Ideal Gas Law
The ideal gas law, pV = nRT (equivalently pV = NkBT), is the equation of state of an ideal gas, a model gas of point particles that interact only through elastic collisions. It relates pressure p, volume V, amount n (or number N) and absolute temperature T through the molar gas constant R = 8.314462618 J mol−1 K−1. Real gases approach it at low density and at temperatures well above their condensation point.
Curious
The law combines several older observations: Boyle's law (1662), that at fixed temperature pressure times volume stays constant; Charles's and Gay-Lussac's laws, that volume or pressure grows in proportion to absolute temperature; and Avogadro's hypothesis (1811), that equal volumes of gases at the same temperature and pressure contain the same number of molecules. Émile Clapeyron combined them in 1834.
The same equation works whether the gas is helium, nitrogen or carbon dioxide, as long as the molecules are far apart compared with their size. It explains why tyres lose pressure on cold mornings, why hot-air balloons rise, and how many molecules fill a room (see Temperature).
Exploring
| Symbol | Quantity | SI unit |
|---|---|---|
| p | Pressure | Pa |
| V | Volume | m³ |
| n | Amount of substance | mol |
| R | Molar gas constant, 8.314462618 | J mol−1 K−1 |
| T | Absolute temperature | K |
Deep Dive
Kinetic derivation. Treating molecules as point particles that collide elastically with the walls, the pressure is p = (1/3)(N/V)m⟨v²⟩. Combining with ⟨Ek⟩ = (3/2)kBT gives pV = NkBT, and R = NAkB. Since the 2019 SI redefinition both NA and kB are exact, so R is exact too. Bernoulli sketched this argument in 1738; Clausius, Maxwell and Boltzmann developed it into kinetic theory.
Where it fails. Near condensation or at high density, molecular size and attractions matter. The van der Waals equation, (p + an²/V²)(V − nb) = nRT, adds an attraction parameter a and an excluded volume b and predicts a critical point. A practical measure of deviation is the compressibility factor Z = pV/nRT, equal to 1 for an ideal gas; for nitrogen at room temperature Z stays within a few tenths of a percent of 1 up to about 10 atm but departs strongly at hundreds of atmospheres. The virial expansion Z = 1 + B(T)/Vm + … systematises these corrections.
Quantum gases. At very low temperature or high density, the thermal de Broglie wavelength becomes comparable to the spacing between particles (see de Broglie Hypothesis) and quantum statistics take over: electrons in metals form a degenerate Fermi gas (see Fermi Energy) and bosons can undergo Bose-Einstein condensation. The ideal gas law is the high-temperature limit of both.
Concept Relationship Map
Requires: temperature, pressure
Enables: kinetic theory, van der Waals equation, stellar structure
Commonly confused with: Celsius or gauge-pressure use in the formula
Sources
- Clapeyron, É. (1834). Journal de l'École Polytechnique, 14, 153–190.
- Tiesinga, E., et al. (2021). “CODATA recommended values of the fundamental physical constants: 2018.” Reviews of Modern Physics, 93, 025010.
- Schroeder, D.V. (2000). An Introduction to Thermal Physics. Addison-Wesley.