This AP Chemistry 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: The ideal gas law, PV = nRT, links pressure, volume, amount and absolute temperature of a gas. Kinetic molecular theory explains it: pressure comes from particle collisions, and temperature measures average kinetic energy. Partial pressures add up in mixtures (Dalton’s law), and real gases deviate at high pressure and low temperature.
Level: AP Chemistry · Curriculum: AP Chem Unit 3
Ideal gas law notes: the key ideas
What does the ideal gas law say?
An ideal gas is a model in which particles have no volume and exert no attractive or repulsive forces on one another. Real gases behave almost ideally at high temperature and low pressure.
PV = nRT
P = pressure (kPa or atm), V = volume (L), n = amount of gas (mol), T = temperature (K), R = 8.314 L·kPa/(mol·K) = 0.08206 L·atm/(mol·K). Choose the R whose units match your pressure and volume.
The ideal gas law combines older laws. Doubling the pressure of a fixed sample at constant temperature halves its volume; heating at constant pressure expands it.
| Law | Held constant | Relationship | Form |
|---|---|---|---|
| Boyle | n, T | P and V inversely proportional | P₁V₁ = P₂V₂ |
| Charles | n, P | V proportional to T (kelvin) | V₁/T₁ = V₂/T₂ |
| Gay-Lussac | n, V | P proportional to T (kelvin) | P₁/T₁ = P₂/T₂ |
| Avogadro | P, T | V proportional to n | V₁/n₁ = V₂/n₂ |
Diego in Texas fills a rigid 4.00 L cylinder with 0.250 mol of helium at 25 °C. Find the pressure.
T = 25 + 273.15 = 298.15 K
P = nRT/V = (0.250 × 8.314 × 298.15) / 4.00
P = 155 kPa
How do gases in a mixture share the pressure?
Gas particles are so far apart that each type behaves as if the others were absent. Dalton’s law states that the total pressure of a mixture is the sum of the partial pressures. The partial pressure of gas A is PA = xA × Ptotal, where the mole fraction xA = nA/ntotal.
A 10.0 L flask at 300 K holds 0.400 mol N₂ and 0.100 mol O₂. Find the total pressure and the partial pressure of N₂.
ntotal = 0.500 mol, so Ptotal = 0.500 × 8.314 × 300 / 10.0 = 124.7 kPa
x(N₂) = 0.400/0.500 = 0.800, so P(N₂) = 0.800 × 124.7
P(N₂) = 99.8 kPa, Ptotal = 125 kPa
What does kinetic molecular theory explain?
The kinetic molecular theory (KMT) says that gas particles move randomly in straight lines, collide elastically with the walls and each other, and have an average kinetic energy proportional to the absolute temperature. Pressure arises from the sum of countless wall collisions. At the same temperature all gases have the same average kinetic energy, so lighter molecules move faster.
Average molar kinetic energy = (3/2)RT, and the root-mean-square speed is urms = √(3RT/M), with M in kg/mol, T in K and R = 8.314 J/(mol·K).
| Gas | M (kg/mol) | urms at 300 K (m/s) |
|---|---|---|
| H₂ | 0.002016 | 1930 |
| He | 0.004003 | 1370 |
| N₂ | 0.028014 | 517 |
| CO₂ | 0.04401 | 412 |
When do real gases deviate from ideal behaviour?
Two assumptions fail. At high pressure the particles take up a noticeable share of the volume, which makes the real volume larger than predicted. At low temperature attractions pull particles together and make the pressure smaller than predicted. The van der Waals equation corrects both: (P + an²/V²)(V − nb) = nRT, where a measures attractions and b the particle volume.
Never put the temperature in °C into PV = nRT. Always convert to kelvin. Also remember that gas collected over water contains water vapour, so subtract its partial pressure before using the gas law.
Write the units next to every number first. If kPa and L go in, use R = 8.314; if atm and L go in, use 0.08206. When units cancel to leave mol, you are safe.
- PV = nRT with R = 8.314 L·kPa/(mol·K) or 0.08206 L·atm/(mol·K); T must be in kelvin.
- Dalton: P_total = sum of partial pressures; P_A = x_A × P_total.
- Root-mean-square speed u_rms = √(3RT/M), M in kg/mol; lighter gases move faster at the same T.
- Real gases deviate at high pressure (particle volume) and low temperature (attractions).
- 1 atm = 101.3 kPa = 760 mmHg.
Gas laws and particle motion: frequently asked questions
Why must gas temperature be in kelvin?
The gas laws are proportionalities to absolute temperature, which is zero when particle motion stops. A Celsius temperature of 0 does not mean zero kinetic energy, so substituting °C would give wrong results.
Which value of R should I use?
Use the one whose units match your data. With pressure in kPa and volume in litres use 8.314 L·kPa/(mol·K); in atm and litres use 0.08206 L·atm/(mol·K); for speeds and energies use 8.314 J/(mol·K).
Why do lighter gases move faster at the same temperature?
At a given temperature all gases have the same average kinetic energy, ½mv². With a smaller mass, the speed must be larger to give the same energy, which is why helium atoms are faster than nitrogen molecules.
What is gas collected over water?
A gas bubbled into an inverted container of water is saturated with water vapour. The total pressure equals the partial pressure of the gas plus the vapour pressure of water at that temperature, so subtract it first.
When does a real gas behave most ideally?
At high temperature and low pressure. Particles are then far apart and moving fast, so their own volume and the attractions between them are negligible compared with the volume and energy of the sample.
