# What is the relationship between Charles law and the kinetic theory of gases?

Jun 6, 2017

From the kinetic theory, one derives the pressure equation,

$p = \frac{m n {v}^{2}}{2}$ where $m$ is mass of a molecule, $n$ is no. of molecules in unit volume, and $v$ is rms speed.

Thus, $n = \frac{N}{V}$ where, $N$ is total number of gas molecules.

Therefore, one may write,

$p V = \frac{m N {v}^{2}}{3}$

Now, $\frac{m {v}^{2}}{2} = E$ where $E$ is the kinetic energy of a molecule.

Thus, $p V = \frac{2 N E}{3}$

Now from the kinetic interpretation of temperature,

$E = \frac{3 k T}{2}$ where $k$ is the Boltzmann constant.

Thus, $p V = N k T$

Now since, $N$ and $k$ are constants, then for a fixed $p$,

$\frac{V}{T} =$ constant

This is Charles' law from kinetic theory.