We can derive the ideal gas law by looking at the empirical laws of Charles, Boyle and Avogadro, but we can also find it theoretically using the kinetic theory of gases.
Hooke, an assistant of Boyle was the first to propose the idea that gas pressure is a result of collisions of randomly moving gas molecules/atoms with the sides of the container, (the explanation we use today). He didn’t advertise this idea though, as Newton had proposed an alternative, (and wrong) theory and Newton was a man that you simply didn’t disagree with. Hooke’s reputation isn’t all that great though – he may have been a brilliant man, (he developed the microscope, coined the term “cell” to describe… cells, developed Hooke’s law to describe the forces in a spring as well as contributions to astronomy, mechanics and the theory of gravitation. It was the latter that got him into trouble with Newton) but was apparently not a very nice one.
The kinetic theory is based on the idea that gas molecules/atoms are constantly, randomly moving and relies on the following postulates:
1. Gases are composed of molecules whose size is negligible, (compared to the distance between them).
2. Molecules move about randomly, but in straight lines and all at different speeds.
3. They don’t interact (attract or repulse) with each other.
4. When they do collide, the interaction is elastic (the total kinetic energy is maintained)
5. The average kinetic energy (KEav= 3/2RT) is proportional to the temperature.
To get the ideal gas equation, we remember that pressure is the result of a collision with the sides of the container and is therefore proportional to the frequency of the collisions and the average force exerted by the molecule during a collision.
P α frequency of collisions x force
The force depends on its momentum (its mass (m) x its velocity (u) – the bigger and faster it is, the more force it exerts. In other words, it hurts more to be hit with a fast bowling ball than a slow table tennis ball). The frequency of the collisions is dependant on the size of the container – a larger container will have fewer collisions, and on the speed of the molecules – the faster they are, the more often they will collide with the walls. Also, obviously, the more molecules you put in the container, the more often they will collide.
So… putting all of those things together:
P α (u x 1/V x N) x mu
If you put volume on the left:
PV α Nmu2
Kinetic energy depends on the mass and velocity of the molecules too (KE = 1/2mu2, which is also proportional to temperature).
PV α nT
(We can change N (number of molecules) to n (moles of molecules)) and when we put a proportionality constant in there we get the ideal gas equation!
PV = nRT.
The postulates for kinetic theory and the ideal gas equation are all well and good for ideal gases but they fall down for real gases. We know from experience that Boyle’s law is not valid had high pressure, (the gas will liquefy and the relationship between pressure and volume will no longer hold) and Charles’ law requires temperatures above the point where the gas will become liquid. These are true because in real life, gas particles will interact with one another, and are generally attracted to each other. They also have a real, measurable volume and cannot be compressed indefinitely. When considering real gases, we need to take these things into account and the ideal gas equation becomes the Van der Waals equation:
This version of the ideal gas equation has a correction factor (a) for the attractive forces between the molecules/atoms and (b) the real volume of the molecules/atoms.