Thermal expansion coefficient and third law of thermodynamics Thermal expansion coefficient is defined as:
\begin{equation}
\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_p
\end{equation}
We can prove through the third law of thermodynamics that:
\begin{equation}
\lim_{T\to0}\alpha=0
\end{equation}
Now, consider ideal gases equation:
\begin{equation}
pV=nRT\implies \alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_p=\frac{1}{V}\left(\frac{\partial}{\partial T}\frac{nRT}{p}\right)_p=\frac{nR}{pV}=\frac{1}{T}
\end{equation}
So we have:
\begin{equation}
\lim_{T\to0}\alpha=\lim_{T\to0}\frac{1}{T}=+\infty
\end{equation}
How do we justify this contradiction? As far as I know the Third Law is strongly based on experimental evidences and experiment-based assumptions (well, this is what I know from an introdoctury physics course). Also since ideal gases are a good approximation for real gases only at high temperatures, can we conclude this is the reason why $\alpha$ does not go to $0$?
On the other hand, if the third law had been based on ideal models first, then what I found above would be contradictory. Is this right?
 A: A real gas will condense if you get close enough to $T=0$, no matter how low the pressure, so in practice there won't be a conflict. But this is an unsatisfying answer.
The basis of the third law is that a system at absolute zero has zero entropy, because there is only one possible lowest-energy state. In the case of an ideal gas at absolute zero, all its point-like, non-interacting particles will be collapsed at a single point with zero volume. This is an unphysical state.
The specific heat of a gas is another example where you would expect $C_p\rightarrow 0$ whereas it is constant for a perfect gas (a perfect gas is an ideal gas with a constant specific heat, so it's even more restricted than an ideal gas).
The escape from this contradiction according to The Wiki (no references provided unfortunately) is that Bose-Einstein or Fermi-Dirac statistics start becoming relevant for ideal/perfect gases near absolute zero.
The surprise is that statistical mechanics and thermodynamics are not complete without quantum mechanics.
