Kinetic Theory
How molecules in perpetual motion give rise to pressure, temperature, and the behaviour of gases
Table of Contents
12.1 Introduction
Kinetic theory explains the behaviour of gases based on the idea that gas consists of rapidly moving atoms or molecules. This is possible as the inter-atomic forces, which are short range and important for solids and liquids, can be neglected for gases.
"All things are made of atoms — little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another." — Richard Feynman
12.2 Molecular Nature of Matter
The discovery that matter is made up of atoms is one of the most significant in all of science. Dalton's atomic theory explained the laws of definite and multiple proportions. Avogadro's hypothesis states that equal volumes of all gases at equal temperature and pressure have equal number of molecules.
12.3 Behaviour of Gases
Gases obey several empirical laws that relate pressure, volume, and temperature:
Boyle's Law
PV = constant (at constant T). Pressure inversely proportional to volume. Double the pressure → half the volume.
Charles's Law
V/T = constant (at constant P). Volume directly proportional to absolute temperature. Heat a gas → it expands.
Gay-Lussac's Law
P/T = constant (at constant V). Pressure directly proportional to absolute temperature.
Avogadro's Law
V ∝ n (at constant T, P). Equal volumes of gases at same T, P contain equal number of molecules.
12.4 Kinetic Theory of an Ideal Gas
The kinetic theory derives the pressure of a gas from molecular collisions with the container walls. The key result connects macroscopic quantities (P, T) to microscopic properties (molecular mass and speed):
Molecular Speeds
From kinetic theory, we get three characteristic speeds:
| Speed | Formula | Description |
|---|---|---|
| Most probable (vₚ) | √(2kBT/m) | Speed at peak of distribution curve |
| Average (vₐᵥ) | √(8kBT/πm) | Mean of all molecular speeds |
| RMS (vᵣₘₛ) | √(3kBT/m) | Root mean square — relates to KE |
The average kinetic energy of a molecule is (1/2)mv²rms = (3/2)kBT. Temperature is a measure of the average translational kinetic energy per molecule. At the same temperature, all gases (monoatomic, diatomic) have the same average translational KE.
12.5 Law of Equipartition of Energy
The law of equipartition of energy states that in thermal equilibrium, the total energy is equally distributed among all the degrees of freedom, with each degree of freedom contributing (1/2)kBT of energy.
| Molecule Type | Degrees of Freedom (f) | Total Energy (U) | Cᵥ | Cₚ | γ = Cₚ/Cᵥ |
|---|---|---|---|---|---|
| Monoatomic (He, Ne, Ar) | 3 (trans only) | (3/2)nRT | (3/2)R | (5/2)R | 5/3 ≈ 1.67 |
| Diatomic (H₂, O₂, N₂) | 5 (3 trans + 2 rot) | (5/2)nRT | (5/2)R | (7/2)R | 7/5 = 1.40 |
| Polyatomic (CO₂, CH₄) | 6 (3 trans + 3 rot) | 3nRT | 3R | 4R | 4/3 ≈ 1.33 |
12.6 Specific Heat Capacity
Kinetic theory correctly predicts the specific heat capacities of gases. The molar specific heat at constant volume is:
Since Cₚ > Cᵥ (heat at constant pressure must also do work of expansion), γ = Cₚ/Cᵥ is always greater than 1. For monoatomic gases γ ≈ 1.67, diatomic γ ≈ 1.40, polyatomic γ ≈ 1.33. The more complex the molecule, the lower γ.
At very high temperatures, vibrational modes also get activated, increasing f. For diatomic molecules: f = 7 at high T (adding 2 vibrational DOF). This is why Cₚ and Cᵥ increase with temperature — kinetic theory predicts this correctly!
12.7 Mean Free Path
The mean free path (λ) is the average distance a molecule travels between successive collisions:
At STP, air molecules have a mean free path of about 70 nm, while the molecular diameter is about 0.2 nm. So a molecule travels about 350 times its own size between collisions!
Why Air is Transparent
Molecules are far apart compared to their size. Light travels through air without hitting molecules because the mean free path for light is much larger than for gas collisions.
Diffusion
Gas molecules spread by random collisions (mean free path determines diffusion rate). Perfume spreads through a room in minutes because molecules make millions of collisions per second.
The viscosity of a gas is related to mean free path: η = (1/3)ρvₐᵥλ. This explains why gas viscosity increases with temperature (molecules move faster) while liquid viscosity decreases (molecules have more energy to overcome intermolecular bonds).