The Ideal Gas Law

Where does $PV=nRT$ come from? The combined gas law says that for a sealed sample $\frac{PV}{T}$ is a constant. Avogadro's law says that at the same temperature and pressure, doubling the number of moles doubles the volume, so that constant must be proportional to $n$. Writing the constant as $n\times R$ gives $\frac{PV}{T}=nR$, which is the ideal gas law. So $R$ is simply the value of $\frac{PV}{nT}$ for any ideal gas, measured once and used everywhere: for exactly 1 mol at 273.15 K and 101.325 kPa, the volume is 22.4 L, and $\frac{(101.325)(22.4)}{(1)(273.15)}\approx 8.31\text{.}$ The same number works for hydrogen, oxygen or carbon dioxide, because under the KMT assumptions the identity of the particles does not matter; only how many there are.

Because $PV=nRT$ relates four quantities (with $R$ fixed), you can algebraically solve for whichever one is unknown, as long as you know the other three. The units of $R$ are a built-in checklist: $\text{kPa}\cdot \text{L}/(\text{mol}\cdot \text{K})$ means pressure MUST be in kPa, volume in L (so divide mL by 1000), amount in mol (so divide grams by molar mass), and temperature in K. How do you choose between this law and the combined gas law? If the problem describes ONE gas sample at ONE set of conditions, or involves moles or grams, use $PV=nRT$. If it describes the same sealed sample moving from a before-state to an after-state, use the combined gas law, where $n$ and $R$ cancel.

• Solve for pressure: $P=\frac{nRT}{V}$
• Solve for volume: $V=\frac{nRT}{P}$
• Solve for moles: $n=\frac{PV}{RT}$
• Solve for temperature: $T=\frac{PV}{nR}$

The ideal gas law can identify an unknown gas. Weigh a sample (mass $m$), measure its $P$, $V$ and $T$, and calculate $n=\frac{PV}{RT}$. Then the molar mass is simply $M=\frac{m}{n}$, which you can compare with the molar masses of likely candidates. Combining the two steps gives $M=\frac{mRT}{PV}$. The same idea predicts gas density ($d=m/V$): substituting $n=m/M$ into $PV=nRT$ and rearranging gives $d=\frac{PM}{RT}$. At the same temperature and pressure, a gas's density is proportional to its molar mass. Air averages about 29 g/mol, so helium (4.00 g/mol) floats up, while carbon dioxide (44.01 g/mol) sinks and collects in low spots, which is why fire extinguishers using ${\text{CO}}_2$ smother flames and why ${\text{CO}}_2$ can build up dangerously in basements and cellars. The formula also shows that heating a gas at constant pressure lowers its density, the principle behind hot-air balloons.

Recall that the ideal gas law assumes the KMT postulates hold exactly — negligible particle volume and no intermolecular attractions. Real gases follow this closely under everyday conditions, but deviate in these circumstances:

• High pressure: particles are forced close together, so their own (non-negligible) volume starts to matter, and the gas doesn't compress quite as much as the ideal law predicts.
• Low temperature: particles move slower, giving intermolecular attractive forces enough time to actually affect motion (this is also, not coincidentally, how gases eventually condense into liquids as they cool).
• Strongly attracting particles: polar molecules such as ${\text{H}}_2\text{O}$ and ${\text{NH}}_3$ attract each other more than small nonpolar ones such as He or ${\text{H}}_2$, so they deviate more under the same conditions. Helium is the most nearly ideal real gas.