Key Idea: This topic is about a gas's pressure P, volume V, temperature T and how much gas there is — and the handful of equations that tie them together. Hold one quantity fixed and the others are linked (the gas laws); count the particles and the ideal gas law PV = nRT = N kB T does it all at once; zoom in and the kinetic model explains why — pressure is particles hitting the walls and temperature is their average kinetic energy. It is examined on both papers — quick Paper 1A multiple-choice (compare two samples, P-against-1/V graphs, 'same T → same average KE') and longer Paper 2 structured questions (a before/after gas-law calculation, find moles or molecules, or explain the particle picture in words). One trap runs through the whole topic: temperature must be in kelvin.
📋 Key formulas
Every equation here is given in the data booklet — look for the booklet badge. There is nothing to memorise; the skill is choosing the right form and putting T in kelvin.
- pressure of the gas (Pa)
- volume of the gas (m³)
- absolute temperature (K) — always in kelvin
- pressure (Pa)
- volume (m³)
- amount of gas, in moles (mol)
- molar gas constant, 8.31 J K⁻¹ mol⁻¹ (given)
- number of molecules (a plain count, no unit)
- Boltzmann constant, 1.38 × 10⁻²³ J K⁻¹ (given)
- absolute temperature (K — kelvin)
- amount of gas, in moles (mol)
- number of molecules (a plain count)
- Avogadro constant, 6.02 × 10²³ mol⁻¹ (given)
- average kinetic energy of ONE particle (J)
- Boltzmann constant, 1.38 × 10⁻²³ J K⁻¹ (given)
- absolute temperature (K — kelvin)
⚖️ The three gas laws
Each gas law is the combined law PV ÷ T = constant with one quantity held fixed so it cancels. Boyle's needs no kelvin (no T in it); the other two must use kelvin.
| Law | Held fixed | What stays constant | In words |
|---|---|---|---|
| Boyle's | temperature T | P × V | squeeze it (V down) → pressure up. Halve V → double P. |
| Charles' | pressure P | V ÷ T | heat it → it expands. Double the kelvin T → double V. |
| Gay-Lussac's | volume V | P ÷ T | heat a sealed rigid can → pressure up. Double the kelvin T → double P. |
Boyle's law P V = K rearranges to P = K × (1/V). So a graph of P against 1/V (at fixed temperature) is a straight line through the origin whose slope is K. Its SI unit is pressure × volume = Pa m³ = J (the joule). This is the classic Paper 1B data-handling task.
🔢 The three constants (don't mix them up)
| Constant | Symbol & value | Goes with… | What it does |
|---|---|---|---|
| Molar gas constant | R = 8.31 J K⁻¹ mol⁻¹ | the moles form: PV = nRT | Links P, V, T to the amount in moles. |
| Boltzmann constant | kB = 1.38 × 10⁻²³ J K⁻¹ | the molecules form: PV = NkBT, and Ēₖ = (3/2)kBT | Links to the number of molecules, and to the energy of one particle. |
| Avogadro constant | NA = 6.02 × 10²³ mol⁻¹ | the bridge n = N ÷ NA | How many particles in one mole — converts moles ↔ molecules. |
R and kB describe the same gas, so the two forms PV = nRT and PV = NkBT must agree. Since N = n NA, that forces R = NA × kB. Check: 6.02 × 10²³ × 1.38 × 10⁻²³ = 8.31 ✓ — exactly the molar gas constant.
✏️ Worked exam-style questions
A balloon holds 2.0 m³ of gas at a pressure of 100 kPa and a temperature of 27 °C. It is taken to a place where the pressure is 150 kPa and the temperature is 177 °C. Find the new volume of the gas.
🔒 Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
An aerosol can holds gas at 200 kPa at 27 °C. It is left in the sun and warms to 87 °C. The can is rigid (fixed volume). (a) Find the new pressure. (b) Find the percentage increase in pressure.
🔒 Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
A flask contains an ideal gas at a pressure of 2.0 × 10⁵ Pa in a volume of 3.0 × 10⁻⁴ m³ at a temperature of 27 °C. Taking kB = 1.38 × 10⁻²³ J K⁻¹ and NA = 6.02 × 10²³ mol⁻¹, find (a) the number of molecules N and (b) the amount of gas in moles.
🔒 Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
(a) A sample of neon is at 227 °C. With kB = 1.38 × 10⁻²³ J K⁻¹, find the average kinetic energy of one neon atom. (b) A sample of argon (heavier atoms) sits beside it at the same 227 °C. Compare the average kinetic energy of an argon atom with that of a neon atom.
🔒 Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
🧠 Quick self-check
Tap each card to reveal the answer.
Halve a gas's volume at constant temperature — what happens to its pressure? It doubles. Boyle's law: P × V stays constant, so V → V/2 forces P → 2P.
What unit must temperature be in for any gas-law formula? Kelvin (K). Convert from Celsius by adding 273 (27 °C = 300 K). This is the single most-common slip in the whole topic.
Which constant goes with the number of molecules N, and which with moles n? N pairs with the Boltzmann constant kB; n pairs with the molar gas constant R. Never mix them in one equation.
How many particles are in one mole, and how do you get N from n? 6.02 × 10²³ (the Avogadro constant NA). Multiply: N = n × NA.
What does the pressure of a gas actually come from? Particles colliding with the walls — billions of tiny pushes add up to a steady force per unit area. Faster or more frequent collisions → higher pressure.
Two different gases at the same temperature — same average kinetic energy? Yes. Ēₖ = (3/2)kB T depends only on T, not on the gas. The heavier gas's particles just move slower.
🎯 Exam tips
Exam Tips
- Temperature ALWAYS in kelvin (K = °C + 273) before any gas-law or kinetic formula. Scaling on the Celsius numbers is the classic trap — pressure and volume scale with the kelvin temperatures.
- For a before/after change, write the combined law P₁V₁ ÷ T₁ = P₂V₂ ÷ T₂. Whatever is held fixed cancels: fixed T → Boyle (P₁V₁ = P₂V₂); fixed P → Charles; fixed V → Gay-Lussac.
- Ideal gas law: use n with R (8.31), or N with kB (1.38 × 10⁻²³) — never mix the two forms in one equation.
- To compare two samples, write PV = NkT (or nRT) for each and DIVIDE one by the other — any equal quantity (P, V or T) cancels, leaving a clean ratio.
- Convert moles ↔ molecules with n = N ÷ NA (NA = 6.02 × 10²³). To get N from n you multiply: N = n × NA.
- A graph of P against 1/V (fixed temperature) is a straight line through the origin; its slope is the Boyle constant K, with SI unit Pa m³ = J.
- Average kinetic energy Ēₖ = (3/2)kB T depends ONLY on the kelvin temperature — same T means the same average KE for any gas; heavier particles just move slower.
- Pressure comes from particles hitting the walls; compressing a gas quickly does work on it, raising the particles' average KE — so a higher temperature and faster particles.