The ideal gas law works well when gas particles are far apart and moving fast, but real gases are made of particles with size and attractions. At high pressure, particles are crowded, so their own volume is no longer negligible. At low temperature, particles move more slowly, so attractive forces can pull them together and reduce the pressure they exert.
These effects explain why real gases deviate from PV = nRT in predictable ways.
The van der Waals equation modifies the ideal gas law by correcting both pressure and volume. The volume correction subtracts nb from the container volume because particles take up space, while the pressure correction adds a(n/V)^2 because attractions make the measured pressure lower than it would be without attractions. The equation is (P + a(n/V)^2)(V - nb) = nRT.
The compressibility factor Z = PV/nRT compares real behavior to ideal behavior, with Z = 1 for an ideal gas, Z < 1 when attractions dominate, and Z > 1 when particle volume dominates.
Key Facts
- Ideal gas law: PV = nRT
- van der Waals equation: (P + a(n/V)^2)(V - nb) = nRT
- Pressure correction: Pideal = Preal + a(n/V)^2
- Volume correction: Vfree = Vcontainer - nb
- Compressibility factor: Z = PV/nRT
- Real gases behave most ideally at low pressure and high temperature
Vocabulary
- Real gas
- A gas whose particles have finite volume and exert intermolecular forces, so it does not always obey the ideal gas law exactly.
- van der Waals equation
- An equation of state that corrects the ideal gas law for particle attractions and particle volume.
- Intermolecular attraction
- A force that pulls gas particles toward one another and can lower the pressure measured on the container walls.
- Excluded volume
- The portion of a container's volume that is unavailable for particle motion because gas particles themselves occupy space.
- Compressibility factor
- A dimensionless ratio, Z = PV/nRT, that shows how much a real gas deviates from ideal gas behavior.
Common Mistakes to Avoid
- Using PV = nRT at very high pressure without checking assumptions is wrong because particle volume becomes important when molecules are close together.
- Forgetting to subtract nb from V is wrong because the gas particles occupy some of the container space, leaving less free volume for motion.
- Subtracting the attraction correction from pressure is wrong because measured real pressure is lower than ideal pressure, so the correction is added inside the equation.
- Assuming Z is always greater than 1 is wrong because attractions can make Z less than 1, especially at moderate pressure and low temperature.
Practice Questions
- 1 A gas has P = 8.00 atm, V = 2.50 L, n = 1.00 mol, and T = 300 K. Using R = 0.0821 L atm mol^-1 K^-1, calculate Z = PV/nRT and state whether the gas is ideal, has Z < 1, or has Z > 1.
- 2 For 2.00 mol of a gas in a 5.00 L container with b = 0.0391 L/mol, calculate the van der Waals free volume V - nb.
- 3 Explain why a gas is more likely to deviate from ideal behavior when it is compressed and cooled. Include both particle volume and intermolecular attractions in your answer.