# How can Gay-Lussac's law can be derived from the combined gas law?

Jul 24, 2017

If memory serves, that is the $\frac{P}{T}$ relation, which is at constant volume and mols of gas.

We could start from the "combined gas law", but it came from the ideal gas law, so it would be simpler to start from the ideal gas law:

$P V = n R T$

Define the initial and final states, with constant $V$ and $n$
(${V}_{1} = {V}_{2} \equiv V$, ${n}_{1} = {n}_{2} \equiv n$):

${P}_{1} V = n R {T}_{1}$ $\text{ } \boldsymbol{\left(1\right)}$

${P}_{2} V = n R {T}_{2}$ $\text{ } \boldsymbol{\left(2\right)}$

By division,

$\frac{\left(2\right)}{\left(1\right)} = \frac{{P}_{2} \cancel{V}}{{P}_{1} \cancel{V}} = \frac{\cancel{n R} {T}_{2}}{\cancel{n R} {T}_{1}}$

Therefore,

${P}_{2} / {P}_{1} = {T}_{2} / {T}_{1}$

or

$\textcolor{b l u e}{{P}_{2} / {T}_{2} = {P}_{1} / {T}_{1}}$

which is Gay-Lussac's Law.