Q.Which of the following is the correct expression for the equation of state of van der Waals gas?

J. D. van der Waals corrected the ideal gas equation for the two assumptions kinetic theory makes that no real gas actually satisfies -- negligible molecular volume and zero intermolecular attraction -- by adding a pressure correction and a volume correction.

Pressure correction. A molecule about to strike the container wall is pulled back very slightly by the attraction of its neighbours (a molecule surrounded on all sides instead feels balanced, cancelling attraction), so it hits the wall a little less forcefully than it would with no attraction at all -- the measured pressure of a real gas is therefore somewhat lower than the true "ideal" pressure. This attractive effect scales with the square of the gas density, giving a correction term $\dfrac{an^2}{V^2}$ (where a is the first van der Waals constant, larger for more strongly attracting molecules), so $P_{ideal}=P+\dfrac{an^2}{V^2}$.

Volume correction. Because real molecules occupy actual space and cannot overlap, the volume genuinely available for them to move in is less than the container volume V. Modelling molecules as hard spheres of radius r, the excluded volume around a colliding pair works out to $8V_m$ (eight times a single molecule's own volume $V_m$), giving an excluded volume per molecule of $4V_m$; for n moles this totals $nb$, where $b=4V_m$ is the second van der Waals constant (larger for physically bigger molecules). The corrected, genuinely available volume is $V_{ideal}=V-nb$.

The van der Waals equation. Substituting both corrections into $PV=nRT$ gives

Both a and b are gas-specific constants: a larger 'a' means stronger intermolecular attraction (and, in particular, a gas is more easily liquefied the larger its 'a' value is, since strong attraction is exactly what pulls molecules together into a liquid); a larger 'b' means physically bigger molecules. Since $P'=an^2/V^2$ has units of pressure, 'a' carries units of $\text{atm}\,\text{L}^2\text{mol}^{-2}$ (or $\text{L}^2\text{atm}\,\text{mol}^{-2}$), and since $nb$ has units of volume, 'b' carries units of $\text{L}\,\text{mol}^{-1}$. The van der Waals equation is a substantial improvement over the plain ideal gas equation for describing real-gas behaviour, though it remains an approximate, not exact, description.

"Van der Waals equation of state derivation and constants a and b" and "van der Waals equation class 11 chemistry important questions" are frequent searches tied to the States of Matter chapter of the NCERT/CBSE Class 11 Chemistry curriculum, a topic tested regularly in JEE Main and NEET real-gas questions. Understanding why a larger 'a' value means easier liquefaction is exactly the kind of conceptual link competitive-exam MCQs like to test alongside the raw formula.