12.1 The Molecular Nature of Matter and the Behaviour of Gases

Put a drop of ink into a glass of still water and, without any stirring, the colour slowly spreads through the whole glass. Squeeze the air in a bicycle pump with your finger over the outlet and it pushes back harder the more you squeeze; let it go and it springs back. A sealed packet of crisps carried up a mountain puffs up like a pillow. These everyday observations are hard to explain if matter is a smooth, continuous jelly. They are easy to explain if matter is made of an enormous number of tiny particles that are always moving — and, in a gas, are mostly empty space with a few particles flying about in it.

This section collects the evidence for that molecular picture and puts numbers on it: how big molecules are, how far apart they sit in a solid, a liquid and a gas, and how to count them with the mole and the Avogadro constant. Then it re-reads the gas laws you met with the ideal-gas equation in Chapter 10 (Thermal Properties of Matter) — Boyle's law, Charles's law, Avogadro's hypothesis and Dalton's law of partial pressures — in the language of molecules, through the molecular form of the gas equation, $pV=Nk_BT$. Section 12.2 will then explain why gases obey it.

The molecular nature of matter

The idea that matter is made of indivisible particles is very old — it appears in ancient Indian philosophy (Kanada's paramanu) and in Greek thought (Democritus's atomos) — but for two thousand years it was a speculation. It became science around 1800, when chemists found that elements always combine in fixed, simple ratios by mass (Dalton's atomic theory), and that gases react in simple ratios by volume (Gay-Lussac). Today the evidence is overwhelming: the jittering of pollen grains in water (Brownian motion, explained by Einstein in 1905 as the grains being knocked about by water molecules), X-ray diffraction from crystals, and microscopes that image individual atoms.

Atoms are about $10^{-10}\ \text{m}$ ($1$ ångström, $1$ Å) across. In solids they are packed tightly, at a spacing of about $2$ Å, held near fixed positions by strong forces, though they vibrate about them. In liquids the spacing is about the same, but the molecules are not tied to fixed positions and can slide past one another. In gases at ordinary pressures the molecules are about ten times further apart, tens of ångströms, so the forces between them are negligible except during brief collisions: each molecule travels freely in a straight line until it hits another molecule or a wall. That is why gases are easy to compress, fill any container and mix freely, and why their behaviour is so simple.

The gas equation in molecular form

Experiments on a fixed amount of any dilute gas give three simple laws, which you met with the ideal-gas equation in Chapter 10 (Thermal Properties of Matter). They are worth restating, because each says something about molecules.

The molecular reading is natural. In Boyle's law, squeezing the gas into half the volume packs twice as many molecules into each cubic metre, so twice as many hit each square metre of wall every second: the pressure doubles. In the pressure law, heating the gas makes the molecules move faster, so they hit the walls harder and more often. The constant $\dfrac{pV}{T}$ is found to be proportional to the amount of gas — and, remarkably, it is the same for every gas when the amount is measured in moles, or simply by counting molecules.

The equation contains no property of the particular gas. A cubic metre of hydrogen and a cubic metre of carbon dioxide at the same temperature and pressure hold the same number of molecules, even though each $\mathrm{CO_2}$ molecule is $22$ times heavier. Notice the care needed with symbols: $n$ is the number of moles, $N$ the number of molecules, and we write $n_V$ for the number density.

Avogadro's hypothesis and Dalton's law of partial pressures

Since $pV=Nk_BT$ contains no property of the gas, equal values of $p$, $V$ and $T$ force equal values of $N$. Avogadro guessed this in 1811, from the simple volume ratios in which gases react, long before anyone could count molecules.

When several gases share a container — air is mainly nitrogen, oxygen and argon — each kind of molecule behaves as if the others were not there, because in an ideal gas the molecules do not interact. Each gas hits the walls and contributes its own share of the pressure.

Real gases and the limits of the ideal-gas model

How good is $pV=nRT$? Measure $\dfrac{pV}{nRT}$ for a real gas over a range of pressures. For an ideal gas this ratio would be exactly $1$ at every pressure. For real gases it is very close to $1$ at low pressures — within a fraction of a per cent at atmospheric pressure — but deviates at hundreds of atmospheres. Two molecular effects compete:

• Attraction between molecules pulls each molecule near a wall back towards the rest of the gas, so it hits the wall less hard: the pressure is lower than ideal and $\dfrac{pV}{nRT}\lt1$. This matters most at low temperatures, when the molecules move slowly, and it is what eventually makes a gas liquefy.
• The finite size of molecules takes up part of the container, leaving less free space, so at very high densities the pressure is higher than ideal and $\dfrac{pV}{nRT}\gt1$.

Both effects vanish as the gas is made more dilute. So real gases approach ideal behaviour at low pressures and high temperatures, and every gas law in this chapter is a low-density law. Air at room conditions obeys it to better than $0.1\%$.

Exercises

Take $R=8.31\ \text{J mol}^{-1}\text{K}^{-1}$, $k_B=1.38\times10^{-23}\ \text{J/K}$, $N_A=6.02\times10^{23}\ \text{mol}^{-1}$, $1\ \text{atm}=1.013\times10^5\ \text{Pa}$, STP $=273\ \text{K}$ and $1\ \text{atm}$, and $g=9.8\ \text{m/s}^2$. Treat all gases as ideal unless told otherwise.