Ideal gas law

Imagine a cube of volume $V = L^3$, containing some $N$ particles of gas, each with mass $M$ and velocity $\mathbf v_i$.

The gas particles don’t interact with each other, and just collide elastically with the walls of the box.

Since particles are reflected elastically, each collision (in this case in the $x$ direction) has a change in momentum $\Delta \mathbf p_{i,x} = 2 M v_{i,x}$. Since the walls have length $L$, the time between collisions is $\Delta t_{i,x} = 2L/v_{i,x}$.

This gives us an average force in the $x$ direction by the $i^\text{th}$ particle $F_{i,x} = \Delta p_{i,x} / \Delta t_{i,x} = \frac ML v_{i,x}^2$. The total force from all $N$ particles is then

We can use this to find the pressure,

This is the ideal gas law, also known as the state equation for an ideal gas.

Note that the box doesn’t need to be a cube for this to hold. It’s generally a good approximation for gasses at every-day temperatures and pressures.