Gas laws
Gas laws are different views of one simple model: many rapidly moving particles in a volume. The ideal-gas equation connects pressure, volume, temperature and amount; kinetic theory explains why those variables are linked.
The ideal-gas equation
PV = nRT
P is absolute pressure, V volume, n amount in moles, T kelvin temperature and R the gas constant. Units must match the chosen value of R.
The molecular form is PV = NkBT. It shows directly that temperature measures the energy scale of molecular motion.
Ideal-gas calculator
Use SI units: P in Pa, V in m³, T in K.
P =
Boyle, Charles and Avogadro as special cases
If n and T are fixed, PV is constant: pressure rises when volume falls (Boyle). If n and P are fixed, V/T is constant (Charles). If P and T are fixed, V is proportional to n (Avogadro).
These relations are not separate laws that need independent memorization; they fall out of PV = nRT when the appropriate variables are held constant.
Kinetic molecular theory
For an ideal gas, particles are treated as point-like, their collisions are elastic, and intermolecular forces are neglected except during collision. The average translational kinetic energy depends only on absolute temperature:
⟨Etrans⟩ = 3kBT/2.
Different gases at the same T have the same average translational kinetic energy, but lighter molecules move faster on average.
Real gases and the compressibility factor
Define Z = PV/(nRT). An ideal gas has Z = 1. Attractions can make Z < 1 because molecules pull one another away from the walls; short-range repulsion and finite molecular size can make Z > 1 at high density.
The van der Waals equation is one historical correction, but modern equations of state are chosen for accuracy over particular pressure and temperature ranges.
Mixtures and partial pressure
For an ideal mixture, Dalton’s law gives Ptotal = ΣPi, and Pi = xiPtotal. Partial pressure links composition to gas-phase equilibrium and to gas exchange in physical and biological systems.
Always use absolute temperature. Substituting °C into gas-law proportionalities gives physically wrong results.
Gas-law reasoning in experiments
Density and molar mass
Combining ρ = m/V with PV = nRT and n = m/M gives ρ = PM/(RT). At the same temperature and pressure, a gas with larger molar mass is therefore denser in the ideal approximation.
Conversely, measuring gas density can estimate molar mass if P and T are known accurately and the gas is sufficiently ideal.
Collection over water
A gas collected over liquid water contains both the desired gas and water vapor. Dalton’s law gives Ptotal = Pgas + PH₂O. The water-vapor pressure must be subtracted before using the dry-gas pressure in a stoichiometric calculation.
The correction grows as temperature rises because water vapor pressure increases strongly with temperature.
Mean speed and diffusion
For an ideal gas, root-mean-square speed is vrms = √(3RT/M). Lighter molecules therefore move faster at the same T. Graham’s law captures a related trend in effusion rates, which vary approximately as 1/√M.
Real diffusion through air is slower than free molecular motion because molecules undergo enormous numbers of collisions.
Worked example: one mole near room conditions
For n = 1.00 mol, T = 298 K and P = 1.00 atm, the ideal-gas law gives V = nRT/P ≈ 24.5 L using R = 0.082057 L atm mol⁻¹ K⁻¹.
This familiar molar volume is not a universal constant. At 273.15 K and 1 atm it is about 22.4 L, and at high pressure a real gas can deviate substantially from either ideal prediction.
Unit discipline
If R = 8.314 J mol⁻¹ K⁻¹ is used, remember that 1 J = 1 Pa·m³. Pressure must then be in pascals and volume in cubic metres. Mixing litres, atmospheres and the SI form of R is one of the most common gas-law errors.
Gas stoichiometry
Because equal volumes of ideal gases at the same T and P contain equal numbers of molecules, balanced gas-phase equations can sometimes be read directly as volume ratios. For N₂ + 3H₂ → 2NH₃, one volume of N₂ reacts with three equal-condition volumes of H₂ to form two volumes of NH₃ if all species can be treated as gases under the stated conditions.
This shortcut fails when temperatures or pressures differ, when gases are nonideal, or when a product condenses.
Reference conditions must be stated
“Molar volume of a gas” has no single value without T and P. Values near 22.4 L mol⁻¹ refer to 273.15 K and 1 atm; near room temperature and 1 atm the value is closer to 24–25 L mol⁻¹. Always state the conditions rather than memorizing one volume as universal.
Exercises
Compression
At constant T, a gas is compressed from 2.0 L to 1.0 L. What happens to pressure?
Solution
It doubles, assuming ideal behavior and fixed amount.
Temperature
Why must T be in kelvin?
Solution
Gas-law proportionalities refer to an absolute energy scale with zero at absolute zero.
Real gas
Where are ideal-gas deviations usually greatest?
Solution
At high pressure and low temperature, where molecular volume and attractions matter most.