Why is the pressure correction in the van der Waals equation proportional to (n/V)^2?
Summing up the argument from the Wikipedia derivation, the reason it is proportional to $(\frac{1}{V_\mathrm m})^2$ where $V_\mathrm m=\frac Vn$ (the molar volume) comes from how the molecules experience the attractive force. The only molecules that experience a net attractive force are those close to the edge of the container because as we move toward the center of the container, the attractive forces on either side of a molecule cancel out. The force toward the center experienced by a molecule at the edge should be proportional to $(\frac{1}{V_\mathrm m})$. But we can also say that the number of molecules at the edge of the container should be proportional to $(\frac{1}{V_\mathrm m})$. So, by combining the proportionality constants into a single constant $a$, we obtain that the correction to the pressure should be $(\frac{a}{V_\mathrm m^2})=\frac {an^2}{V^2}$ .