Now that we can relate temperature to kinetic energy and gas velocity, we can use these idea to derive a relationship between all the other gas properties (specifically volume, pressure, and the number of particles).
To this end we start out with a few assumptions.
Now we simply have to do a fairly difficult physics problems to solve for the pressure felt by the walls of the container that results from the collision of the molecules. This can be found in detail here . However, without going through all the math we can arrive at a few conclusions ourselves. First, the pressure results from the number of collisions that occurs per unit time as the "impact" of those collisions. The number of collisions will be proportional to the velocity of the particles and the impact depends on their momentum (the product of mass and velocity).
Let's use this idea to examine the relationship between \(P\) and \(n\). The impact of the collision will be unchanged as we change the number of molecules. However, the number of collisions will scale directly with \(n\). Therefore we find
\[P \propto n\]
Similarly the volume,\(V\), will have no effect on impact, but as the volume increases the number of collisions will decrease proportionally. So we find that
\[P \propto {1 \over V}\]
Finally, we can look at \(P\) and \(T\). Changing the temperature changes velocity that affects both the number of collisions as well as the "impact". However since velocity is proportional to \(\sqrt {T}\) we find that
\[P \propto T\]
When we put this all together (and we actually do the math/physics part) we get a quantitative result that yields
\[PV=nRT\]
This is the ideal gas law now derived as a physical model rather than simply arrived at from empirical data at low pressures. Most interesting is now it provides us some "physical insight" into gas behavior. When gases are behaving ideal, our assumptions that lead us to our derivation must be quite accurate. Thus at low pressures where the ideal gas law holds the best, the molecules must be far apart and essentially occupying a negligible volume (assumption 1). In addition, they must not be interacting at all (assumption 3). What happens at high pressure when the ideal gas law begins to fail? The molecules get closer and closer together. Now their volume cannot be ignored (assumption 1 fails) and they begin to interact (assumption 3 fails). Thus even in the failures of the ideal gas law, we can learn something about the behavior of gases.