Real vs Ideal Gas

Why the ideal-gas model exists

The kinetic theory of gases asks us to picture molecules as point particles bouncing elastically off the container walls and each other. Two simplifying assumptions make the algebra tractable: each particle has zero volume, and the only force between particles is the contact-collision impulse. With those, three lines of derivation give PV = nRT.

The model is shockingly accurate. Air at sea level — 1 atm, 25 °C — is described by PV = nRT to within roughly 0.1%. That is why engineers, biologists, meteorologists, and high-school students alike reach for it first. But it is a low-pressure, high-temperature limit. As you compress a gas or chill it toward its boiling point, the two assumptions both break, and corrections enter.

Two reasons real gases misbehave

Excluded volume. Real molecules are not points — a CO₂ molecule has a kinetic diameter of about 3.3 Å, an Ar atom 3.4 Å. When a gas is compressed enough that the molecules collectively fill a non-trivial fraction of the container, the volume available for them to wander in is V − nb, not V. This makes the gas less compressible than ideal: at very high pressure, Z > 1.

Intermolecular attractions. Even noble-gas atoms feel weak London dispersion forces; polar molecules feel dipole–dipole, and water feels hydrogen bonds. These attractions slow down molecules just before they hit the wall, so each collision delivers slightly less momentum than the ideal value. Result: lower pressure than PV = nRT predicts. Z < 1 over a wide range of intermediate pressures.

Each effect dominates in a different regime, and they fight each other:

Z = PV/(nRT)

           Low P:                Moderate P:           Very high P:
       Z ≈ 1                Z < 1 (attraction)    Z > 1 (repulsion)

   ──────────●─────────────●──────────────●──────────→  P
            ideal       attraction     excluded
                          minimum       volume wins

Worked example — CO₂ at 100 atm, 273 K

How much volume does 1 mol of CO₂ occupy at 100 atm, 273 K? Compare ideal gas, van der Waals, and the experimental value.

Ideal gas:

V_ideal = nRT/P = (1)(0.0821)(273) / 100 = 0.2241 L

Van der Waals (a = 3.592 L²·atm/mol², b = 0.04267 L/mol):

(P + a/V²)(V − b) = RT          (n = 1)
(100 + 3.592/V²)(V − 0.04267) = 22.41

Solving iteratively starting from V = V_ideal = 0.2241:
  Iteration 1:  V = 0.0837 L
  Iteration 2:  V = 0.0775 L
  Iteration 3:  V = 0.0762 L  (converged)

V_vdW ≈ 0.076 L

Experimental value at 100 atm, 273 K: V ≈ 0.0703 L (CO₂ at this state is on the verge of liquefaction; T_c = 304 K, P_c = 73 atm). The ideal gas overestimates the volume by a factor of three. Van der Waals gets within 8% of the right answer — a useful approximation, but not a substitute for tabulated PVT data near critical points.

Compressibility factor at this state: Z_exp = (100)(0.0703) / (0.0821)(273) ≈ 0.314. The gas is severely non-ideal — attractions are pulling it toward the liquid phase.

Real-world deviations

Gas	Conditions	Z (compressibility)	Note
He	25 °C, 1 atm	1.0005	Essentially ideal
Air	25 °C, 1 atm	0.9994	Slight attraction
N₂	25 °C, 100 atm	1.001	Repulsion offsets attraction
CO₂	0 °C, 100 atm	0.31	Severe attraction; near liquefaction
NH₃	25 °C, 10 atm	0.88	Hydrogen bonding
H₂O (steam)	200 °C, 50 atm	0.78	H-bonding dominates
H₂	25 °C, 100 atm	1.06	Repulsion dominates (small a)

Industrial design relies on tabulated Z(T,P) values from generalized compressibility charts using reduced coordinates T_r = T/T_c, P_r = P/P_c. Two unrelated gases at the same reduced state have nearly identical Z — the principle of corresponding states.

Ideal vs real, side by side

	Ideal gas	Real gas
Molecular volume	Zero (point particles)	Finite (kinetic diameter)
Intermolecular force	Zero except at collision	Dispersion + dipole + H-bond
Equation of state	PV = nRT	Van der Waals, virial, R-K, etc.
Compressibility factor Z	Exactly 1.0	Varies with T, P
Liquefies on cooling?	No (no attractions)	Yes (attractions overcome KE)
Internal energy	Function of T only	Function of T and V
Joule–Thomson coefficient	Zero (no temperature change on expansion)	Non-zero (heating or cooling)
Where it works	High T, low P	Always (with right parameters)

Variants of real-gas equations

• Van der Waals (1873). Two parameters per gas. Captures liquefaction qualitatively, fails near the critical point. Still the most used pedagogically because the corrections have transparent physical meaning.
• Virial expansion. Z = 1 + B/V + C/V² + ..., truncated at second order. B(T) is calculable from intermolecular potentials, fits experimental data better than van der Waals at low to moderate density.
• Redlich-Kwong (1949) and modifications. Add temperature dependence to the attraction term; widely used in petroleum and chemical engineering software (HYSYS, Aspen).
• Peng–Robinson (1976). Cubic equation of state preferred for hydrocarbons and natural gas; combines accuracy near critical with computational simplicity.
• Tabulated steam tables / IAPWS-IF97. For water/steam, no equation of state matches a measurement-based table. Power plants run on these tables to ±0.05% accuracy.

An equation-of-state hierarchy: ideal gas → van der Waals → virial → cubic (R-K, P-R) → multi-parameter (BWR, IAPWS). Each step trades simplicity for accuracy in a wider state-space.

Common pitfalls

• Using PV = nRT inside a refrigerator. Refrigerants operate near their boiling point on purpose — the phase change is what moves heat. Ideal-gas calculations get the cycle efficiency badly wrong.
• Treating compressed-gas-cylinder pressures as ideal. A nominal 200-bar SCBA cylinder of compressed air is roughly 1.5% off ideal at room temperature; for breathing-air calculations you take the correction. Compressed natural gas (CNG) at 250 bar is 10% off — engineers always use compressibility-factor tables.
• Plugging Celsius into PV = nRT. A perennial student error, but worth restating: T must be Kelvin. The temperature units of R fix this — using R = 8.314 J/mol·K with T in °C silently produces nonsense.
• Assuming "real-gas" always means lower pressure than ideal. At very high pressures, the excluded-volume term dominates and Z > 1: the gas is harder to compress than ideal. Hydrogen at 100 atm is a textbook example.
• Forgetting van der Waals fails near the critical point. The equation predicts a smooth, single-phase fluid where reality has a phase transition. For supercritical CO₂ extraction or rocket-propellant tank design, only multi-parameter equations of state suffice.
• Comparing Z(P) curves at different temperatures. Compressibility behavior is strongly T-dependent. He at 4 K is wildly non-ideal; at 300 K it's textbook ideal. Always quote the temperature alongside Z.