Real and Ideal Gases: MDCAT Chemistry notes

Ideal gas and the two faulty assumptions

An ideal gas obeys $PV = nRT$ exactly at every temperature and pressure. The kinetic molecular theory gets this result from two assumptions that real gases do not satisfy:

1. The actual volume of the gas molecules is negligible compared with the volume of the container.
2. There are no forces of attraction or repulsion between the molecules. In an ideal gas these forces are absent, not merely weak.

No real gas is perfectly ideal. Gases such as $\mathrm{H_2}$, $\mathrm{He}$, $\mathrm{N_2}$ and $\mathrm{O_2}$ come close under ordinary conditions. Easily liquefied gases such as $\mathrm{NH_3}$, $\mathrm{SO_2}$ and $\mathrm{CO_2}$ deviate more because their intermolecular forces are stronger.

When do real gases deviate?

• At low pressure the molecules are far apart, so their own volume is a tiny fraction of the total volume. At about 1 to 2 atm it can be ignored; at hundreds of atmospheres it cannot. The lower the pressure, the more valid the "negligible volume" postulate.
• At high temperature the molecules move so fast that their attractions have little effect on their motion.
• At high pressure the "negligible volume" postulate fails; at low temperature the "no attractions" postulate fails. Intermolecular forces become significant at low temperature and high pressure.

Compressibility factor

The deviation is measured by the compressibility factor $Z = \dfrac{PV}{nRT}$.

• $Z = 1$: ideal behaviour.
• $Z \lt 1$ (negative deviation): attractive forces dominate, so the gas is more compressible than an ideal gas. Seen for gases like $\mathrm{CO_2}$ and $\mathrm{NH_3}$ at moderate pressures and low temperatures.
• $Z \gt 1$ (positive deviation): at very high pressure the molecules are pressed so close that repulsive forces and their own finite volume dominate. The gas is harder to compress and its volume is larger than predicted. $\mathrm{H_2}$ and $\mathrm{He}$ show only positive deviation at ordinary temperatures.
• As temperature is raised, the $Z$ versus $P$ curves move closer to the ideal line $Z = 1$.

Van der Waals equation

Van der Waals corrected the ideal gas equation for both faulty assumptions.

• Volume correction: $b$ is the excluded volume per mole, about four times the actual volume of the molecules. The free volume available for motion is $V - nb$.
• Pressure correction: a molecule about to hit the wall is pulled back by the molecules behind it, so the observed pressure is less than the ideal pressure. The term $\dfrac{an^2}{V^2}$ is added to $P$. A larger $a$ means stronger attractions and a gas that is easier to liquefy.

Key formulas

• Ideal gas equation: $PV = nRT$, with $R = 0.0821\ \mathrm{atm\,dm^3\,K^{-1}\,mol^{-1}} = 8.314\ \mathrm{J\,K^{-1}\,mol^{-1}}$
• Compressibility factor: $Z = \dfrac{PV}{nRT}$
• Van der Waals equation: $$\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$$
• Units: $a$ in $\mathrm{atm\,dm^6\,mol^{-2}}$, $b$ in $\mathrm{dm^3\,mol^{-1}}$

Common MDCAT traps

• Ideal behaviour needs both high temperature and low pressure. Options that mix them (high T with high P) are wrong.
• The postulate that fails at high pressure is "molecular volume is negligible", not "molecules move in straight lines".
• Intermolecular forces in an ideal gas are absent, not "weak".
• Positive deviation at high pressure comes from repulsive forces, not attractive forces. Attractions cause negative deviation.
• The compressibility factor is the ratio $PV/nRT$, not the equation $PV = nRT$.

Quick revision

• Real gases deviate most at low temperature and high pressure.
• $Z = 1$ ideal, $Z \lt 1$ attractions dominate, $Z \gt 1$ repulsions dominate.
• $\mathrm{H_2}$ and $\mathrm{He}$ show positive deviation at room temperature.
• Van der Waals constant $a$ corrects pressure; $b$ corrects volume.
• Molecular volume matters least at the lowest pressure.