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Chemistry 2e · Gases

Non-Ideal Gas Behavior

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Constants cross-checked against OpenStax Chemistry 2e Table 9.3 and NIST-referenced data (2026-08).
Want it in plain words first? Jump to Eli explains — the same idea, no jargon.
On this page 9 sections
  1. In 30 seconds
  2. Why this matters
  3. The college version
  4. Eli explains
  5. Worked example
  6. Key takeaway
  7. Check yourself
  8. Study tools
  9. Sources & references

In 30 seconds

The law, PV = nRT, treats gas particles as point masses with zero volume that never attract one another. That model is remarkably accurate at ordinary temperatures and pressures. But it is a model, not reality: real molecules have finite size and attract each other weakly. When a gas is compressed to high pressure or cooled to low temperature, those simplifications stop being harmless, and the gas behaves non-ideally.

This topic covers when and why real gases deviate from the ideal gas law, the compressibility factor Z that measures deviation, and the , a corrected equation of state.

Why this matters

Most real-world gas problems fall where the ideal gas law is "close enough" — but close enough is not exact. Engineers designing high-pressure cylinders, scuba tanks, and industrial ammonia synthesis must know when ideal-gas estimates fail, because a 10–20% pressure error can mean the difference between a safe vessel and a rupture. Scientists modeling planetary atmospheres and chemists studying gases near need corrected equations. Non-ideal behavior also connects to the next chapter: the forces behind deviations are the forces behind liquids and solids.

The college version

Core Concepts

Where the ideal gas law breaks down

The ideal gas law assumes particles have negligible volume and exert no forces on each other. At low pressure, molecules are far apart: their own volume is a tiny fraction of the container, and attractions are rare. At high pressure or low temperature, molecules are crowded: the space they occupy matters, and attractions pull them closer than the ideal model expects.

The compressibility factor

The compressibility factor compares a real gas's measured pressure to the ideal prediction:

Z = PVnRT

For an ideal gas, Z = 1 exactly. For a real gas:

  • Z < 1: the gas is more compressible than ideal — attractions pull molecules together, so pressure is lower than ideal.
  • Z > 1: the gas is less compressible than ideal — finite molecular volume resists compression, so pressure is higher than ideal.

At very high pressure, Z rises well above 1 for all gases because molecular volume dominates. At moderate pressure, Z dips slightly below 1 because attractions dominate.

The van der Waals equation

The van der Waals equation modifies the ideal gas law with two correction terms:

(P + an2V2)(V - nb) = nRT

The attraction correction adds an2V2 to the pressure: molecules feel a net inward pull that reduces the pressure the gas exerts on the walls. Constant a measures the strength of intermolecular attraction (units L2 · atm/mol2). The volume correction subtracts nb: each mole occupies volume b unavailable for compression (units L/mol). Values are measured experimentally: for CO₂, a = 3.59 and b = 0.0427; for N₂, a = 1.39 and b = 0.0391 (OpenStax Chemistry 2e, Table 9.3). CO₂'s larger a reflects its stronger intermolecular attraction.

Conditions that push gases off ideal behavior

  • High pressure: molecules are forced close together, so both corrections matter.
  • Low temperature: slow-moving molecules let attractions act; gases approach liquefaction.
  • Polar or large molecules: strong intermolecular forces (large a) cause larger deviations.

Helium has tiny a and b values and stays nearly ideal over a wide range. Ammonia and water vapor deviate noticeably even at moderate pressures.

How It Works / Step-by-Step Process

  1. Write the ideal gas law and compute the ideal pressure.
  2. Look up the van der Waals constants a and b for the specific gas.
  3. Subtract nb from the container volume V.
  4. Add an2V2 to the pressure term.
  5. Solve for the corrected pressure.
  6. Compare with the ideal result and compute Z to state whether attractions or volume dominate.

Common Confusions

Common ConfusionCorrect Understanding
"The ideal gas law is wrong."It is highly accurate at ordinary conditions; it fails systematically only at high pressure or low temperature.
"Z > 1 always means attractions."No — Z > 1 means molecular volume dominates; Z < 1 means attractions dominate.
"The a term adds pressure to the gas."It adds a term to the equation to correct for the fact that attractions reduce measured pressure.
"b is the volume of one molecule."It is the excluded volume per mole — space unavailable because molecules occupy it.
"The van der Waals equation is exact for real gases."It is an improvement, not an exact law; other equations of state exist because no simple equation is perfect.
Eli, the EliExplains learning guide

Eli explains

The same idea, in plain words

Explain it like I’m 10

Imagine a crowded school bus. The ideal gas rule says every student takes up zero space and never bothers anyone — that works when the bus is almost empty. When the bus is packed, students take up real room and tug on each other. The van der Waals equation says: subtract the space students actually fill, and add back the tugging they do.

