Q.Write the van der Waals equation for $n$ moles of a real gas, and explain the physical significance of the constants $a$ and $b$.

The ideal gas equation, $PV=nRT$, describes real gases only approximately, and the approximation breaks down

noticeably at high pressure and low temperature — precisely where the kinetic theory's assumptions of negligible

molecular volume and zero intermolecular attraction stop being reasonable. The compressibility factor,

$Z = \dfrac{PV}{nRT}$, quantifies this deviation: $Z=1$ exactly for an ideal gas at every condition, while for a

real gas $Z$ typically dips below $1$ at low-to-moderate pressure (intermolecular attraction dominates, pulling

molecules closer together than the ideal model predicts) and rises above $1$ at high pressure (the molecules' own

finite volume dominates, resisting further compression).

Johannes van der Waals corrected the ideal gas equation for both effects, producing the van der Waals equation

for $n$ moles of a real gas:

The pressure-correction term, $\dfrac{an^2}{V^2}$, is added to the measured pressure $P$: because

intermolecular attraction pulls molecules near the container wall slightly inward, a real gas strikes the walls

somewhat less forcefully than an ideal gas would, so the measured pressure is lower than the "true" value, and

this term compensates. The constant $a$ measures the strength of intermolecular attraction — larger for gases with

stronger dispersion forces or permanent dipoles (e.g. $\text{NH}_3$, $\text{CO}_2$), smaller for weakly attracting

gases (e.g. $\text{H}_2$, $\text{He}$).

The volume-correction term, $nb$, is subtracted from the measured volume $V$: since the gas molecules …