1The idea in one paragraph

A gas is mostly empty space with particles flying through it. If you pretend the particles take up no room at all and ignore any pull between them, the behaviour of every gas can be predicted by one equation, pV = nRT, whatever the gas is. That pretend gas is the ideal gas. No real gas is ideal, but at room temperature and ordinary pressures most gases are close enough that the equation gives answers within a few per cent. The model breaks down when its two big pretences stop being reasonable: at high pressure, where the particles are squeezed so close that their own size matters, and at low temperature, where they move so slowly that their attractions for each other matter. Knowing why it fails is as much a part of this subtopic as using it.

2The ideal gas model

Figure 1 sets out the assumptions. You must be able to recognise them and state them.

An ideal gas consists of moving particles with negligible volume and no intermolecular forces. All collisions between particles are elastic.

Each assumption has a job.

• Constant random motion. The particles move in straight lines until they hit another particle or a wall. Their average kinetic energy is proportional to the kelvin temperature (S1.1).
• Negligible volume. The particles themselves are treated as points. All of the gas's volume is the space between them, so the volume available to the particles is the whole container.
• No intermolecular forces. Particles neither attract nor repel one another except at the instant of collision. Nothing slows a particle as it heads for a wall.
• Elastic collisions. No kinetic energy is lost in collisions, so a gas left alone does not slow down and settle.
• Pressure is the result of particles hitting the walls. More collisions per second, or harder collisions, mean higher pressure.

Every gas law in section 4 follows from these. Squeeze the same gas into half the volume at the same temperature and particles hit each area of wall twice as often, so the pressure doubles. Heat it and the particles move faster and hit harder and more often, so either the pressure rises or, if the container can expand, the volume does.

3Real gases and the limits of the model

Real particles do have a size, and they do attract one another. Under most conditions neither matters much. Figure 2 shows when each one starts to.

At high pressure, particle volume is no longer negligible. Compress a gas and the empty space shrinks while the particles stay the same size. The particles now occupy a real fraction of the container, so the space they can move in is less than the container's volume. The model's assumption of point particles fails, and the real gas takes up more volume than pV = nRT predicts.

At low temperature, intermolecular forces are no longer negligible. Slow particles spend longer near each other, and their attractions have time to act. A particle about to hit the wall is pulled back slightly by its neighbours, so it hits more softly, and some particles cluster. The real pressure is lower than the model predicts. Cool far enough and the attractions win entirely: the gas condenses to a liquid, which an ideal gas can never do.

Real gases deviate most from ideal behaviour at high pressure (particle volume matters) and low temperature (intermolecular forces matter). They behave most ideally at high temperature and low pressure.

A full answer names the condition and the assumption that fails and why. "Real gases are not ideal at high pressure" states; "at high pressure the particles are close together, so their own volume is a significant fraction of the container and can no longer be ignored" explains.

How big is the deviation? Figure 3 is a sketch, not required by the guide as mathematics, that makes both effects visible at once. For an ideal gas, pV ÷ nRT = 1 at every pressure. For a real gas it starts near 1 at low pressure, dips below 1 as attractions pull the particles together, and rises above 1 at high pressure as particle volume takes over.

Some gases deviate more than others. The guide's linking question points to Structure 2.2. Under the same conditions, a gas whose particles are larger or attract each other more strongly deviates more. Helium, with tiny atoms and very weak attractions, is one of the most nearly ideal gases. Ammonia, whose molecules attract each other strongly (by hydrogen bonding, which Structure 2.2 explains), deviates much more and is easy to liquefy.

4Pressure, volume and temperature

The guide asks you to investigate how pressure, volume and temperature are related for a fixed mass of gas, and to analyse the graphs. The names of the individual gas laws are not assessed; the relationships and the graph shapes are. In each case one variable is held constant.

Pressure and volume, at constant temperature. Pressure is inversely proportional to volume: halve the volume and the pressure doubles, so p × V stays constant. A graph of p against V is a curve that falls ever more slowly; a graph of p against 1/V is a straight line through the origin, which is the easy way to show the relationship from data. Figure 4 draws both.

