Ideal gas

Remarks

In the microscopic picture, the gas consists of point particles with no interactions besides elastic collisions; with $N$ particles and the Boltzmann constant $k$, the equation of state reads $pV = NkT$. In thermodynamics, clause 2 follows from clause 1: the identity $(\partial U / \partial V)_T = T (\partial p / \partial T)_V - p$ gives $T \cdot nR/V - p = 0$. An ideal gas whose heat capacities are constant is called calorically perfect; for a monatomic gas $U = \tfrac{3}{2} nRT$.

Example: helium at room temperature and atmospheric pressure satisfies $pV = nRT$ to within a small fraction of a percent, so it is well described as an ideal gas.

Non-example: a van der Waals gas, with $(p + a n^2 / V^2)(V - nb) = nRT$ and $a, b > 0$, fails clause 1, and its internal energy $U = U_{\mathrm{ideal}}(T) - a n^2 / V$ depends on $V$, so it also fails clause 2.