1The idea in one paragraph

A gas has four things you can measure from outside: its pressure P, volume V, temperature T and amount n. Experiments from the 1660s onwards showed that, for a fixed amount, any two of P, V and T fix the third, and all three laws fold into one equation, PV = nRT. The kinetic model explains why. It treats the gas as enormous numbers of tiny molecules flying about at random and bouncing off the walls. Each bounce changes a molecule's momentum, so the wall feels a force; spread over the wall's area, that force is the pressure. Work the model through and it predicts P = ⅓ρv̄², and matching that to PV = nRT shows that temperature measures the average kinetic energy of the molecules. The model describes an ideal gas. Real gases obey it well when they are at low pressure and high temperature, which is when their molecules are far apart and barely feel each other.

2Pressure and the amount of substance

Pressure is the force acting at right angles to a surface, per unit area:

P = F ÷ A

Its unit is the pascal, Pa, which is one newton per square metre. Atmospheric pressure at sea level is about 1.01 × 10⁵ Pa. That is larger than it sounds: on a window 1.20 m by 0.80 m the air pushes with a force F = PA = 1.01 × 10⁵ × 0.96 = 9.7 × 10⁴ N, roughly the weight of ten small cars. The window survives because the air on the other side pushes back just as hard.

The F in P = F/A must be the component perpendicular to the surface. A force along the surface is friction or shear, not pressure.

Gases contain unimaginable numbers of molecules, so we count them in moles. The amount of substance n is

n = N ÷ N_A

where N is the number of molecules and N_A = 6.02 × 10²³ mol⁻¹ is the Avogadro constant. A balloon holding 3.0 × 10²³ molecules therefore holds (3.0 × 10²³) ÷ (6.02 × 10²³) = 0.50 mol. The mass of one mole in grams is the molar mass: 4.0 g mol⁻¹ for helium, 32 g mol⁻¹ for oxygen gas, O₂.

3The ideal gas: a model, not a substance

An ideal gas is a model system, not something you can buy. It is defined by the assumptions of the kinetic model, which Figure 1 sets out.

1. A gas consists of a very large number of identical molecules moving at random, in all directions, with a range of speeds.
2. The volume of the molecules themselves is negligible compared with the volume of the container.
3. All collisions, between molecules and with the walls, are elastic: kinetic energy is conserved.
4. There are no forces between molecules except during collisions, and collisions last a negligible time.

Assumption 4 has a consequence you will use constantly. With no intermolecular forces there is no intermolecular potential energy, so the internal energy of an ideal gas is entirely the random kinetic energy of its molecules.

Why use a gas that does not exist? Because real gases behave almost exactly like it under ordinary conditions, and it turns 10²³ tracked molecules into a few lines of algebra. A simplified model earns its place by predicting what we measure; section 9 shows where this one breaks.

4The three empirical gas laws

An empirical law is one found by measurement rather than derived from theory. Each gas law holds one quantity fixed, for a fixed mass of gas, and relates the other two.

Constant temperature: Boyle's law. Pressure is inversely proportional to volume.

PV = constant (fixed mass, constant T)

Figure 2 shows the two ways to plot it. P against V is a curve (a hyperbola): halve the volume and the pressure doubles. P against 1/V is a straight line through the origin, and that is the graph to draw if an examiner asks you to show the relationship is an inverse proportion.

In terms of molecules: at constant temperature the molecules keep the same average speed, so each collision with the wall delivers the same average momentum change. Squeeze the gas into a smaller volume and there are more molecules in each cubic metre, and each one crosses the container and returns to a wall sooner. More collisions per second on each square metre of wall means a larger force per unit area, so a larger pressure.

Constant pressure: Charles's law. Volume is proportional to absolute temperature: V ∝ T, or V ÷ T = constant.

Constant volume: the pressure law. Pressure is proportional to absolute temperature: P ∝ T, or P ÷ T = constant.

Figure 3 shows both. Plotted against temperature in degrees Celsius, the volume (or pressure) gives a straight line that does not pass through the origin. Extend it backwards and it reaches zero at about −273 °C, whatever gas you use and whatever mass you start with. That temperature is absolute zero, 0 K. Plotted in kelvin, the line passes through the origin, which is what "proportional" means.

In terms of molecules: raise the temperature and the molecules move faster. At constant volume, faster molecules hit the walls more often and with a larger momentum change each time, so the pressure rises. At constant pressure, the only way to stop the pressure rising is for the gas to expand, so that collisions on each square metre become less frequent; the volume increases.

Every one of these proportionalities uses kelvin. P ∝ θ in degrees Celsius is wrong, because doubling 10 °C to 20 °C does not double the pressure; doubling 283 K to 566 K does.

5Putting the three together: the ideal gas law

The guide asks you to see how the three laws combine into one. Take a fixed mass of gas from state 1 (P₁, V₁, T₁) to state 2 (P₂, V₂, T₂) in two stages.

The route did not matter, so for a fixed amount of gas, PV ÷ T has the same value in every state. Experiment then shows that this constant is proportional to the amount of gas: twice as many moles, twice the constant. Writing the constant as nR gives the ideal gas law:

PV = nRT, or equivalently PV = Nk_BT

R = 8.31 J K⁻¹ mol⁻¹ is the molar gas constant. The second form counts molecules instead of moles, and k_B = 1.38 × 10⁻²³ J K⁻¹ is the Boltzmann constant. They are the same equation because nR = Nk_B, which means R = N_A k_B: check it, 6.02 × 10²³ × 1.38 × 10⁻²³ = 8.31.