Volume and temperature, at constant pressure. Heat a gas in a container that can expand, such as a syringe with a free plunger, and its volume rises in a straight line with temperature. Plotted against temperature in °C, the line does not go through the origin; extend it backwards and it reaches zero volume at about −273 °C. That temperature is absolute zero, 0 K. Plotted against temperature in kelvin, the same data give a straight line through the origin: V is directly proportional to T. Figure 5, panels (a) and (b), shows the two plots.

Pressure and temperature, at constant volume. Heat a gas in a sealed rigid container and its pressure rises in the same way: p is directly proportional to T in kelvin, panel (c).

That is why every gas calculation uses kelvin. In °C, doubling the number does not double anything: 20 °C to 40 °C is only 293 K to 313 K.

Sketched or plotted? The guide's Nature of science question asks when to sketch a graph and when to plot one. A sketch shows the shape and the key features (straight, through the origin, curving towards the axis) without values; use it to explain a relationship. A plotted graph uses measured points on a scale, with a line of best fit, and lets you read values, find a gradient or extrapolate, as in Try it Q5. The first communicates the idea; the second tests it.

5Molar volume

Avogadro's law (S1.4) says that equal volumes of gases at the same temperature and pressure contain equal numbers of particles. Turn it round and one mole of any ideal gas occupies the same volume at a given temperature and pressure: the molar volume. It is a constant only once the temperature and pressure are fixed.

The data booklet gives the value at standard temperature and pressure (STP), which is 273.15 K (0 °C) and 100 kPa:

molar volume of an ideal gas at STP = 22.7 dm³ mol⁻¹ (2.27 × 10⁻² m³ mol⁻¹)

Worked example. Calcium carbonate decomposes on heating: CaCO₃(s) → CaO(s) + CO₂(g). Calculate the volume of carbon dioxide, at STP, released by 5.00 g of calcium carbonate.

At any other temperature or pressure, 22.7 dm³ mol⁻¹ is wrong and you must use pV = nRT instead. At room temperature (298 K and 100 kPa) the molar volume is about 24.8 dm³ mol⁻¹, which pV = nRT gives you in one line.

6The ideal gas equation

All the relationships of section 4, and the molar volume, combine into one equation:

pV = nRT p in Pa, V in m³, n in mol, T in K, R = 8.31 J K⁻¹ mol⁻¹

The guide requires SI units, and R has the value 8.31 only in SI units. So convert before you substitute:

Worked example 1: find a pressure. 2.00 g of oxygen gas is held in a 1.50 dm³ flask at 25.0 °C. Calculate the pressure.

Worked example 2: find a molar mass. The guide's linking question asks how pV = nRT gives the molar mass of a gas from experimental data. Figure 6 shows one method: inject a weighed mass of a volatile liquid into a gas syringe inside an oven hot enough to vaporise it, and read the volume of vapour.

Since n = m ÷ M, the equation rearranges to M = mRT ÷ pV. A student injects 0.500 g of liquid; the vapour occupies 333 cm³ at 100.0 °C and 101 kPa.

The value matches ethanol, C₂H₅OH (46.08 g mol⁻¹). The main errors in this experiment come from the model itself: near its boiling point the vapour is not ideal, because attractions between molecules still act, so the measured volume tends to be too small and the molar mass too large.

The combined gas law. When a fixed amount of gas moves from one set of conditions to another, n and R cancel:

p₁V₁ ÷ T₁ = p₂V₂ ÷ T₂

Here the units of p and V need not be SI, as long as they are the same on both sides. T must still be in kelvin, because the equation relies on proportionality to absolute temperature.

Worked example 3. A weather balloon holds 2.50 dm³ of helium at 101 kPa and 20.0 °C at ground level. It rises to where the pressure is 35.0 kPa and the temperature is −40.0 °C. Calculate its new volume.

Check the direction: the pressure fell to about a third, which alone would nearly triple the volume; the cold shrinks it a little. 5.74 dm³ is a little over twice the start. Sensible.