The ideal gas law is great in many situations, but under many real world conditions it does not accurately describe gas properties. One of the first problems we can imagine for the ideal gas law is what happens to the volume in the limit of high pressure. As the pressure gets very very high the volumes get very very small. In the limit of infinite pressure the volume should actually go to zero! This clearly cannot be the case in the real world. This is because the particles that make up the gas take up some volume themselves. The ideal gas law assumes the particles are not interacting. This means they can all occupy the same space. However, this is not realistic. This is because the gas particles themselves occupy some volume. In most circumstances, this volume is a tiny fraction of the total volume and thus it can be ignored. However as the volume shrinks this "volume of the particles" becomes a larger and larger fraction as depicted below.
The ideal gas does not account for this affect. We can modify the ideal gas equation to deal with this by replacing the volume of the container with the volume of the space between the particles. This is the volume of the container minus the volume physically occupied by the molecules. This is the hard sphere gas model
\[P(V-nb)=nRT\]
\(b\) is a constant that is the molar volume of the gas particles themselves (the actual volume of 1 mole of the atoms or molecules with no spaces in between). This constant will be different for different gases. In particular we would expect large molecules to have a large \(b\) since they will occupy more space than small molecules or atoms. If we rearrange this equation to solve for the volume we find
\[V={nRT \over P} + nb=V_{IG} + nb\]
That is the volume of the gas. In this case it is the volume predicted by the ideal gas law, \(V_{IG}\) plus some small correction that is the actual size of the molecules. At low pressures when \(V_{IG} >> nb\) this correction won't matter and you'll get a nearly identical result to the ideal gas law. However, when the pressure is high and \(V_{IG}\) becomes smaller this little correction will becomes increasingly more important.
Thus the hard sphere model attempts to deal with one aspect that is ignored in the ideal gas law: the repulsive forces between the gas particles. The particles themselves cannot occupy the same space. Under standard conditions, the repulsive forces are not relevant as the particles are sufficiently far apart. However, as the pressure is increased (the volume is decreased) the assumption that the molecules are not occupying any physical space becomes unrealistic. The hard sphere model corrects this by adding this new term to the ideal gas law.
Hard Sphere Model