van der Waals Equation — Definition

Definition

Imagine an 'ideal gas' as a theoretical concept where gas particles are tiny points with no volume and absolutely no attraction or repulsion between them. They just bounce around perfectly elastically.

The 'ideal gas law', $PV = nRT$, works great for these imaginary gases, especially at low pressures and high temperatures, where real gases *act* pretty ideally. However, real gases are not ideal. Their molecules *do* have a finite size, and they *do* exert attractive forces on each other.

These two factors cause real gases to deviate from ideal behavior.

This is where the van der Waals equation comes in. It's like an upgrade to the ideal gas law, trying to make it more realistic for actual gases. Johannes Diderik van der Waals realized that to describe real gases better, we need to make two crucial corrections to the ideal gas law:

1
Volume Correction (The 'b' term): — In an ideal gas, the volume 'V' in $PV=nRT$ is the entire volume of the container. But real gas molecules themselves take up space. This means the actual volume available for the molecules to move around in is *less* than the container volume. Van der Waals introduced a term, 'b', which represents the 'excluded volume' per mole of gas. So, for 'n' moles, the effective volume available for movement becomes $(V - nb)$. This 'b' value is related to the size of the gas molecules – larger molecules have a larger 'b'.

1
Pressure Correction (The 'a' term): — Ideal gas molecules don't attract each other. But real gas molecules do. When a gas molecule is about to hit the wall of the container (which causes pressure), it's pulled back by other molecules behind it. This 'pull' reduces the force of its impact on the wall, meaning the observed pressure is *less* than what it would be if there were no attractive forces. Van der Waals added a term, $a\frac{n^2}{V^2}$, to the observed pressure 'P' to account for these attractive forces. This 'a' value is a measure of the strength of intermolecular attractive forces – stronger attractions mean a larger 'a'.

So, by applying these two corrections, the ideal gas law $P_{ideal}V_{ideal} = nRT$ transforms into the van der Waals equation:

$$(P + a\frac{n^2}{V^2})(V - nb) = nRT$$

This equation helps us understand why real gases can be liquefied and why they don't always follow the simple $PV=nRT$ relationship. It's a cornerstone for understanding the behavior of real gases in chemistry.