10.3 Real gas behavior and equations of state

Real Gas Behavior

Real gases don't always follow the ideal gas law. As pressure rises and temperature drops, intermolecular forces and the physical size of molecules start to matter, causing measurable deviations from ideal predictions. To handle this, we use more sophisticated equations of state like the van der Waals equation and the virial equation.

Ideal vs. Real Gas Behavior

The ideal gas law treats gas molecules as point particles with no intermolecular forces:

$PV = nRT$

This works well at low pressures and high temperatures, where molecules are far apart and moving fast enough that attractions between them are negligible. Helium near room temperature and atmospheric pressure behaves almost ideally.

Real gas behavior shows up when those assumptions break down. At high pressures, molecules are packed closer together, so their finite volume matters and attractive forces become significant. At low temperatures, molecules move more slowly and those attractive forces have more time to act.

Intermolecular Forces in Gases

Three main types of intermolecular forces affect gas behavior:

• London dispersion forces are present in all molecules but are the only intermolecular force in nonpolar gases like $CH_4$. They arise from temporary fluctuations in electron distribution.
• Dipole-dipole interactions occur between polar molecules like $HCl$, where permanent partial charges attract neighboring molecules.
• Hydrogen bonding is a particularly strong type of dipole-dipole interaction found in molecules like $H_2O$, where hydrogen is bonded to a highly electronegative atom (O, N, or F).

The strength of these forces depends on molecular size (larger electron clouds are more polarizable, giving stronger dispersion forces) and polarity (polar molecules experience additional attractive forces beyond dispersion).

These forces affect real gas behavior in two key ways:

• Attractive forces pull molecules toward each other, reducing the pressure the gas exerts on container walls compared to ideal predictions.
• Finite molecular volume means the actual space available for molecular motion is less than the total container volume.

At sufficiently high pressures and low temperatures, these effects can cause a gas to condense into a liquid. Propane stored in a tank at moderate pressure is a familiar example.

Equations of State for Real Gases

Van der Waals Equation of State

The van der Waals equation modifies the ideal gas law with two correction terms:

$\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$

Here's what each correction does:

• The $\frac{an^2}{V^2}$ term adds to the measured pressure to account for intermolecular attractions. Attractive forces reduce the force of molecular impacts on the container walls, so the "effective" pressure driving the gas behavior is higher than what you measure. The parameter $a$ reflects the strength of those attractions.
• The $nb$ term subtracts from the total volume to account for the finite size of molecules. The parameter $b$ represents the volume excluded per mole of gas because molecules can't overlap.

Both $a$ and $b$ are experimentally determined constants specific to each gas. For example, $CO_2$ has $a = 3.59 \text{ L}^2\text{atm/mol}^2$ and $b = 0.0427 \text{ L/mol}$. A gas with stronger intermolecular forces will have a larger $a$ value; a gas with larger molecules will have a larger $b$ value.

The van der Waals equation is a significant improvement over the ideal gas law because it can predict the existence of a critical point (the temperature and pressure above which distinct liquid and gas phases no longer exist) and approximate liquid-vapor phase transitions. That said, it's still a relatively simple model and loses accuracy very close to the critical point.

Virial Equation of State

The virial equation takes a different approach. Instead of adding correction terms to $P$ and $V$ directly, it expresses the compressibility factor $Z$ as a power series. The compressibility factor is defined as:

$Z = \frac{PV}{nRT}$

For an ideal gas, $Z = 1$ exactly. Deviations from 1 tell you how "non-ideal" the gas is. If $Z < 1$, attractive forces dominate (the gas is more compressible than ideal). If $Z > 1$, repulsive forces and molecular volume dominate.

The virial equation can be written in terms of pressure:

$Z = 1 + B(T)\frac{P}{RT} + C(T)\left(\frac{P}{RT}\right)^2 + \cdots$

or in terms of molar density $\rho$:

$Z = 1 + B(T)\rho + C(T)\rho^2 + \cdots$

The virial coefficients $B(T)$, $C(T)$, etc., are temperature-dependent and specific to each gas. Each coefficient captures a different level of molecular interaction:

• $B(T)$ (second virial coefficient) accounts for pairwise interactions between two molecules.
• $C(T)$ (third virial coefficient) accounts for three-body interactions.

For nitrogen at 300 K, $B = -123.2 \text{ cm}^3/\text{mol}$. The negative sign indicates that attractive forces dominate at that temperature.

These coefficients can be determined experimentally from $PVT$ data or calculated theoretically from intermolecular potential models. In practice, you truncate the series at the level of accuracy you need. For moderate pressures, keeping just the $B(T)$ term often gives good results. Higher-order terms become necessary at higher pressures where multi-body interactions matter.