Molecules and the behaviour of gases

Matter is made of molecules

Narrated video. Transcript:

Put a drop of ink into a glass of still water. Without any stirring, the colour slowly spreads everywhere. Something far too small to see must be moving and mixing it.

The idea that matter is made of tiny particles is very old. In India, Kanada suggested it more than two thousand years ago, and in Greece, so did Democritus.

In 1803, John Dalton turned the idea into science. His atomic theory explained why elements always combine in fixed proportions to form compounds.

Gay-Lussac then found that gases react in simple ratios by volume. Watch two litres of hydrogen and one litre of oxygen combine into two litres of steam.

Avogadro explained why: equal volumes of all gases, at the same temperature and pressure, contain the same number of molecules. This is Avogadro’s law.

Atoms are tiny, about an ångström across: 10⁻¹⁰ metre. In a solid they are packed a few ångströms apart and can only vibrate about fixed positions.

In a liquid they are just as close, but free to move around. In a gas they are tens of ångströms apart, and fly freely in straight lines until they collide.

How empty is a gas? Take oxygen at 0 °C and normal atmospheric pressure, with molecules 3 ångströms across. One mole fills 22.4 litres, but the molecules themselves take up only about 8.5 cubic centimetres.

That is less than four parts in ten thousand of the space. A gas is mostly empty, so its molecules hardly feel each other. That is why gases are the easiest state of matter to understand.

The ideal gas equation

Narrated video. Transcript:

Trap some air in a syringe, block the nozzle, and push the piston in slowly so the temperature stays the same. The smaller the volume, the harder the air pushes back.

Robert Boyle measured this. At constant temperature, pressure times volume stays the same: P V is constant. Halve the volume and the pressure doubles. This is Boyle’s law.

Now keep the pressure fixed and warm the gas. It expands. Jacques Charles found that the volume is proportional to the absolute temperature: V over T is constant. This is Charles’ law.

Put the two together: for a fixed amount of gas, P V over T stays constant. And at the same pressure and temperature, twice as much gas fills twice the volume.

So P V equals μ R T, where μ is the number of moles and R is the universal gas constant, 8.314 joules per mole per kelvin. This is the ideal gas equation.

The number of moles μ is the number of molecules N divided by Avogadro’s number, 6.02 × 10²³. It is also the mass of the gas divided by its molar mass.

Write μ as N over N A, and the equation becomes P V equals N k B T. Here k B is R divided by N A, called Boltzmann’s constant: 1.38 × 10⁻²³ joules per kelvin.

Try it for one mole at 273 kelvin and 1.013 × 10⁵ pascals. V equals R T over P: 8.314 times 273, divided by 1.013 × 10⁵, gives 0.0224 cubic metres, or 22.4 litres.

The answer is the same for hydrogen, oxygen or nitrogen. It is called the molar volume at S T P, standard temperature and pressure.

Real gases and Dalton’s law

Narrated video. Transcript:

Is the ideal gas equation exact? Measure P V over μ T for some real gases at a fixed temperature, and plot it against the pressure.

For an ideal gas the graph is a flat line at R. Real gases bend away from it at high pressures, but every curve comes back to R as the pressure falls towards zero.

So real gases behave like an ideal gas at low pressures and high temperatures. Then the molecules are far apart and fast, and the forces between them hardly matter.

Now plot Boyle’s law: P against V at a fixed temperature. Each curve is a hyperbola, and a hotter gas gives a higher curve.

Plot V against temperature at constant pressure, and you get a straight line. Extend it backwards: the volume would reach zero at minus 273.15 °C, absolute zero. A real gas turns liquid long before that.

Finally, mix two gases that do not react: μ 1 moles of one and μ 2 moles of the other, in a volume V at temperature T.

The total pressure is μ 1 plus μ 2, times R T over V. That splits into μ 1 R T over V plus μ 2 R T over V: the pressure each gas would exert if it were alone.

These are the partial pressures, P 1 and P 2. The total pressure of a mixture of ideal gases is the sum of the partial pressures. This is Dalton’s law of partial pressures.

Air is about 78 percent nitrogen and 21 percent oxygen by number of molecules. So of the 101 kilopascals of air pressure at sea level, about 79 come from nitrogen and 21 from oxygen.

Try it: gas in a box

Interactive simulation. Things to try:

• Drag the amber handle (or use the slider) to move the piston. With the heat bath on, halve V and watch P double: Boyle’s law.
• Press “Record” at a few volumes and temperatures. PV/T stays the same, μR = 0.333 kPa·L/K.
• Charles’ law: at 0.6 L and 300 K note P. Now set T = 600 K and pull the piston out until P is back to that value. V has doubled, to 1.2 L.
• Switch the heat bath off and push the piston in. The gas heats up, because the piston does work on the molecules.

Try it: air is a mixture

Interactive simulation. Things to try:

• Blue dots are nitrogen and purple dots are oxygen: 80% and 20% of the molecules.
• Each partial pressure is μRT/V for that gas alone. They add up to the total (Dalton’s law).
• Set the oxygen to zero: the total pressure drops by exactly the oxygen’s share.

Why does P double?

A gas is squeezed to half its volume while its temperature stays fixed, and its pressure doubles. What is the best molecular explanation?

1. The molecules move twice as fast
2. There are twice as many molecules in each cubic centimetre, so the walls are hit twice as often
3. The molecules swell up and push on each other
4. The molecules attract each other more strongly

A rising bubble

An air bubble of volume 1.0 cm³ rises from the bottom of a lake 40 m deep, where the temperature is 12 °C, to the surface, where it is 35 °C. Atmospheric pressure is 1.01 × 10⁵ Pa and g = 9.8 m/s². What is its volume at the surface?

1. 4.2 cm³
2. 4.9 cm³
3. 5.3 cm³
4. 14 cm³

Summary

• Boyle’s law (T fixed): PV = constant
• Charles’ law (P fixed): V / T = constant
• Ideal gas equation: PV = μRT = Nk_BT
• Constants: R = 8.314 J mol⁻¹ K⁻¹, k_B = R/N_A = 1.38 × 10⁻²³ J K⁻¹
• Dalton’s law: P = P₁ + P₂ + …

• Matter is made of molecules. In a gas they are far apart (tens of ångströms) and fill only a tiny fraction of the volume.
• Equal volumes of all gases at the same T and P hold equal numbers of molecules (Avogadro). One mole fills 22.4 L at STP.
• Real gases obey PV = μRT best at low pressure and high temperature.
• In a mixture, each gas exerts its own partial pressure as if it were alone; these add up to the total.
• Always use kelvin and absolute (not gauge) pressure in the gas laws.