Units. Use P in Pa, V in m³, T in K. Volumes are often given in cm³ or litres: 1 cm³ = 10⁻⁶ m³ and 1 litre = 10⁻³ m³.

Worked example 1: two states of the same gas. A syringe holds 300 cm³ of air at 1.0 × 10⁵ Pa and 17 °C. The air is compressed to 120 cm³ and warms to 47 °C. Calculate the new pressure.

The volumes can stay in cm³ here, because they appear as a ratio and the units cancel. Temperatures cannot stay in °C, ever.

Worked example 2: how much gas is there? A helium balloon has a volume of 0.012 m³ at 20 °C and 1.10 × 10⁵ Pa. Find the amount of helium, the number of atoms, and the mass (molar mass 4.0 g mol⁻¹).

6Changes of state on a P–V diagram

The guide says changes of state of an ideal gas can be represented on a P–V diagram: pressure on the vertical axis, volume on the horizontal. Every point on it is one state of the gas, and for a fixed amount of gas, PV = nRT fixes its temperature. Figure 4 shows the three simple changes.

• A curve PV = constant is an isotherm: every point on it has the same temperature. Isotherms further from the origin are at higher temperature, because PV is larger. In Figure 4, T₁ < T₂ < T₃.
• Constant pressure is a horizontal line. From A to B the gas expands at constant pressure, so its temperature rises (it moves out to the T₃ isotherm).
• Constant volume is a vertical line. From B to C the pressure falls at constant volume, so the temperature falls (back to T₁).
• Constant temperature runs along an isotherm, as from C back to A.

To read a temperature off a P–V diagram, compare PV at the points. At A and C, PV is the same, so T is the same. HL students will draw these diagrams again in B.4, where the area under a line becomes the work done.

7Where pressure comes from: collisions with the walls

The kinetic model predicts the pressure of an ideal gas, and the guide asks you to understand how. Figure 5 walks through it for one molecule in a cube of side L, then scales up.

1. A molecule of mass m moves towards the shaded wall with velocity component vₓ. The collision is elastic, so it rebounds with −vₓ. Its momentum changes by 2mvₓ, and by Newton's third law the wall receives an equal and opposite impulse.
2. The molecule must cross to the far wall and back before it hits the shaded wall again: a distance 2L at speed vₓ, which takes 2L ÷ vₓ.
3. The average force on the wall from this one molecule is momentum change ÷ time between hits = 2mvₓ ÷ (2L ÷ vₓ) = mvₓ² ÷ L.
4. For N molecules, add them up, using the mean of vₓ². Because the motion is random, no direction is special, so the mean of vₓ² is one third of the mean of v², because v² is the sum of the squares of three equal-on-average components.
5. Divide the total force by the wall's area L², and note that Nm ÷ L³ is the density ρ of the gas.

The result is the equation in the data booklet:

P = ⅓ ρ v̄²

Here v̄² is the mean square speed: square every molecule's speed, then average. It is not the square of the mean speed. Its square root is the root mean square (rms) speed, v_rms = √(v̄²), a sensible single "typical speed" for the molecules.

Worked example 3: how fast are air molecules? Air at sea level has a pressure of 1.01 × 10⁵ Pa and a density of 1.20 kg m⁻³.

About half a kilometre per second: faster than sound in air, which the molecules themselves carry.

8Temperature, speed and internal energy

Now join the two routes, as Figure 6 shows. Multiply P = ⅓ρv̄² by V, and use ρV = Nm:

The left side is the average translational kinetic energy of one molecule. So the absolute temperature of an ideal gas is a direct measure of the average kinetic energy of its molecules, which is the B.1 statement Eₖ = (3/2)k_BT, now proved. At 0 K the molecules of an ideal gas would have no kinetic energy at all. Double the kelvin temperature and you double the mean kinetic energy, but the rms speed rises only by a factor of √2.

For a monatomic ideal gas (single atoms, such as helium or argon), the only energy the particles have is this translational kinetic energy, so the internal energy is N times it:

U = (3/2) N k_B T = (3/2) n R T

Notice what U does not depend on: pressure and volume. For an ideal gas, internal energy depends on temperature alone. That fact drives most of B.4.

Worked example 4: argon at room temperature. A cylinder holds 2.0 mol of argon at 300 K. Find the mean kinetic energy of an atom and the internal energy of the gas.

9When a real gas behaves like an ideal one

Real gases break two of the model's assumptions: their molecules do take up some room, and they do attract each other slightly at short range. Whether that matters depends on how close together the molecules are and how fast they are moving. Figure 7 sketches the pattern using PV ÷ nRT, which equals exactly 1 for an ideal gas.

A real gas is well approximated by an ideal gas at:

• Low pressure and low density. The molecules are far apart, so their own volume is a tiny fraction of the container's and they spend almost no time close enough to attract each other.
• High temperature, well above the temperature at which the gas would condense. The molecules move so fast that the brief attractions barely change their paths or their energy.

At high pressure the molecules' own volume is no longer negligible, and the gas resists compression more than PV = nRT predicts. Close to condensing, attractions lower the pressure, and at the extreme they win and the gas becomes a liquid, which an ideal gas can never do. Air at room temperature and pressure is comfortably ideal.