# Question a334f

Jun 3, 2015

Think of the ideal gas law as being a general model that encompasses all the other gas laws.

For example, you know that the ideal gas law looks like this

$P V = n R T$, where

$P$ - the pressure of a gas;
$V$ - the volume it occupies;
$n$ - the number of moles of gas present in that volume;
$T$ - the temperature of the gas.

Let's say that a have number of moles of gas that occupies a volume ${V}_{1}$ under conditions ${P}_{1}$ and ${T}_{1}$. The ideal gas law equation for this gas would be

P_1V_! = nRT_1# $\text{ } \textcolor{b l u e}{\left(1\right)}$

If I keep the number of moles of gas unchanged, i.e. I don't add or remove any molecules of gas from my sample, but change the volume to ${V}_{2}$ and the conditions to ${P}_{2}$ and ${T}_{2}$, the ideal gas law equation would be

${P}_{2} {V}_{2} = n R {T}_{2}$ $\text{ } \textcolor{b l u e}{\left(2\right)}$

If I want to express the number of moles of gas I have using the ideal gas lw equation, I can write

$n = \frac{{P}_{1} {V}_{1}}{R {T}_{1}}$ $\to$ from equation $\textcolor{b l u e}{\left(1\right)}$

and

$n = \frac{{P}_{2} {V}_{2}}{R {T}_{2}}$ $\to$ from equation $\textcolor{b l u e}{\left(2\right)}$

This means that I get

$\frac{{P}_{1} {V}_{1}}{\cancel{R} \cdot {T}_{1}} = \frac{{P}_{2} {V}_{2}}{\cancel{R} \cdot {T}_{2}} \iff {\underbrace{\frac{{P}_{1} {V}_{1}}{T} _ 1 = \frac{{P}_{2} {V}_{2}}{T} _ 2}}_{\textcolor{g r e e n}{\text{combined gas law}}}$

Both the ideal gas law and the combined gas law can be used to describe the behavior of an ideal gas under certain conditions for pressure and temperature.

However, the ideal gas law still stands if all the parameters change, whereas the combined gas law can only be used when the number of moles of gas is constant when going from one set of conditions to another.

Each of the gas laws describes the behavior of an ideal gas when at least one parameter of the ideal gas law equation is constant

• Boyle's Law $\to$ $n$ and $T$ constant;
• Charles' Law $\to$ $P$ and $n$ constant;
• Avogadro's Law $\to$ $P$ and $T$ constant;

and so on.