# How does the ideal gas law differ from the combined gas law?

May 17, 2014

Both laws can be used to determine the pressure, volume or temperature of a gas at certain conditions, but there are some differences between them.

First, the combined gas law assumes you are taking a sample of a gas at one set of conditions and changing $P$, $V$, or $T$. The number of gas molecules, and therefore the number of moles, $n$, must stay constant. The ideal gas law doesn't need this restriction. The ideal gas law can also be used to determine the density of a gas, something that the combined gas law can't do.

Just to show you, if we keep the number of molecules constant, we can derive the combined gas law from the ideal gas law.

$P V = n R T$

Solve for the number of moles, $n$
$n = P \frac{V}{R T}$

Since the combined gas law is used for changes in conditions, we'll set it equal to itself, with subscripts for each set of conditions.

$\frac{{P}_{1} {V}_{1}}{{T}_{1} R} = \frac{{P}_{2} {V}_{2}}{{T}_{2} R}$

R is a constant so we can drop it out and we're left with the combined gas law

$\frac{{P}_{1} {V}_{1}}{{T}_{1}} = \frac{{P}_{2} {V}_{2}}{{T}_{2}}$