistical Mechanics.
