This physics grades 11-12 lesson on ideal gas equation notes gives the key ideas first, then worked examples and diagrams. After that, practise with the worksheet on ideal gas equation and check yourself with the quiz.
Quick answer: An ideal gas obeys pV = nRT, where pressure, volume and kelvin temperature are linked through the amount of gas. Kinetic theory explains this: the mean kinetic energy of a molecule is (3/2)kT, and the internal energy of a monatomic ideal gas is (3/2)nRT. Added heat changes temperature by Q = mcΔT or the state by Q = mL.
Level: Physics grades 11-12 (A Level · IB · Class 11-12) · Curriculum: IB B.1-B.3
Ideal gas equation notes: the key ideas
What makes a gas ideal?
An ideal gas is a model that lets physicists predict how the pressure, volume and temperature of a gas change together. It works well for air, helium or nitrogen at room conditions, and it fails only when a gas is squashed hard or cooled close to its boiling point.
An ideal gas is a gas whose particles are point-like, never attract or repel each other except during collisions, and collide elastically with the walls and with each other.
These assumptions matter because they remove any potential energy between particles. The only energy stored in the gas is the random kinetic energy of its particles, so the internal energy depends on temperature alone.
How do you use pV = nRT?
The equation of state links the four measurable quantities of a gas. A sealed tyre, a diver’s cylinder and a balloon all obey it to a good approximation.
pV = nRT (also written pV = NkT)
p = pressure (Pa), V = volume (m³), n = amount of gas (mol), N = number of molecules, T = temperature (K), R = 8.31 J/(mol K), k = 1.38 × 10⁻²³ J/K, and R = NAk with NA = 6.02 × 10²³ mol⁻¹.
A 5.0 L cylinder holds 0.40 mol of gas at 27 °C. Calculate the pressure.
Convert first: T = 27 + 273 = 300 K and V = 5.0 L = 5.0 × 10⁻³ m³. Then p = nRT / V = (0.40 × 8.31 × 300) / (5.0 × 10⁻³) = 1.99 × 10⁵ Pa, so p ≈ 2.0 × 10⁵ Pa.
Never put a Celsius temperature or a volume in litres into pV = nRT. The answer is wrong by a factor of 300 or 1000 and still looks like a sensible number.
How are temperature and molecular motion linked?
Kinetic theory connects the large-scale temperature to the motion of individual molecules. The hotter the gas, the faster its molecules move on average.
The mean kinetic energy of a molecule is ½m⟨c²⟩ = (3/2)kT. For n moles of a monatomic ideal gas the internal energy is U = (3/2)nRT. The root-mean-square speed is crms = √(3RT / M), with M the molar mass in kg/mol.
Find the r.m.s. speed of nitrogen molecules (M = 0.028 kg/mol) at 300 K and their mean kinetic energy.
crms = √(3 × 8.31 × 300 / 0.028) = √(2.67 × 10⁵) = 517 m/s. Mean kinetic energy = (3/2) × 1.38 × 10⁻²³ × 300 = 6.21 × 10⁻²¹ J.
Notice that doubling the kelvin temperature doubles the mean kinetic energy but multiplies the r.m.s. speed by only √2, about 1.41.
How much energy changes the temperature or the state?
Thermal energy added to a solid or liquid raises its temperature or changes its state, never both at once. While a substance melts or boils, the energy goes into breaking bonds, so the temperature stays flat.
Q = mcΔT and Q = mL
Q = energy transferred (J), m = mass (kg), c = specific heat capacity (J/(kg K)), ΔT = temperature change (K or °C), L = specific latent heat (J/kg).
| Substance | Value | Meaning |
|---|---|---|
| Water (liquid) | 4180 J/(kg K) | specific heat capacity |
| Ice | 2100 J/(kg K) | specific heat capacity |
| Aluminium | 900 J/(kg K) | specific heat capacity |
| Water, melting | 3.34 × 10⁵ J/kg | latent heat of fusion |
| Water, boiling | 2.26 × 10⁶ J/kg | latent heat of vaporisation |
How much energy turns 0.10 kg of ice at −20 °C into water at 0 °C?
Warming the ice: Q₁ = 0.10 × 2100 × 20 = 4200 J. Melting it: Q₂ = 0.10 × 3.34 × 10⁵ = 33 400 J. Total = 4200 + 33 400 = 37 600 J, matching the curve below.
Write the plan before you calculate: warm, melt, warm, boil. Each stage has its own formula and the stages simply add.
Boiling needs far more energy than melting because the molecules must be separated completely and must push back the atmosphere. For 0.10 kg of water the vaporisation stage costs 226 000 J, nearly seven times the melting stage.
- pV = nRT with p in Pa, V in m³, T in K and R = 8.31 J/(mol K)
- Mean kinetic energy of a molecule = (3/2)kT; internal energy U = (3/2)nRT
- r.m.s. speed c = √(3RT / M), with M in kg/mol
- Q = mcΔT changes temperature; Q = mL changes state at constant temperature
Gas laws, kinetic theory and thermal energy: frequently asked questions
What is an ideal gas in physics?
It is a model gas whose particles have negligible volume and exert no forces on each other except in collisions. Real gases such as air behave almost ideally at normal pressure and temperature.
Why must temperature be in kelvin in pV = nRT?
The equation is a proportionality between pV and absolute temperature, which starts from zero when molecular motion is minimal. Celsius values can be zero or negative while the gas still has pressure, so they give wrong results.
What is the internal energy of an ideal gas?
It is the total random kinetic energy of all its particles, since there is no potential energy between them. For a monatomic gas U = (3/2)nRT, so it depends only on temperature.
What is the difference between specific heat capacity and latent heat?
Specific heat capacity is the energy needed to raise 1 kg of a substance by 1 K without changing its state. Latent heat is the energy needed to change the state of 1 kg at constant temperature.
Why does the temperature stay constant during melting?
The energy supplied increases the potential energy of the particles as bonds are broken, not their kinetic energy. Temperature measures mean kinetic energy, so it does not change until all the solid has melted.