Worked example

Example 1: Is 1.00 mol of CO₂ ideal in a 22.4 L container at 273 K?

Ideal gas law:

P = nRTV = (1.00 mol)(0.08206 L · atm/(mol · K))(273 K)22.4 L = 1.00 atm

Van der Waals, solved for P:

P = nRTV - nb - an2V2

Substitute a = 3.59 L2 · atm/mol2, b = 0.0427 L/mol:

P = (1.00)(0.08206)(273)22.4 - (1.00)(0.0427) - (3.59)(1.00)2(22.4)2

Volume correction: 22.4 - 0.0427 = 22.3573 L, so

nRTV - nb = 22.402 L · atm22.3573 L = 1.0020 atm

Pressure correction:

an2V2 = 3.59 L2 · atm/mol2 · (1.00 mol)2(22.4 L)2 = 0.00716 atm

So P = 1.0020 - 0.00716 = 0.995 atm, only about 0.5% below ideal — CO₂ behaves essentially ideally here. The two corrections act in opposite directions and nearly cancel.

Example 2: The same 1.00 mol of CO₂ squeezed into 0.500 L at 273 K

Ideal gas law first:

P = (1.00)(0.08206)(273)0.500 = 44.8 atm

Van der Waals with V = 0.500 L:

P = (1.00)(0.08206)(273)0.500 - (1.00)(0.0427) - (3.59)(1.00)2(0.500)2

Volume correction: 0.500 - 0.0427 = 0.4573 L, so

nRTV - nb = 22.4020.4573 = 48.99 atm

Pressure correction:

an2V2 = 3.590.25 = 14.36 atm

P = 48.99 - 14.36 = 34.6 atm

Now the real pressure is about 23% lower than ideal (34.6 vs 44.8 atm): at high pressure, attractions pull CO₂ molecules together, making the gas far more compressible than ideal. The compressibility factor confirms it:

Z = PVnRT = (34.6)(0.500)(1.00)(0.08206)(273) = 17.322.4 = 0.77 < 1

A Z below 1 means attractions dominate — why ideal-gas numbers fail for high-pressure CO₂ systems.

Key takeaways

  • The ideal gas law is exact only for a hypothetical gas; real gases deviate most at high pressure and low temperature.
  • Z = PV/(nRT). Z = 1 ideal; Z < 1 attractions dominate; Z > 1 molecular volume dominates.
  • Van der Waals: (P + an2V2)(V - nb) = nRT. a corrects attraction; b corrects molecular volume.
  • Gases with strong intermolecular forces (large a) deviate more; helium deviates least.
  • At very high pressure the V - nb correction dominates (Z > 1); at moderate pressure attractions dominate (Z < 1).
  • The van der Waals equation reduces to the ideal gas law when a = b = 0.

Check yourself

5 review questions from the chapter. Try each one, then open the answer.

  1. Under what two conditions does a real gas deviate most from the ideal gas law?

    Show answer

    High pressure (molecules crowded together) and low temperature (slow motion lets attractions act).

  2. A gas has Z = 1.2 at high pressure. Which correction dominates — attraction or molecular volume — and why?

    Show answer

    Molecular volume. Z > 1 means the gas is less compressible than ideal because the molecules' own volume resists squeezing.

  3. Write the van der Waals equation and identify what a and b correct for.

    Show answer

    (P + an2V2)(V - nb) = nRT. a corrects for intermolecular attraction; b corrects for molecular volume.

  4. In Example 2, why is the real pressure of compressed CO₂ lower than the ideal pressure?

    Show answer

    Attractive forces between CO₂ molecules pull them inward, reducing the force on the container walls — the an2/V2 correction is large at small volume (14.4 atm), far larger than the small volume correction.

  5. Helium (a = 0.0342) vs CO₂ (a = 3.59) at the same conditions: which deviates more, and why?

    Show answer

    CO₂ deviates far more, because its much larger a constant means stronger intermolecular attractions.

Keep learning

Ready to build on this? Continue to the next lesson.

Study tools & related lessonsKey vocabulary · Related

Key vocabulary

ideal gas
A hypothetical gas with zero-volume particles and no intermolecular attractions
compressibility factor, Z
The ratio PV/(nRT); how far a gas is from ideal behavior
van der Waals equation
A corrected gas equation adding attraction and volume corrections to the ideal gas law
van der Waals constant a
A measure of intermolecular attraction strength in a gas
van der Waals constant b
Volume occupied by one mole of the gas molecules themselves
liquefaction
The change of a gas into a liquid

Sources & references

  1. openstax.org — Chemistry 2e

This lesson was adapted from the open educational references above; their licenses and attributions are preserved. See Copyright & Licensing.

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