Ideal Gas Law

Derivation

Starting with a picture of molecules bouncing around a container, as the molecule hits the wall with some force, it is reflected causing some pressure against the walls of the container. The definition of pressure is

$$P=\frac{F}{A}[\mathrm{Pa}]$$

with pressure in units of Pascals $[\mathrm{Pa}]=[N/{\mathrm{m}}^2]$, $F$ being the force, and $A$ being the pressure exerted on the wall. There must be some sense of how many molecules there are so we use the number of mols which is defined as

$$N_{mols}=\frac{N}{N_{Av}}$$

with $N_{Av}=6.023\times 10^{23}[\frac{\mathrm{molecules}}{\mathrm{mol}}]$ being Avogadro’s number and $N$ being the number of molecules in the container. The ideal gas law can then be defined as

$$PV=N_{mols}RT$$

where $R=8.314[\frac{\mathrm{J}}{\mathrm{mol}\cdot \mathrm{K}}]$ is the universal gas constant, $T$ is the temperature, and $V$ is the volume of the container. This says as temperature rises, pressure rises.

Can make a simplification and define Boltzmann’s constant

$$k=\frac{R}{N_{Av}}=1.3807\times 10^{-23}[\frac{\mathrm{J}}{\mathrm{K}}]$$

making the ideal gas law that is typically used in Physics become

$$PV=NkT$$ $$P=\frac{N}{V}kT$$ $$P=nkT$$

where $n=\frac{N}{V}$ is the number density.