The compressibility factor Z = PV/nRT is the most important tool for understanding real gas deviations.

For a perfectly ideal gas: Z = 1 at all pressures. The PV vs P graph is a horizontal line at PV = nRT.

For a real gas — the competition between two effects:

Effect 1 — Intermolecular Attraction (a term):

• Molecules attract each other
• Molecules near walls are pulled back by inner molecules
• They hit walls with less force → observed P < ideal P
• Equivalently: at given P, actual V < ideal V
• This pushes Z BELOW 1

Effect 2 — Molecular Volume / Excluded Volume (b term):

• Molecules occupy space; cannot overlap
• Free volume = V - nb (less than total volume)
• Effective pressure is higher because molecules are packed into smaller free volume
• This pushes Z ABOVE 1

Pressure dependence:

• At LOW P: V is large → b/V is negligible, a/$V^{2}$ is negligible → Z ≈ 1 for all gases.
• At MODERATE P: a/$V^{2}$ term becomes significant → attraction dominates → Z < 1 (for most gases).
• At VERY HIGH P: V becomes small → b/V becomes large → volume correction dominates → Z > 1 for ALL gases.

Temperature dependence:

• At LOW T: slow molecules → attractions have more effect relative to KE → Z dips more below 1.
• At HIGH T: fast molecules → KE dominates over attractions → gas approaches ideal → Z closer to 1.
• At T_B (Boyle temperature): the attractive and repulsive corrections exactly cancel at moderate P → Z ≈ 1 over wide P range.

The $H_{2}$/He exception explained: For $H_{2}$: a = 0.244 $L^{2}$·atm/$mol^{2}$ (very small). For He: a = 0.034 $L^{2}$·atm/$mol^{2}$ (almost zero). The b correction (increases Z) always outweighs the tiny a correction (decreases Z) → Z > 1 at all pressures. The crossover that other gases show (from Z < 1 to Z > 1) never happens for $H_{2}$ and He under normal conditions.