Ideal Gases

This guide covers Topic 15, Ideal gases, in full — subtopics 15.1 The mole, 15.2 Equation of state and 15.3 Kinetic theory of gases — from Cambridge International AS & A Level Physics 9702, 2025–2027 series. This is A Level content and follows directly on from thermal equilibrium and temperature scales.

Before studying this

This resource assumes temperature scales from Temperature, and momentum from Dynamics: Newton’s Laws and Momentum.

Syllabus coverage

CAMBRIDGE INTERNATIONAL AS & A LEVEL PHYSICS 9702 — A Level, Topic 15

15.1 The mole — understanding that amount of substance is a fundamental quantity measured in moles; using molar quantities where one mole of any substance contains a number of particles equal to the Avogadro constant Nₐ.

15.2 Equation of state — understanding that a gas obeying pV = nRT at all pressures, volumes and temperatures is known as an ideal gas; recalling and using pV = nRT, where n is the number of moles; recalling and using pV = NkT, where N is the number of particles and k is the Boltzmann constant, and relating k to the molar gas constant by k = R/Nₐ.

15.3 Kinetic theory of gases — stating the basic assumptions of the kinetic theory of gases; explaining how molecular movement causes the pressure exerted by a gas, and deriving and using pV = (1/3)Nm⟨c²⟩, where ⟨c²⟩ is the mean-square speed (a simple model considering one-dimensional collisions and then extending to three dimensions using (1/3)⟨c²⟩ = ⟨cₓ²⟩ is sufficient); understanding that the root-mean-square speed cᵣₘₛ is given by √⟨c²⟩; and comparing pV = (1/3)Nm⟨c²⟩ with pV = NkT to deduce that the average translational kinetic energy of a molecule is (3/2)kT, and recalling and using this expression. (Why a mean square is used: the particles move randomly, so their velocity components average to zero — positive and negative values would cancel — whereas the squares of the velocity components are all positive and have a non-zero mean.)

The mole and the Avogadro constant

Amount of substance, measured in moles, is a fundamental quantity in the SI system. One mole of any substance contains a fixed number of particles — the Avogadro constant, Nₐ = 6.02 × 10²³ mol⁻¹.

The equation of state

A gas that obeys the relationship pV = nRT under all conditions of pressure, volume and temperature is called an ideal gas:

pV = nRT

where p is pressure, V is volume, n is the number of moles, R is the molar gas constant (8.31 J mol⁻¹ K⁻¹), and T is the thermodynamic temperature in kelvin. Temperature in this equation must always be in kelvin, not Celsius.

Worked example. 2.0 mol of an ideal gas at 300 K occupies a volume of 0.020 m³. Its pressure:

p = nRT/V = (2.0 × 8.31 × 300) / 0.020 = 249,300 Pa ≈ 2.49 × 10⁵ Pa

The equation of state in terms of molecules

The equation of state can equally be written in terms of the number of particles N rather than the number of moles n, using the Boltzmann constant k in place of the molar gas constant R:

pV = NkT

The Boltzmann constant is related to the molar gas constant by:

k = R / Nₐ

so k is, in effect, “the gas constant per particle” rather than per mole. The two forms, pV = nRT and pV = NkT, describe exactly the same physical law — pick whichever matches the quantity (moles or number of particles) given in a question.

Kinetic theory of gases

The kinetic theory of gases models a gas as a large number of identical point particles in continuous random motion, colliding elastically with each other and with the walls of their container, with negligible forces between particles except during collisions. Gas pressure arises from the cumulative effect of many particles colliding with the container walls, transferring momentum on each collision.

Deriving the kinetic theory equation

The syllabus prescribes a specific route for this derivation, based on considering the change in momentum of particles colliding with one wall of a cube-shaped container of side L:

1. A single particle of mass m moving with velocity component vₓ perpendicular to one wall rebounds elastically, reversing its velocity, so its change in momentum per collision with that wall is 2mvₓ.
2. Between successive collisions with the same wall, the particle travels a distance 2L (there and back), taking time 2L/vₓ — so the rate of momentum transfer (force) from this one particle is 2mvₓ ÷ (2L/vₓ) = mvₓ²/L.
3. Summing over all N particles in the container, the total force on the wall is (m/L)Σvₓ² = (Nm/L)⟨vₓ²⟩, where ⟨vₓ²⟩ is the mean square velocity component in the x-direction.
4. Because the particles move randomly in three dimensions, ⟨c²⟩ = ⟨vₓ²⟩ + ⟨vy²⟩ + ⟨vz²⟩ = 3⟨vₓ²⟩, so ⟨vₓ²⟩ = ⟨c²⟩/3.
5. Pressure is force per unit area, and the wall has area L², so p = F/L² = (Nm/L³)⟨vₓ²⟩ = (Nm/L³) × ⟨c²⟩/3. Since L³ is the volume V of the container, this rearranges to the kinetic theory equation:

pV = (1/3) Nm⟨c²⟩

where N is the number of particles, m is the mass of one particle, and ⟨c²⟩ is the mean square speed of the particles.

