Animated Solution for Chemistry - States of Matter: In van der Waals' equation of state of the gas law, the constant 'b' is a measure of

Welcome, future engineers and scientists! Today, we are going to shatter a perfect illusion that we often rely on in physics and chemistry. We are going to talk about why the ideal gas law isn't always so ideal, and how a brilliant scientist named Johannes van der Waals fixed it.

In the realm of the Kinetic Theory of Gases, we often make a very convenient, yet physically impossible assumption. We assume that gas molecules are point masses. This means we pretend they have zero volume—just tiny mathematical dots bouncing around in a container.

Under this assumption, the entire volume of the container, $V$, is completely free and available for any molecule to move through. This leads us to the beautiful and simple ideal gas equation:

But nature doesn't work like that. Atoms and molecules are real, physical entities. They have electron clouds. They take up space.

Molecules Have Size

Imagine you are in an empty room. You can walk anywhere freely. Now, imagine filling that room with a hundred giant yoga balls. Your 'free volume' to move has drastically decreased because the yoga balls are occupying physical space.

This is exactly what happens in a real gas. Because the molecules themselves have a finite volume, the actual free space available for them to move is less than the total volume of the container.

To understand how much space is lost, let's dive into the geometry of a molecular collision. Imagine two spherical gas molecules, each with a radius $r$. When they collide, how close can their centers get?

Because they are rigid spheres, their centers can only get as close as $2r$. They cannot overlap. This means that around every molecule, there is a spherical 'no-entry' zone of radius $2r$ where the center of another molecule simply cannot enter.

This forbidden zone is called the excluded volume.

Johannes van der Waals realized this flaw in the ideal gas equation. He knew that the volume $V$ in $pV=nRT$ should actually be the free volume, not the total volume.

He introduced a volume correction term. Instead of the total container volume $V$, the molecules only have $(V-nb)$ available to them, where $n$ is the number of moles and $b$ is the excluded volume per mole.

The modified equation for one mole of gas looks like this:

But what exactly is this constant '$b$'? Let's calculate the volume of that 'no-entry' sphere we talked about. The volume of a sphere with radius $2r$ is:

Since this excluded volume is shared by two colliding molecules, the excluded volume per molecule is half of that:

Notice that $\frac{4}{3}\pi r^3$ is the actual volume of one single molecule. So, the excluded volume is exactly four times the actual volume of the molecule!

For one mole of gas, we multiply this by Avogadro's number, $N_A$. This gives us our constant $b$:

This beautiful geometric derivation proves that the van der Waals constant '$b$' is a direct measure of the effective volume occupied by the gas molecules themselves. It accounts for the finite size of the particles, correcting the overly simplistic assumption of the ideal gas law.

Therefore, the correct answer is that '$b$' is a measure of the volume occupied by the molecules.