Ideal Gas Equation

Boyle's law ($V \propto 1/P$ at constant $n,T$), Charles's law ($V \propto T$ at constant $n,P$), and Avogadro's

law ($V \propto n$ at constant $P,T$) can each be combined into a single proportionality:

Introducing a proportionality constant $R$, called the universal gas constant, converts this into an equation:

This is the ideal gas equation, the single most important equation of state in this chapter, relating pressure

$P$, volume $V$, number of moles $n$, and absolute temperature $T$ for an ideal gas. The value of $R$ depends only

on the units chosen for $P$, $V$, and $T$: $R = 0.0821\ \text{L atm K}^{-1}\text{mol}^{-1}$ when pressure is in

atmospheres and volume in litres, or $R = 8.314\ \text{J K}^{-1}\text{mol}^{-1}$ (equivalently

$8.314\ \text{Pa m}^3\text{K}^{-1}\text{mol}^{-1}$) in SI units.

Worked example — volume from the ideal gas equation. Find the volume occupied by $2\ \text{mol}$ of an ideal

gas at $300\ \text{K}$ and $1\ \text{atm}$. Rearranging, $V = \dfrac{nRT}{P} = \dfrac{2 \times 0.0821 \times 300}{1} = 49.26\ \text{L}$.

Density and molar mass from the ideal gas equation. Since $n = \dfrac{\text{mass}}{M} = \dfrac{m}{M}$, the

ideal gas equation can be rewritten as $PV = \dfrac{m}{M}RT$, and rearranged to give

where $\rho = m/V$ is the density of the gas. This is an extremely useful relationship: measuring the density of an

unknown gas at a known temperature and pressure lets a chemist calculate its molar mass directly, without needing

to know its chemical identity in advance. For example, a gas with a density of $1.964\ \text{g L}^{-1}$ at STP

($0^\circ\text{C} = 273\ \text{K}$, $1\ \text{atm}$) has molar mass

which is the molar mass of $\text{CO}_2$ ($12 + 2\times16 = 44$) — a good illustration of how a single density

measurement can identify (or at least narrow down) an unknown gas.

At STP (Standard Temperature and Pressure, $0^\circ\text{C}$ and $1\ \text{atm}$), one important consequence of …