The van der Waals Equation, Made Simple

The van der Waals equation looks like one of those cryptic formulas the JEE syllabus throws at you to scare you off:

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

Stay with me. By the end of this lesson, every symbol in there will make sense — and you'll see it's just the ideal gas law with two honest repairs bolted on.

First, why we even need it

To study gases, scientists built a model: the ideal gas. It obeys one famous law you already know.

$PV = nRT$

And real gases go along with it happily — as long as the pressure stays low and the temperature stays high, so the molecules are spread far apart.

But squeeze a real gas, or cool it down, and it pushes back: "Welcome to the real world. We don't follow your ideal rules here."

That happens because the ideal model is built on two convenient lies:

• Lie 1: molecules don't attract each other. The model assumes every bit of pressure on the wall is the real, full pressure.
• Lie 2: molecules take up no space. The model treats them as tiny dots with zero volume.

Both are fine when molecules are far apart and rarely meet. Both fall apart when you pack them close. So a real gas needs two corrections — one for pressure, one for volume. That's the whole idea behind van der Waals.

The pressure correction — the a term

Think about a molecule deep inside the gas. Its neighbours pull on it from every side, so the pulls cancel out.

Now think about a molecule about to slam into the wall. There are no gas molecules outside the wall pulling it forward — only molecules behind it, pulling it back inward. So it hits the wall a little softer than it would if nothing tugged on it.

Every molecule near the wall gets held back like this. The result: a real gas pushes on the wall with less pressure than an ideal gas would. The measured pressure is too low.

To get back the "ideal" pressure, we add the missing bit back on:

$P_{ideal} = P_{real} + \frac{an^2}{V^2}$

Read the correction term piece by piece:

• a measures how strongly the molecules attract each other. Sticky gases (like water vapour or ammonia) have a big a. Barely-interacting gases (like helium) have a tiny a.
• n^2/V^2 is the concentration squared. Why squared? Because attraction needs two molecules — one being pulled, one doing the pulling. Pack them tighter (more n, less V) and both the pulled and the pullers crowd together, so the effect grows with the square.

In short: more molecules in less space means a stronger inward tug, so a bigger correction.

The volume correction — the b term

Now the second lie. Real molecules are not points — they have actual size. So the room a molecule can roam in is not the full container.

$\text{free volume} = \text{total volume} - \text{volume blocked by the molecules}$

Each mole of gas blocks out a fixed chunk of space, written b. For n moles, the blocked space is nb. So everywhere the ideal law used $V$, the real gas only gets to use $V - nb$:

$V_{free} = V - nb$

Read b like this: it's the "personal space" of one mole of the gas — roughly the volume the molecules themselves occupy. Big, fat molecules have a big b. Small ones have a small b.

Putting both repairs together

Take $PV = nRT$. Swap the honest pressure in for $P$, and the free volume in for $V$, and the two corrections click into place:

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

For exactly one mole, set $n = 1$ and it shrinks to the version you'll often see in problems:

$\left(P + \frac{a}{V^2}\right)(V - b) = RT$

That's it. The same ideal gas law, with a fixing the attraction lie and b fixing the size lie.

Your turn. Two gases sit in identical containers at the same temperature and number of moles. Gas X has a large a; gas Y has a tiny a. Which one pushes on the walls with more pressure?

Check: Gas Y. A large a means strong inward attraction, which holds molecules back from the wall and lowers the real pressure. Weak attraction (small a) lets the molecules hit nearly as hard as an ideal gas — so gas Y reads higher.

What JEE actually asks

A few high-value points the examiners come back to:

• Units of a and b. To work them out, always use the full n-mole equation, not the one-mole shortcut. From $\frac{an^2}{V^2}$ having units of pressure, you get a in $\text{atm·L}^2\text{·mol}^{-2}$. From $nb$ having units of volume, b comes out in $\text{L·mol}^{-1}$.
• At low pressure, the volume is large, so $nb$ is negligible next to $V$ and the gas behaves nearly ideal — the a term dominates the small deviation. Real pressure ends up a touch lower than ideal.
• At high pressure, the volume is squeezed small, so the b term takes over — molecular size matters most, and the gas is harder to compress than ideal.
• A gas behaves most ideally when both a and b are small — which is why helium and hydrogen come closest to ideal.

The one-line summary

The ideal gas law assumes molecules neither attract nor take up space. Both are false for a real gas. The a term adds back the pressure lost to attraction; the b term subtracts the volume the molecules occupy. Fix both, and $PV = nRT$ becomes the van der Waals equation. As simple as that.