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The ideal gas law $PV = nRT = Nk_BT$ connects macroscopic state variables. It emerges from kinetic theory and encapsulates four empirical gas laws: Boyle's ($P \propto 1/V$ at constant $T$), Charles's ($V \propto T$ at constant $P$), Gay-Lussac's ($P \propto T$ at constant $V$), and Avogadro's (equal volumes at same $T, P$ contain equal numbers of molecules).

Important consequences: (1) At constant $T$, $P\rho$ is constant (since $\rho \propto P$). (2) At constant $P$, $\rho \propto 1/T$. (3) The number density $n = N/V = P/(k_BT)$ depends on $P$ and $T$ but not on the type of gas.

Real gases deviate from ideality at high pressure (molecules too close, volume matters) and low temperature (intermolecular forces significant). The van der Waals equation $(P + a/V^2)(V - b) = nRT$ corrects for both: $a$ accounts for attractive forces (reduces effective pressure), $b$ for molecular volume (reduces available volume). For JEE, know when ideal behavior breaks down and qualitatively what corrections are needed, but quantitative van der Waals calculations are rare.