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General Chemistry I · Gases

Real Gases and the van der Waals Equation

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On this page 7 sections
  1. In 30 seconds
  2. Why this matters
  3. The college version
  4. Eli explains
  5. Key takeaway
  6. Study tools
  7. Sources & references

In 30 seconds

Real gases deviate from the ideal gas law at high pressure and low temperature, where the two assumptions of the kinetic molecular theory fail: molecules have a nonzero volume, and they attract one another. The van der Waals equation, (P + a(n/V)²)(V − nb) = nRT, corrects for both effects with two gas-specific constants.

Why this matters

Real-gas corrections matter wherever gases are compressed or cooled: high-pressure industrial reactors, CO₂ in fire extinguishers and dry-ice systems, liquefied gases, and deep-sea diving (where gas mixtures deviate measurably from ideality). Understanding the van der Waals equation shows why the ideal gas law is an approximation and when it is safe to use it.

The college version

Key Ideas

  • Deviation at high P: molecules are crowded, so their own volume matters and attractions matter.
  • Deviation at low T: slow-moving molecules are more affected by intermolecular attractions.
  • Constant a corrects for intermolecular attractions (lowers pressure).
  • Constant b corrects for the finite volume of molecules (excluded volume).
  • Ideal-gas limit: at low P and high T, the van der Waals equation reduces to PV = nRT.

Equations and Variables

  • van der Waals equation: (P + a(n/V)²)(V − nb) = nRT.
    • P = pressure, V = volume, n = moles, R = 0.08206 L·atm/(mol·K), T = kelvin.
    • a = attraction constant (L²·atm/mol²); b = excluded-volume constant (L/mol).
  • Solved for pressure: P = nRT/(V − nb) − a(n/V)².
  • Ideal gas law for comparison: P = nRT/V.

How It Works

  1. The ideal gas law assumes point particles (no volume) and no attractions.
  2. At high pressure, molecules are packed close together: their actual volume is no longer negligible, so the space they can move in is less than the container volume. The van der Waals equation replaces V with (V − nb), shrinking the "free" volume and raising the pressure.
  3. At the same time, attractions between molecules pull them inward as they approach the walls, reducing the force (pressure) they exert. The equation adds the a(n/V)² term to pressure to compensate.
  4. At low temperature, slow molecules feel attractions more strongly, so the a correction dominates.
  5. At low pressure and high temperature, both corrections vanish and ideal behavior returns.

Worked Example

Compute the pressure of 1.00 mol CO₂ in a 0.100 L container at 0 °C using (a) the ideal gas law and (b) the van der Waals equation (a = 3.59 L²·atm/mol², b = 0.0427 L/mol). (a) Ideal: P = nRT/V = (1.00 mol)(0.08206 L·atm/mol·K)(273.15 K)/(0.100 L) = 224 atm. (b) van der Waals: P = nRT/(V − nb) − a(n/V)² = (0.08206)(273.15)/(0.100 − 0.0427) − (3.59)(1.00/0.100)² = 22.41/0.0573 − 3.59(100) = 391 − 359 = 32 atm. Under these conditions the real pressure (32 atm) is far below the ideal prediction (224 atm) — attractions and volume dominate.

Common Confusions

  • "Real gases always have higher pressure than ideal" — not always; the attraction term (a) lowers pressure while the volume term (b) raises it, and either can dominate depending on conditions.
  • "a and b are the same for every gas" — they are gas-specific constants that depend on molecular size and attraction strength.
  • "The ideal gas law is wrong" — it is an excellent approximation at low pressure and high temperature; it only breaks down under extreme conditions.
  • "At low temperature gases become ideal" — the opposite: low temperature makes attractions more important and deviations larger.
Eli, the EliExplains learning guide

Eli explains

The same idea, in plain words

Explain it like I’m 10

The ideal gas law pretends molecules are invisible specks that ignore each other. That's fine when they're far apart, like people spread across a big empty gym. But squeeze everyone into a phone booth and the pretense fails: people take up space (the "b" correction) and grab each other (the "a" correction). The van der Waals equation is just the gas law with those two real-world fixes added in. The analogy's limit: molecular "grabbing" is a weak electromagnetic attraction, not a handshake — but "crowd them and they stop acting invisible" is exactly the idea.

Key takeaways

  • Real gases deviate from ideal at high P and low T.
  • a accounts for attractions (tends to lower pressure); b accounts for molecular volume (tends to raise pressure).
  • van der Waals: (P + a(n/V)²)(V − nb) = nRT.
  • At low P and high T, real gases approach ideal behavior.
  • a and b are specific to each gas (e.g., CO₂: a = 3.59, b = 0.0427).
  • Real gases deviate at high P and low T.
  • a corrects attractions; b corrects molecular volume.
  • van der Waals: (P + a(n/V)²)(V − nb) = nRT.
  • At low P and high T, real ≈ ideal.
  • The two corrections push pressure in opposite directions.

Keep learning

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

Practice General Chemistry I

This lesson has no separate scored set. Practice draws from the subject’s question bank.

Study tools & related lessonsYou’ll learn to · Related

You’ll learn to

  • Explain when and why real gases deviate from ideal behavior.
  • Interpret the van der Waals constants a and b.
  • Use the van der Waals equation to compute a real-gas pressure.
  • Compare ideal and real pressures under extreme conditions.

Sources & references

  1. OpenStax, "9.6 Non-Ideal Gas Behavior," Chemistry 2e.
  2. Petrucci et al., "6.9 Non-ideal (Real) Gases," Chemistry LibreTexts.
  3. NIST CODATA, "molar gas constant."

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

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