Kinetic Molecular Theory

So, the Kinetic Molecular Theory(KMT) is a framework that explains why gases, like the ones above, behave in roughly the way they do. Note that the gas molecules in the GIF above have been slowed down a lot and in reality, move a lot faster than shown.

Notice that not every molecule in the GIF has the same speed. The speeds of the molecules vary from molecule to molecule but the KMT assumes there are enough molecules such that the average velocity can be discerned. This is a fair assumption to make because even in just a mole of molecules, there's much well over billions of particles, so we're not exactly lacking particles.

Since the velocities vary, they follow a statistical distribution known as a Maxwell-Boltzmann Distribution:

The higher a point is, the more molecules have a given speed. Now, we'll take the ideas communicated in this graph and relate it to the temperature of a gas. However, we first make some assumptions:

1) The gas particles are very small, and so they can be assumed to have effectively no volume. In other words, if you subtracted the total volume of the molecules in a gas from the volume of the container tha gas is in, there would be pretty much no difference.

This is a fair assumption because molecules are extremely small compared to real-life containers.

2) Particles don't exert forces on each other outside when they collide with each other. This means any form of electromagnetic(like intermolecular forces) and gravitational interaction is negligible.

This is a fair assumption because the particles are so fast and they are essentially massless.

3)The particles have elastic collisions with the walls of the container so their momentum and kinetic energy is conserved.

This is a fair assumption to make because many molecules are rounded out in general.

The final equation for the KMT uses this assumptions to formulate the following relation:

Notice that the left side is the kinetic energy of the gas, where m is the total mass of the gas. But what is v_rms? Well, this is a quantity known as the root-mean-square speed, which is actually exactly what it sounds like. The root-mean-square speed of a gas is given by:

where the v terms squared are on top. Their sum is divided by the number of particles n. This gives the mean of squares of velocities. Then, you square root this and get the root-mean-square speed of the gas. This quantity is important because velocity is a vector, so the total velocity would be somewhere around 0 if we took into account particle directions. However, squaring velocities gives us pure speeds which we want.