This AP Physics 2 lesson on ideal gas law notes gives the key ideas first, then worked examples and diagrams. After that, practise with the worksheet on ideal gas law and check yourself with the quiz.
Quick answer: An ideal gas is a collection of tiny, randomly moving particles whose average kinetic energy is proportional to the kelvin temperature. Wall collisions create pressure, and the gas obeys pV = nRT. Heat conducts through solids at a rate H = kAΔT/L, so good insulators have a low thermal conductivity k.
Level: AP Physics 2 (algebra-based) · Curriculum: AP P2 Unit 9
Ideal gas law notes: the key ideas
What does kinetic theory say about a gas?
Kinetic theory pictures a gas as a vast number of tiny particles that move randomly and almost never feel each other except in brief collisions. The particles are far apart compared with their own size, so a gas fills any container and is easy to compress. Collisions with the walls are elastic on average, which means the gas does not lose energy by simply sitting still.
An ideal gas is a model in which particles have negligible volume, exert no forces except during collisions, and collide elastically with each other and the walls. Real gases such as air at room conditions behave this way very closely.
Pressure appears because every collision with a wall delivers a small impulse. Trillions of impacts per second blend into a steady force on each square metre of wall. More collisions or harder collisions both raise the pressure.
How are temperature and molecular energy connected?
The absolute temperature measures the average translational kinetic energy of the particles. Doubling the kelvin temperature doubles the average energy per particle, whatever the gas. Lighter particles reach the same energy by moving faster, so at a given temperature hydrogen moves much faster than oxygen.
Kavg = (3/2) kBT vrms = √(3kBT/m) = √(3RT/M)
Kavg is the average kinetic energy per particle (J), kB = 1.38 × 10⁻²³ J/K, T is the temperature in kelvin (K), m is the mass of one particle (kg), M is the molar mass (kg/mol) and R = 8.31 J/(mol·K).
Find the root-mean-square speed of oxygen (M = 0.032 kg/mol) at 300 K.
vrms = √(3 × 8.31 × 300 / 0.032) = √(2.337 × 10⁵) = 483 m/s.
How do you use the ideal gas law?
pV = nRT = NkBT
p is the absolute pressure (Pa), V is the volume (m³), n is the amount of gas (mol), N is the number of particles, T is the temperature in kelvin (K). Convert litres with 1 L = 1.0 × 10⁻³ m³.
The law links four variables, so fixing two of them leaves a simple relationship between the other two. The table lists the three cases that AP questions use most.
| Fixed quantities | Relationship | Graph that is a straight line |
|---|---|---|
| n and T | pV = constant (Boyle) | p against 1/V through the origin |
| n and V | p/T = constant | p against T (kelvin) through the origin |
| n and p | V/T = constant | V against T (kelvin) through the origin |
Wanjiru in Nairobi seals 0.062 mol of air in a rigid 1.5 L bottle at 20 °C (293 K). Find the pressure, then the pressure after the bottle warms to 55 °C (328 K) in a car.
p1 = nRT/V = (0.062 × 8.31 × 293)/(1.5 × 10⁻³) = 1.0 × 10⁵ Pa. Volume and amount are fixed, so p2 = p1 × 328/293 = 1.1 × 10⁵ Pa.
Temperatures in the gas law must be in kelvin, and the pressure must be absolute. Using 25 instead of 298 for T, or a gauge pressure instead of an absolute one, gives results that are wrong by a large factor.
How does heat move through a solid by conduction?
In conduction, energy passes from fast-vibrating particles to their slower neighbours without any bulk movement of the material. Metals conduct best because free electrons carry energy quickly. Trapped air, as in foam or wool, conducts poorly and makes an excellent insulator.
H = Q/Δt = kAΔT/L
H is the rate of heat flow (W), k is the thermal conductivity (W/(m·K)), A is the cross-sectional area (m²), ΔT is the temperature difference across the material (K) and L is the thickness (m).
| Material | k (W/(m·K)) | Role |
|---|---|---|
| Copper | 400 | cooking pans, heat sinks |
| Aluminium | 205 | pans, radiators |
| Glass | 0.80 | windows |
| Wood | 0.15 | doors, handles |
| Polystyrene foam | 0.033 | cool boxes, insulation |
A foam cool box lid has k = 0.033 W/(m·K), area 0.30 m² and thickness 0.040 m. Inside it is 4 °C and outside 30 °C. Find the heat flow rate.
H = kAΔT/L = (0.033 × 0.30 × 26)/0.040 = 6.4 W. Over 2.0 hours this lets in 4.6 × 10⁴ J.
For any gas question, write the known values in a column with units, convert to kelvin and cubic metres first, and only then pick the formula. Doubling or halving checks catch most slips.
How do AP questions combine these ideas?
A typical free-response question asks you to translate between words, equations and graphs. You may need to explain, in a short paragraph, why the pressure rises when a sealed container is heated: faster particles hit the walls more often and harder, and the pressure is proportional to the kelvin temperature. Another style asks you to design an experiment, naming the variable you change, the variable you measure and the quantities you hold fixed.
Quantities stay linked: if the volume doubles at constant temperature, each particle travels farther between wall hits, so collisions happen half as often and the pressure halves. Reasoning like this earns marks even before any numbers appear.
- pV = nRT = Nk_BT, with T in kelvin and p absolute
- Average kinetic energy per particle: K = (3/2) k_B T, with k_B = 1.38 × 10⁻²³ J/K
- v_rms = √(3RT/M), so lighter gases move faster at the same temperature
- Conduction rate: H = kAΔT/L in watts
- At constant T and n, a graph of p against 1/V is a straight line through the origin
Gas particles, pressure and heat conduction: frequently asked questions
Why must temperature be in kelvin in the ideal gas law?
The law is a proportion between pressure, volume and the average kinetic energy of the particles. That energy is zero at 0 K, not at 0 °C, so only the kelvin scale makes the proportions work.
Do heavier gas molecules have more kinetic energy?
No. At the same temperature all ideal gas particles have the same average translational kinetic energy, (3/2) k_B T. Heavier particles simply move more slowly.
What is the difference between k_B and R?
k_B = 1.38 × 10⁻²³ J/K applies to a single particle, while R = 8.31 J/(mol·K) applies to a mole. They are related by R = N_A k_B, where N_A = 6.02 × 10²³ per mole.
Why do metals feel colder than wood at the same temperature?
Metal has a much larger thermal conductivity, so it draws heat from your hand faster. The faster loss of heat from your skin is what you sense as cold.
How do you find the number of particles in a gas sample?
Use N = pV/(k_BT) with p in pascals, V in cubic metres and T in kelvin. You can also find n = pV/(RT) first and multiply by 6.02 × 10²³.
