9.6 Non-Ideal Gas Behavior

Factors and Equations Related to Non-Ideal Gas Behavior

The ideal gas law works great under normal conditions, but real gases don't always follow the rules. When pressure gets very high or temperature drops very low, the assumptions behind $PV = nRT$ start to break down. Understanding why this happens and how to correct for it is what this section is all about.

Factors in gas behavior deviation

Two assumptions in the ideal gas law cause problems under extreme conditions: that gas particles have no volume, and that they don't interact with each other.

Particle volume matters at high pressures. When you compress a gas (like $CO_2$ at 100 atm), the particles get squeezed closer together. The space the particles themselves take up becomes a significant fraction of the container's total volume. That means there's less free space for particles to move around in, and collisions happen more often.

Intermolecular forces matter at high pressures and low temperatures. When particles are close together or moving slowly, the attractive forces between them (van der Waals forces) become significant. These attractions pull particles slightly toward each other and away from the container walls, which reduces the pressure below what the ideal gas law would predict. Think of $H_2$ at very low temperatures: the molecules are moving slowly enough that these weak attractions actually affect behavior.

At extremely short distances, repulsive forces also kick in, preventing particles from overlapping.

Van der Waals equation interpretation

The van der Waals equation modifies the ideal gas law to account for both of these real-world factors:

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

• $a$ is the correction factor for intermolecular forces. A larger $a$ value means stronger attractions between particles. For example, $CO_2$ has $a = 3.59 \text{ L}^2\text{·atm/mol}^2$.
• $b$ is the correction factor for particle volume. A larger $b$ means the particles themselves take up more space. For $CO_2$, $b = 0.0427 \text{ L/mol}$.

Here's how each correction works:

• Pressure correction $\frac{an^2}{V^2}$: This term gets added to the measured pressure because attractive forces cause the real pressure to be lower than expected. The correction grows larger when volume is small (high pressure) because particles are closer together.
• Volume correction $(V - nb)$: This term subtracts the volume occupied by the gas particles themselves from the total volume. For 1 mol of $CO_2$, you'd subtract about 0.0427 L from the container volume. That's tiny at low pressures, but it matters a lot when the gas is highly compressed.

Compressibility and Comparison of Ideal and Non-Ideal Gas Behavior

Compressibility as a non-ideal indicator

The compressibility factor ($Z$) gives you a quick way to check whether a gas is behaving ideally:

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

For a perfect ideal gas, $Z = 1$ exactly. Any deviation from 1 tells you the gas is behaving non-ideally.

• $Z < 1$: Attractive forces dominate. The gas occupies less volume than the ideal gas law predicts because particles are being pulled toward each other. This tends to happen at moderate pressures. For example, $N_2$ at 200 atm and 300 K has $Z \approx 0.8$.
• $Z > 1$: Repulsive forces and particle volume dominate. The gas occupies more volume than predicted because particles are being pushed apart or simply can't be compressed further. This happens at very high pressures, like $He$ at 1000 atm.

Ideal vs. van der Waals calculations

When should you use each equation?

• Ideal gas law $PV = nRT$: Use this at low pressures and high temperatures, where particles are far apart and moving fast. Under these conditions (like $N_2$ at 1 atm and 298 K), the assumptions hold well and calculations are simpler.
• Van der Waals equation: Use this at high pressures and low temperatures, where particle volume and intermolecular forces can't be ignored. For $CO_2$ at 50 atm and 250 K, the van der Waals equation gives noticeably more accurate results.

Comparing the two:

1. At low pressures and high temperatures, both equations give nearly the same answer (typically less than 5% difference).
2. At high pressures and low temperatures, the difference can be 20% or more. For $NH_3$ at 100 atm and 200 K, the ideal gas law gives a poor prediction, while the van der Waals equation stays much closer to experimental values.

Advanced Concepts in Non-Ideal Gas Behavior

These terms go a bit beyond intro-level content, but they're worth knowing by name:

• Critical point: The temperature and pressure at which the boundary between liquid and gas phases disappears. Above this point, the substance exists as a supercritical fluid.
• Boyle temperature: The temperature at which a real gas behaves most like an ideal gas across a wide range of pressures ($Z \approx 1$).
• Fugacity: A corrected version of pressure used in thermodynamic equations for non-ideal gases. It accounts for how the gas actually behaves versus how an ideal gas would.
• Virial equation: A more mathematically rigorous alternative to the van der Waals equation. It uses a power series expansion to describe gas behavior and can be more accurate, especially at moderate deviations from ideality.