Q.The van der Waals' equation of state explains the behaviour of which of the following?

Why the Ideal Gas Law Fails

The ideal gas equation $PV = nRT$ assumes molecules are point particles with no interactions — they bounce off walls perfectly and never attract or repel each other. That works fine at low pressures and high temperatures, where molecules are far apart and moving fast. But crank up the pressure or lower the temperature, and reality bites.

Two things break the ideal picture:

1. Molecules have finite size. In the ideal model, the entire container volume $V$ is available for motion. But real molecules occupy some space themselves. The volume actually free for them to move in is less than $V$.

2. Molecules attract each other. At close range, intermolecular forces (van der Waals forces) pull molecules together. This reduces the force with which they hit the walls — the measured pressure is lower than what an ideal gas would exert.

The van der Waals equation corrects both problems in a simple, physically motivated way.

The Corrections, Step by Step

Volume correction — the $b$ term

If each mole of gas molecules occupies a volume $b$ that is excluded (the molecules themselves take up space), then for $n$ moles the free volume is not $V$ but $V - nb$. So we replace $V$ with $V - nb$:

Here $b$ is called the co-volume or excluded volume per mole. It is roughly four times the actual volume of one mole of molecules.

Pressure correction — the $a$ term

Attractive forces between molecules reduce the pressure. A molecule about to hit the wall is pulled back by neighbours. The reduction in pressure is proportional to:

• the number of molecules near the wall (density $n/V$), and
• the number of molecules pulling them back (also density $n/V$).

So the pressure drop is proportional to $(n/V)^2$. We add a correction term $a n^2 / V^2$ to the measured pressure $P$ to recover the pressure an ideal gas would exert:

The constant $a$ measures the strength of intermolecular attraction — larger $a$ means stronger attraction.

The Full Equation

Putting both corrections together:

This is the van der Waals equation for $n$ moles of a real gas.

For one mole ($n = 1$), it simplifies to:

What $a$ and $b$ Mean Physically