9.6 Non-Ideal Gas Behavior

Gas behavior isn't always straightforward. At high pressures or low temps, gases start acting weird. The volume of particles and forces between them become important, making gases deviate from ideal behavior.

Enter the van der Waals equation. It fixes the ideal gas law by considering particle size and interactions. The compressibility factor helps us spot non-ideal behavior. Knowing when to use ideal vs. non-ideal equations is key for accurate predictions.

Factors and Equations Related to Non-Ideal Gas Behavior

Factors in gas behavior deviation

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• Volume of gas particles becomes significant relative to total volume at high pressures (CO2 at 100 atm)
  • Leads to less available volume for gas particles to occupy and increases frequency of particle collisions
• Intermolecular forces between gas particles become more significant at high pressures and low temperatures (H2 at 10 K)
  • Attractive forces (van der Waals forces) cause gas particles to collide less frequently with container walls, reducing pressure
  • Repulsive forces at very short distances prevent gas particles from occupying the same space

Van der Waals equation interpretation

• van der Waals equation: $\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$ corrects for non-ideal behavior by considering volume of gas particles and intermolecular forces
  • $a$: correction factor for intermolecular forces (0.55 L2⋅atm/mol2 for CO2)
  • $b$: correction factor for volume of gas particles (0.03 L/mol for CO2)
• Pressure correction term: $\frac{an^2}{V^2}$ accounts for reduction in pressure due to attractive intermolecular forces
  • Becomes more significant at high pressures and low volumes (compressed gas cylinders)
• Volume correction term: $(V - nb)$ accounts for reduction in available volume due to volume of gas particles
  • Subtracts $nb$ from total volume, where $n$ is number of moles and $b$ is volume correction factor (0.5 L for 1 mol of CO2 at 100 atm)

Compressibility and Comparison of Ideal and Non-Ideal Gas Behavior

Compressibility as non-ideal indicator

• Compressibility factor (Z): $Z = \frac{PV}{nRT}$ is the ratio of actual volume to volume predicted by ideal gas law
  • Deviations from Z = 1 indicate non-ideal behavior (Z = 0.8 for N2 at 200 atm and 300 K)
• Z > 1: gas is less compressible than an ideal gas, occurs at high pressures when intermolecular repulsive forces dominate (He at 1000 atm)
  • Volume is greater than predicted by ideal gas law
• Z < 1: gas is more compressible than an ideal gas, occurs at low pressures when intermolecular attractive forces dominate (Cl2 at 1 atm)
  • Volume is less than predicted by ideal gas law

Ideal vs van der Waals calculations

• Ideal gas law: $PV = nRT$ assumes gas particles have negligible volume and no intermolecular forces
  • Suitable for low pressures and high temperatures (N2 at 1 atm and 298 K)
• van der Waals equation: $\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$ accounts for volume of gas particles and intermolecular forces
  • Provides more accurate results at high pressures and low temperatures (CO2 at 50 atm and 250 K)
• Comparing calculations:
  1. At low pressures and high temperatures, both equations yield similar results (< 5% difference for O2 at 1 atm and 500 K)
  2. At high pressures and low temperatures, van der Waals equation provides more accurate predictions of gas properties, such as pressure, volume, and compressibility factor (20% difference for NH3 at 100 atm and 200 K)

Advanced Concepts in Non-Ideal Gas Behavior

• Critical point: the temperature and pressure at which the liquid and gas phases of a substance become indistinguishable
• Boyle temperature: the temperature at which a real gas behaves most like an ideal gas over a wide range of pressures
• Fugacity: a measure of the tendency of a substance to escape from a phase, replacing pressure in thermodynamic equations for non-ideal gases
• Virial equation: an expansion series that describes the pressure of a gas as a function of its molar volume, providing a more accurate description of non-ideal gas behavior than the van der Waals equation