PV = nRT — the ideal gas law — covers most everyday gas problems. But it's an approximation. At high pressure and low temperature, real gases deviate significantly from "ideal" behavior. Here's when, why, and what to use instead.
The ideal gas assumptions
The ideal gas law works because it assumes:
- Gas molecules have negligible volume compared to the container.
- Molecules don't interact with each other (no attraction or repulsion).
- Collisions are perfectly elastic.
For dilute gases at moderate temperatures, all three assumptions are nearly true. The math works.
Where each assumption breaks
Assumption 1 (molecules have negligible volume): fails at high pressures, where molecules are squeezed close together. The ideal gas law overestimates the available space.
Assumption 2 (no molecular interactions): fails at low temperatures, where molecules move slowly enough for attractive forces to matter. The ideal gas law overestimates pressure.
Assumption 3 (elastic collisions): generally true for non-reactive gases. Doesn't break in everyday situations.
Where the ideal gas law works well
- Atmospheric conditions: 1 atm, 273-300 K. Excellent agreement.
- Most chemistry lab work: moderate temperatures and pressures.
- Stoichiometry: gas mole calculations to within a few percent.
For these, you can use PV = nRT confidently.
Where the ideal gas law fails
High pressure (10+ atm, especially 50+ atm): molecular volumes become comparable to container volume.
Low temperatures (below 0°C, especially below −100°C): kinetic energy decreases; intermolecular attractions become significant.
Near phase changes (liquefaction): the gas is about to become a liquid. The ideal gas law gives nonsensical results near these transitions.
Polar gases at low T (water vapor, ammonia): hydrogen bonding causes significant attraction.
Worked example: nitrogen at high pressure
1 mole of N₂ at 273 K (0°C):
- 1 atm: ideal predicts V = 22.4 L. Actual: 22.4 L. Perfect agreement.
- 10 atm: ideal predicts V = 2.24 L. Actual: ~2.18 L. ~3% error.
- 100 atm: ideal predicts V = 0.224 L. Actual: ~0.21 L. ~6% error.
- 1000 atm: ideal predicts 0.0224 L. Actual: significantly different. Massive error.
For most gases at most temperatures, the error grows roughly linearly with pressure up to ~100 atm.
The compressibility factor (Z)
Real gas behavior is captured by the compressibility factor:
Z = PV / (nRT)
Z = 1 for an ideal gas. Z < 1 when intermolecular attraction dominates. Z > 1 when molecular volume dominates.
Z varies with pressure and temperature. Tables and charts of Z exist for common gases.
The van der Waals equation
The simplest correction to the ideal gas law:
(P + an²/V²)(V − nb) = nRT
Where:
- 'a' corrects for intermolecular attractions (different for each gas)
- 'b' corrects for molecular volume
For nitrogen: a = 1.39 L²·atm/mol², b = 0.0391 L/mol.
The van der Waals equation is much more accurate than ideal gas, especially at moderate pressures (up to ~50 atm). It's still an approximation but a good one.
More sophisticated equations of state
For very precise work or extreme conditions:
- Redlich-Kwong: better than van der Waals across wider P/T range.
- Peng-Robinson: standard in petroleum and chemical engineering.
- Lee-Kesler: highly accurate; used in industrial process design.
- Virial equation: series expansion form. Theoretically rigorous.
These all reduce to the ideal gas law in the limit of low pressure.
Liquefaction
Cool a gas enough or compress it enough, and it liquefies. The ideal gas law doesn't apply at all — liquids have vastly different properties (much higher density, intermolecular bonding, etc.).
Liquefaction temperatures (at 1 atm):
- Water vapor: 100°C (boiling point).
- Ammonia: −33°C.
- CO₂: −78°C (sublimes; doesn't form liquid at 1 atm).
- Air: −195°C.
- Hydrogen: −253°C.
- Helium: −269°C.
Below these, the gas becomes a liquid. The ideal gas law predicts strange results approaching the liquefaction point.
Critical point and supercritical fluids
Beyond the critical point, the distinction between liquid and gas disappears — there's a "supercritical fluid" with properties of both.
Critical temperatures and pressures:
- Water: 374°C, 218 atm.
- CO₂: 31°C, 73 atm.
- Methane: −83°C, 46 atm.
Above these, no amount of compression liquefies the gas. Supercritical CO₂ is used industrially for extraction (decaffeinating coffee, removing oils).
Practical takeaways
Use ideal gas law (PV = nRT) when:
- Pressure under 5–10 atm.
- Temperature well above the boiling point of the gas.
- Non-polar, non-associating gases (N₂, O₂, Ar, He).
- Quick estimates and stoichiometry.
Use a real-gas equation when:
- Pressure exceeds ~10 atm.
- Working near liquefaction point.
- Polar molecules at low temperature.
- Engineering design where precision matters.
Why this matters in real applications
Natural gas pipelines: operating at 50-100 atm. Engineers use real-gas equations.
Industrial cryogenics: liquid air, liquid nitrogen production. Equations of state critical for plant design.
HVAC systems: refrigerant cycles use equations of state, not ideal gas law.
Atmospheric chemistry: ideal gas works well at typical surface conditions but breaks down in upper atmosphere or near volcanic emissions.
Calculate it
Our ideal gas law calculator handles PV = nRT for any of the four variables. Use it for the bulk of practical chemistry calculations. For high-precision or extreme-condition work, you'll want to use a real-gas equation of state — those are typically built into commercial chemical engineering software.