Molecules and the Gas Laws: Quick Revision

A brief note on Molecules and the Gas Laws. Read it before a test. Try each question at the end before you open its answer.

Molecules

• Avogadro number: $N_A = 6.02 \times 10^{23}$ particles in one mole.
• Avogadro's law: equal volumes of gases at the same $p$ and $T$ hold equal numbers of molecules.
• Sizes: an atom is about $10^{-10}$ m across. In a gas at room conditions, molecules are about ten times their size apart.
• Volume for each molecule of a gas: $k_BT/p$. The spacing is its cube root.
• Brownian motion: unequal hits from molecules make a small grain jiggle. Smaller grains and hotter liquids jiggle more.

The gas laws

• Boyle: $pV$ constant at a steady $T$.
• Charles: $V \propto T$ at a steady $p$.
• Pressure law: $p \propto T$ at a steady $V$.
• Ideal gas equation: $pV = nRT = Nk_BT$.
• Constants: $R = 8.31$ J/(mol K), $k_B = R/N_A = 1.38 \times 10^{-23}$ J/K.
• Fixed amount: $p_1V_1/T_1 = p_2V_2/T_2$.
• Density: $\rho = pM/(RT)$.
• Always use kelvin and the absolute pressure.

Mixtures and real gases

• Dalton's law: $p = p_1 + p_2 + \dots$, with $p_i = x_ip$.
• Each gas fills the whole volume.
• A real gas is close to ideal at low pressure and high temperature.
• $pV/(nRT)$ below $1$: the pull between molecules wins. Above $1$: their size wins.

A worked example

A tank holds $2.0$ mol of oxygen and $6.0$ mol of nitrogen at a total pressure of $320$ kPa. Find the partial pressure of the oxygen.

1. Mole fraction. $x = \dfrac{2.0\ \text{mol}}{8.0\ \text{mol}} = 0.25$.
2. Dalton. $p = 0.25 \times 320\ \text{kPa} = 80$ kPa.

Practise with new numbers

Check yourself

1. A gas at $27\ ^\circ\text{C}$ is heated at a steady pressure until its volume doubles. What is its new temperature in degrees Celsius?

2. Which gas is denser at the same $p$ and $T$: helium or argon?

3. Why do equal volumes of hydrogen and oxygen at the same $p$ and $T$ have different masses?