The root-mean-square speed, cᵣₘₛ = √⟨c²⟩, is used rather than the mean speed or the mean velocity. The mean velocity is zero, because velocity is a vector and the particles move randomly in all directions, so positive and negative components cancel — this is precisely why step 4 above squares the velocities before averaging them. The mean speed, by contrast, is not zero (speed is always positive), but working directly with a mean of speeds, rather than a mean of squared speeds, does not lead to the pV = (1/3)Nm⟨c²⟩ result, so cᵣₘₛ is the quantity that emerges naturally from the derivation, not merely a convenient substitute.

Linking kinetic theory to temperature

Comparing pV = (1/3)Nm⟨c²⟩ with pV = nRT shows that the average translational kinetic energy of a gas molecule, ½m⟨c²⟩, is directly proportional to the thermodynamic temperature T. This is a key result: a gas’s temperature is, at the molecular level, a direct measure of the average kinetic energy of its particles.

Root-mean-square speed

Combining pV = (1/3)Nm⟨c²⟩ with pV = NkT gives ½m⟨c²⟩ = (3/2)kT, confirming that mean molecular kinetic energy depends on temperature only — not on pressure, volume, or the identity of the gas. Helium and xenon at the same temperature have the same mean molecular kinetic energy; the heavier xenon molecules simply move more slowly.

Rearranging gives the root-mean-square speed:

c_rms = sqrt(<c^2>) = sqrt(3kT/m) = sqrt(3RT/M)

Worked example. Nitrogen gas (molar mass M = 0.028 kg mol⁻¹) at 300 K:

c_rms = sqrt(3RT/M) = sqrt((3 x 8.31 x 300) / 0.028) = sqrt(267,100) ~= 517 m/s

This is the r.m.s. speed of an average molecule — individual molecules have a whole distribution of speeds around this value.

Internal energy of an ideal gas

The kinetic model assumes negligible intermolecular forces except during collisions, so an ideal gas has no molecular potential energy — its internal energy is entirely kinetic:

U = (3/2) n R T        (for a monatomic ideal gas)

This internal-energy expression is not itself a named recall equation in the 9702 syllabus or data booklet — it follows directly from summing the per-molecule kinetic energy E = (3/2)kT (which is on the syllabus) over N = nNₐ molecules. If a question needs it, expect it to be derived from that per-molecule relation rather than simply quoted.

Internal energy therefore depends only on temperature, which is why an isothermal process (constant T) has ΔU = 0, whatever happens to pressure or volume.

Common mistakes

• Using Celsius temperature in pV = nRT — this equation requires thermodynamic (Kelvin) temperature.
• Confusing mean speed with root-mean-square speed — they are not the same quantity, and cᵣₘₛ is specifically used because it avoids the cancellation that occurs when averaging velocities in random directions.
• Forgetting that the kinetic theory model assumes negligible intermolecular forces except during collisions — this is why it applies well to gases and not to liquids or solids.
• Treating n (moles) and N (number of particles) as interchangeable — they are related by N = nNₐ but appear in different versions of the gas equations.

Quick revision checklist

• One mole = Nₐ particles, Nₐ = 6.02 × 10²³ mol⁻¹
• pV = nRT, with T always in kelvin
• pV = NkT, and k = R/Nₐ, the Boltzmann constant
• Deriving pV = (1/3)Nm⟨c²⟩ from the momentum change of particles colliding with a wall
• Why cᵣₘₛ, not mean speed, is used — mean velocity is zero because directions are random
• Average molecular kinetic energy ∝ thermodynamic temperature

Related resources

• Temperature — the previous A Level topic
• Thermodynamics — the next A Level topic
• Cambridge AS & A Level Physics hub

Written against Cambridge International AS & A Level Physics 9702, 2025–2027 series. Always check the current syllabus for your examination year.