The Ideal Gas Law & Its Limits

Use PV = nRT confidently at low pressure and high temperature — and know when real-gas effects make it fail.

Material & Energy BalancesChemical Engineering Year 1Free preview
⏱️ About 14 min

If you use PV = nRT everywhere, how wrong could you be — and how do you know before it bites a design calculation?

💡
The big idea: The ideal gas law is an excellent approximation at low density (low P, high T), but real gases deviate strongly near the critical region and at high pressure.
🎯 By the end, you'll be able to
  • Write and use PV = nRT with correct units
  • Recognize the physical conditions that make the ideal-gas model accurate
  • Compute molar volume or pressure from PV = nRT in a fully worked example
  • Identify the breakdown regions (high pressure, near the critical point)
📎 Helpful to know first
  • Heats of Solution & Mixing

The Model: An Equation of State (EOS)

An equation of state links pressure P, volume V, temperature T (and amount n) for a fluid. For gases at sufficiently low density, the simplest EOS is the ideal gas law.

Engineers use it constantly for flow-rate conversions, equipment sizing, and back-of-the-envelope checks — but it is still a model with limits.

\[ PV = nRT \]
Ideal gas law. Use consistent units for P, V, n, T, and the gas constant R.

Gas Constant R in Common Unit Systems

These are the same constant expressed in different units (all textbook-standard):

  • R = 8.314 J/(mol·K)
  • R = 0.08314 L·bar/(mol·K)
  • R = 0.08206 L·atm/(mol·K)

Pick the version that matches your pressure unit to avoid hidden conversion errors.

When Is PV = nRT a Good Approximation?

The ideal-gas model works best when gas molecules are far apart, so intermolecular forces and molecular size are negligible. A practical rule of thumb:

  • Low pressure (low density)
  • High temperature (especially high relative to the critical temperature Tc)

Near the critical point and/or at high pressure, real gases can deviate dramatically.

⚠️ Pitfall: Ideal-gas behavior can fail badly near the critical region

Even at moderate temperatures, compressing a gas to high pressure increases density. Then molecular attractions/repulsions matter and PV = nRT can under- or over-predict V and P. This is especially severe near the critical point where properties change rapidly.

📝 Worked example: Compute the molar volume of nitrogen treated as an ideal gas at P = 5.00 bar and T = 350 K. Use R = 0.08314 L·bar/(mol·K).
  1. Given: P = 5.00 bar, T = 350 K, n = 1 mol (molar basis).
  2. Ideal gas law on a molar basis: v = V/n = RT/P.
  3. Compute RT: (0.08314 L·bar/(mol·K))(350 K) = 29.099 L·bar/mol
  4. Divide by P: v = 29.099 / 5.00 = 5.8198 L/mol
  5. Unit check: (L·bar/mol) / bar = L/mol
✓ v_ideal = 5.820 L/mol (at 5.00 bar, 350 K; ideal-gas assumption)
✏️ Practice: Air is approximated as an ideal gas. Compute the pressure (bar) of 2.00 mol of air in a 10.0 L rigid tank at 300 K. Use R = 0.08314 L·bar/(mol·K).
bar
Solution
  1. Use PV = nRT → P = nRT/V.
  2. nRT = (2.00 mol)(0.08314 L·bar/(mol·K))(300 K) = (2.00)(24.942) = 49.884 L·bar
  3. P = 49.884 / 10.0 = 4.9884 bar

Check your understanding

1. The ideal gas law is most accurate when a gas is at:
Low P and high T (relative to critical) imply low density and weaker intermolecular effects, so PV = nRT is accurate.
2. Which value of R is consistent with P in bar and V in liters?
R = 0.08314 L·bar/(mol·K) matches bar and liters directly.
🔎 Related lessons
The Compressibility Factor Z
✅ Key takeaways
  • Ideal gas law: PV = nRT (use consistent units)
  • R can be expressed as 8.314 J/(mol·K) = 0.08314 L·bar/(mol·K) = 0.08206 L·atm/(mol·K)
  • Ideal-gas behavior is best at low pressure and high temperature (relative to critical properties)
  • Near the critical point and/or at high pressure, real gases deviate and need Z or a real-gas EOS
➡️ Next we quantify non-ideality with the compressibility factor Z, which corrects PV = nRT in a simple, engineering-friendly way.
Want to test yourself on this? Try the Chemical Aptitude test →