understand that a gas obeying pV ∝ T, where T is the thermodynamic temperature, is known as an ideal gas

Ideal‑Gas Model – Equation of State

Learning Objective (AO1)

Explain why a gas that obeys the proportionality p V ∝ T (with T the thermodynamic temperature) is described as an ideal gas, and show how this leads to the ideal‑gas equation p V = n R T.

1. Key Concepts (Syllabus 15.1 – 15.2)

QuantitySymbolDefinitionSI Unit
PressurepForce per unit areaPa (N m⁻²)
VolumeVSpace occupied by the gasm³
Thermodynamic temperatureTAbsolute temperatureK
Amount of substancenNumber of molesmol
Universal gas constantRR = 8.314 J mol⁻¹ K⁻¹ = 0.08206 L atm mol⁻¹ K⁻¹J mol⁻¹ K⁻¹
Avogadro’s constantNANumber of particles per mole6.022 × 10²³ mol⁻¹
Boltzmann constantkBEnergy per particle per kelvin1.381 × 10⁻²³ J K⁻¹

Mole‑Concept Box

1 mol = 6.022 × 10²³ particles. The total number of particles in a sample is N = n NA. Because R = NAkB, the macroscopic constant R links the microscopic kinetic‑theory expression to the ideal‑gas law.

2. Ideal Gas as a Model (Key Concept 1)

The ideal‑gas law is a model that simplifies the behaviour of real gases by adopting three explicit assumptions:

  1. Gas particles are point‑like – their own volume is negligible compared with the container volume.

  2. There are no intermolecular forces except during perfectly elastic collisions.

  3. The gas is in thermal equilibrium, so a single temperature T characterises the average kinetic energy of all particles.

These assumptions allow a simple relationship between the macroscopic variables p, V, T and the amount of substance n. The model works well at moderate pressures (≤ 1 atm) and temperatures (≈ 300 K – 1000 K), which covers the majority of Cambridge AS & A‑Level exam questions.

3. Derivation from Kinetic‑Theory (Syllabus 15.3)

Consider N identical particles of mass m moving randomly with average squared speed ⟨c²⟩. Kinetic theory gives

\$pV = \frac{1}{3}\,N\,m\,\langle c^{2}\rangle\$

Equipartition of energy states that the average translational kinetic energy of one particle is

\$\frac{1}{2}m\langle c^{2}\rangle = \frac{3}{2}k_{\mathrm{B}}T\$

Substituting the second expression into the first yields

\$pV = N k_{\mathrm{B}} T\$

Since N = nN{A} and R = N{A}k_{B}, we obtain the macroscopic ideal‑gas law

\$\boxed{pV = nRT}\$

4. Forms of the Ideal‑Gas Equation (AO2)

For a fixed amount of gas (n constant) the equation can be rearranged as needed:

  • Pressure: