Matter is made of molecules

Where does the gas go?

Blow up a balloon and tie it tightly. Look at it the next day. It is smaller. A helium balloon goes down even faster.

The gas got out through the rubber. How can it do that? If a gas were a smooth fluid with no gaps, nothing could pass through solid rubber. It passes because a gas is made of very small particles. The rubber has gaps between its own molecules, and a helium atom is small enough to slip through them.

This idea is old. About 2500 years ago Kanada in India and Democritus in Greece both guessed that matter is made of small pieces that cannot be cut. They called them atoms. Today we have strong evidence for them.

Atoms and molecules

John Dalton built the modern atomic theory around 1800. All atoms of one element are the same. Atoms of different elements differ in mass. A compound is made of molecules, and each molecule has a fixed number of atoms of each kind.

Gases gave the next clue. Gay-Lussac found that gases react in simple ratios of their volumes. Two volumes of hydrogen and one volume of oxygen make two volumes of steam. Avogadro explained it in 1811:

Avogadro's law. Equal volumes of all gases, at the same temperature and pressure, hold the same number of molecules.

One mole of any substance holds $6.02 \times 10^{23}$ particles. This number is the Avogadro number, $N_A$. At $273$ K and $101.3$ kPa, one mole of any gas fills $22.4$ L.

How big are they, and how far apart?

An atom is about $10^{-10}$ m across, that is $0.1$ nm. A small molecule such as nitrogen is about $0.3$ nm across.

In a solid the molecules sit close together, in a regular pattern. In a liquid they are just as close, but they have no fixed pattern and can slide past each other. In a gas they are far apart. At room conditions the spacing is about ten times the size of a molecule.

How do we know the spacing in a gas? Use the gas law of the next lesson, $pV = Nk_BT$. Each molecule gets a small cube of room, of volume $V/N$:

Here $k_B = 1.38 \times 10^{-23}$ J/K is the Boltzmann constant. The side of that cube is the average spacing $d$.

Push the pressure up to $2000$ kPa. Does the spacing ever come close to the size of a molecule? It does not, but it gets nearer. You will see in the third lesson why a gas stops being ideal then.

A worked example

How many molecules are there in one cubic centimetre of air, at $300$ K and $101.3$ kPa?

1. The number in each cubic metre. From $pV = Nk_BT$, we want $N/V$, so $\dfrac{N}{V} = \dfrac{p}{k_BT} = \dfrac{101.3 \times 10^3\ \text{Pa}}{1.38 \times 10^{-23}\ \text{J/K} \times 300\ \text{K}} = 2.45 \times 10^{25}\ \text{m}^{-3}$.
2. One cubic centimetre is $10^{-6}$ m³. So it holds $2.45 \times 10^{19}$ molecules.
3. Check. That is about twenty billion billion. The number is huge because each molecule is so small.

Brownian motion

In 1827 Robert Brown looked at pollen grains in water under a microscope. The grains moved all the time, in a jerky, random way. You can see the same thing with smoke in still air.

Why do they move? Water molecules hit each grain from every side. At any instant the hits on one side do not quite balance the hits on the other side. So the grain is pushed a little this way, then a little that way.

• A smaller grain jiggles more. Fewer hits fall on it, so the imbalance matters more.
• A hotter liquid makes the grain jiggle faster. Its molecules move faster.

In 1905 Albert Einstein worked out how far a grain should wander. Jean Perrin measured it, and from it he found the Avogadro number. This was strong proof that molecules are real.

Practise with new numbers

Try it in Python

The program finds how much room each molecule has in liquid water and in steam. It uses the molar mass and the density for the liquid, and $V/N = k_BT/p$ for the steam.

Common mistakes

• Thinking the molecules of a gas touch. They are far apart. Most of a gas is empty space.
• Using the volume per molecule as the spacing. The spacing is the cube root of the volume per molecule.
• Saying Brownian motion is the motion of molecules. We see the grain move. The molecules that hit it are far too small to see.
• Using grams in place of kilograms for a molar mass. Water is $18$ g/mol, which is $0.018$ kg/mol.

Remember these

• One mole holds $N_A = 6.02 \times 10^{23}$ particles.
• Avogadro's law: equal volumes of gases, at the same $p$ and $T$, hold equal numbers of molecules.
• Size of an atom: about $10^{-10}$ m. Spacing in a gas at room conditions: about ten times larger.
• Volume for each molecule of a gas: $k_BT/p$.
• Brownian motion: unequal hits from molecules make a small grain jiggle.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Why does a helium balloon go down faster than a balloon of air?

2. How many molecules are in $0.75$ mol of oxygen?

3. A drop of milk in water is seen under a microscope. The fat drops jiggle. What happens to the jiggling if the water is warmed?

4. At the same temperature and pressure, $1.0$ L of hydrogen holds $N$ molecules. How many molecules does $3.0$ L of carbon dioxide hold?