As we saw when looking at the hard sphere gas model, the idea gas models does not accurately model the behavior of real gases in all situations. This is because the ideal gas model is that of a gas of completely non-interacting particles. The particles have no attraction for one another and no repulsions for one another. Real gas particles do. However, since these attractions and repulsions typically have a distance dependence that is very short ranged. That means that when the particles are separated by distances that are on par with their size these forces are very small. Therefore, for many gases the ideal gas model works well since under typical conditions (SaTP) the gas particles are separated by distances that are large compared to the particle size. None the less, we can develop a more complex model for a gas that includes both attractions and repulsions. This is the van der Waals (VDW) model for a gas. Like the hard sphere model it is similar to the ideal gas equation except that it modifies both the pressure and the volume.
\[\left(P + {an^2 \over V^2}\right)(V-nb)=nRT\]
\(b\) 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. This is identical to the hard sphere model. \( a\) is a constant that accounts for the attractions between the gas particles. The larger the value of \(a\) the greater the attractions between the particle. Note: that in the limit of very low pressure when the particles are very far apart and the gas behaves ideally, the VDW equation reduces to the ideal gas law. This is because as the pressure gets very low the volume gets very large. Once the volume is very large \(V - nb \approx V\). Similarly, as the volume gets large then \({an^2 \over V^2} \approx 0\). The VDW equation can easily be solved for pressure, but it is difficult to solve for volume since it has three roots.