Kinetic Theory of Gases
The kinetic theory of gases is the model that derives the macroscopic properties of a gas (pressure, temperature, heat capacity, viscosity, thermal conductivity and diffusion) from the random motion of very many molecules that collide with each other and with container walls. Its central result, p = (1/3)nm⟨v²⟩, combined with the ideal gas law, identifies temperature with mean molecular kinetic energy.
Curious
Air seems still, but each cubic metre near the ground contains about 2.5×1025 molecules flying around at hundreds of metres per second, each colliding with others billions of times a second. When they hit a surface they push on it; the sum of all those pushes is pressure. Heat the gas and the molecules speed up, so they hit harder and more often: pressure rises (see Ideal Gas Law).
Daniel Bernoulli proposed the basic idea in 1738, but it was ignored for a century. Clausius, Maxwell and Boltzmann developed it in the 1850s–1870s, and its successful predictions became a key argument that atoms are real (see Ludwig Boltzmann).
Exploring
| Assumption | Justification | Where it fails |
|---|---|---|
| Molecules are point-like | Size « spacing at low density | High pressure (see van der Waals) |
| Collisions are elastic | Energy conserved on average | Very hot gases (excitation, ionisation) |
| No forces except in collisions | Short-range forces | Near condensation |
| Classical motion | de Broglie wavelength « spacing | Very low temperature (quantum gases) |
Deep Dive
Pressure derivation. A molecule of mass m bouncing elastically off a wall transfers momentum 2mvx; averaging over molecules gives p = nm⟨vx²⟩ = (1/3)nm⟨v²⟩. Comparing with p = nkBT yields (1/2)m⟨v²⟩ = (3/2)kBT (see Temperature). Clausius introduced the mean free path in 1858 to answer the objection that fast molecules should make diffusion fast.
Transport and a surprise. Kinetic theory predicts viscosity η ≈ (1/3)nm⟨v⟩λ. Because λ ∝ 1/n, the density cancels: gas viscosity should be independent of pressure. Maxwell made this counterintuitive prediction and confirmed it experimentally in 1866, a striking early success. Loschmidt used kinetic theory in 1865 to make the first reasonable estimate of molecular size and of the number of molecules per unit volume.
Equipartition and its failure. Kinetic theory with equipartition predicted heat capacities that agreed for monatomic gases but not for diatomic ones, where some degrees of freedom seemed to be “frozen”. Maxwell called this the greatest difficulty facing the molecular theory; quantum mechanics later resolved it (see Internal Energy). The Boltzmann transport equation provides the rigorous foundation, and the Chapman–Enskog method derives transport coefficients from it.
Concept Relationship Map
Requires: ideal gas law, linear momentum
Enables: Maxwell-Boltzmann distribution, transport theory, statistical mechanics
Commonly confused with: all molecules having one speed
Sources
- Maxwell, J.C. (1860). Philosophical Magazine, 19, 19–32; 20, 21–37.
- Bernoulli, D. (1738). Hydrodynamica. Strasbourg.
- Chapman, S., Cowling, T.G. (1970). The Mathematical Theory of Non-Uniform Gases (3rd ed.). Cambridge University Press.