The ideal gas law is perhaps the best-known equation of state, and admits both a derivation via the kinetic theory of gases and via statistical mechanics. But these are both microscopic theories, and the ideal gas law is a macroscopic equation.
- Can the ideal gas law be derived from macroscopic thermodynamics alone, without recourse to microscopic theories? If so, how? If not, why not?
Assume now that $$U = \frac{3}{2}Nk_\text{B}T,$$ as is the case for a monatomic ideal gas.
- Can we now derive the ideal gas law?
Perhaps something to consider regarding the second question. From thermodynamics, $$P = -\left(\frac{\partial U}{\partial V}\right)_{S,N} = -\frac{3}{2}Nk_\text{B}\left(\frac{\partial T}{\partial V}\right)_{S,N} \stackrel{?}{=} \frac{Nk_\text{B}T}{V} \quad \Longleftrightarrow \quad \left(\frac{\partial T}{\partial V}\right)_{S,N} \stackrel{?}{=} -\frac{2}{3}\frac{T}{V},$$ where $\stackrel{?}{=}$ is read "is to be shown equal to." This is an interesting identity, I suppose, but I'm not sure if or where I can proceed from here. As usual, any help is appreciated.