$PV = nRT$ assumes two things that are not true: that molecules take up no room, and that they do not pull on one another. At a bar or two nobody notices. At a hundred and fifty bar in a nitrogen bottle, the error is worth real money.

The honest fix is one number. $Z = \dfrac{PV}{nRT}$ — Z equals P V over n R T — the compressibility factor, where $P$, $V$, $n$, $T$ and $R$ are exactly as before and $Z$ itself is dimensionless, because it is a measured PV divided by the PV an ideal gas would have had. $Z = 1$ is ideal. Below 1, attraction is winning: molecules pull each other back from the wall, so the gas is more compressible than ideal and the bottle holds MORE than the ideal law claims. Above 1, the molecules' own volume is winning: there is less free space than the arithmetic assumed. With the correction in place, the working form is $PV = ZnRT$.

Van der Waals built the same two corrections into the equation itself: $\left(P + \dfrac{an^2}{V^2}\right)\left(V - nb\right) = nRT$. Read it as the ideal law with two apologies. $b$ is the excluded volume in m³ per mole — the room the molecules themselves occupy — so the space actually available is $V - nb$, not $V$. $a$ is the attraction constant in Pa·m⁶/mol², and $\dfrac{an^2}{V^2}$ is how much the pull between molecules softens each impact on the wall. Set $a$ and $b$ to zero and the ideal gas law walks back out.

Which correction wins is a property of the gas. Carbon dioxide attracts itself strongly — $a = 0.364$ — so at ordinary densities its real pressure comes out BELOW ideal. Hydrogen barely attracts itself at all — $a = 0.0248$ — so what is left is molecules taking up room, and its pressure comes out ABOVE ideal. Same equation, opposite verdicts, and knowing which to expect before you compute is the skill this lesson is really after.