Real Gases — van der WaalsJEE Advanced

When PV = nRT stops being true — interactive Chemistry simulation for IIT-JEE.

Concept

Real molecules attract each other at a distance (softening the pressure) and occupy volume up close (stiffening it). Van der Waals patches the ideal law with two constants: a for attraction, b for size. The compressibility factor Z = PV/RT scores the deviation: Z < 1 = attraction winning, Z > 1 = repulsion winning.

Key formula

\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT, \qquad Z = \frac{PV_m}{RT}

Derivation

Wall pressure is reduced because molecules near the wall are pulled inward ∝ (density)² → add $a/V_m^2$ to the measured P.

Free volume is less than V because molecules exclude each other → subtract b (≈ 4× actual molecular volume).

Low P: Z dips below 1 (attraction); very high P: (V−b) dominates, Z climbs above 1. H₂ & He (tiny a) show Z > 1 almost throughout.

Scenarios to explore

Real Gases — van der Waals a & b corrections and the Z factor.

Real-world applications

Gas liquefaction (below critical T, isotherms develop the condensation loop).
CO₂ cylinders & supercritical extraction.
Boyle temperature: T_B = a/Rb where the gas plays ideal longest.

JEE exam tips

Critical constants: V_c = 3b, P_c = a/27b², T_c = 8a/27Rb — and Z_c = 3/8 for every vdW gas.
Easily liquefiable gas = big a (NH₃, CO₂) — links to adsorption tendency too.
At Boyle temperature the first-order deviation vanishes: T_B = a/Rb.

Common mistakes

Applying the a-correction with a minus sign to P (it ADDS to measured P).
b is per MOLE excluded volume — 4× the molecular volume, not equal to it.
Assuming all gases dip below Z = 1 — H₂/He barely do (a too small).

Exam traps to avoid

High T + low P → most ideal behaviour (both corrections fade).
Z at fixed T,P differs per gas — deviation is substance-specific through a & b.