Scientist use models to try to understand and predict behaviors in the world. These models can be broadly classified into two types: empirical and physical. An empirical model is one that simply seeks to provide a mathematical relationship between different properties based solely on observation. The combined gas law is an example of an empirical model that is a relationship between the properties of volume, temperature, and pressure for a gas. Based on the combined gas law a chemist could predict the pressure of a known amount of gas given its volume and temperature. A physical model is different in that it seeks to not only predict but to provide some physical insight. The Kinetic Molecular Theory is an example of a physical model. It starts with some assumptions (the particles are small and non-interacting, the particles are in constant motion,...) and then from there seeks to derive a relationship between the properties. In this case, it happens to derive the identical relationship that we refer to as the ideal gas law.
We can therefore either look at the ideal gas law as an empirical model derived from observation (combination of experiments from Boyle, Charles, ....) or a physical model derived from kinetic molecular theory. It is more useful to think the ideal gas law from the perspective of a physical model. This is because one of the most important aspects of models is the insight we gain from their failings. Many scientific models start by simplifying the behaviors observed in the real world. By testing the models versus real data we can see when the assumptions of the model are valid and when the simplifications they make should be called into question.
For example, what do we gain from the fact for typical conditions (1 atm, room temperature) the ideal gas law does a remarkable job of accurately predicting the relationship between the volume, temperature, and pressure of essentially every gaseous substance? This is a remarkable feat. Why should Xe behave like N2? Why should acetone vapor behave anything like ammonia? The fact that all substances appear to be the same is a validation of our small particle kinetic molecular theory model. The KMT model assumes that gas particles are non-interacting and derives the ideal gas law. Non-interacting particles means that the molecules have no intermolecular forces. They are not repulsive nor are they attractive. When the model is correct, its assumptions must be valid. The ideal gas model assumes there are no interactions at all and it predicts the correct behavior (in most conditions). How can it be possible that there are no intermolecular forces? In a gas, under conditions where the ideal gas law is valid, the particles must be sufficiently far apart that they are in fact non-interacting.
This would suggest two conditions for which gases would not behave in this ideal non-interacting fashion. First, when they get very close together. Second, when their intermolecular force are very strong.
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 ModelAs 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.