The Kinetic Molecular Theory of Gases

A gas consists of a very large number of particles whose own volume is negligible compared with the volume of the container. Those particles are in constant, random, straight-line motion. Collisions between particles, and with the walls, are elastic, so no kinetic energy is lost overall. There are no attractive or repulsive forces between the particles. Finally, the average kinetic energy of the particles is proportional to the absolute temperature in kelvin.

The ideal model distinguishes particle size from the container volume. “Negligible particle volume” means the physical volume occupied by particles is small compared with the space between them; the gas sample still fills the whole vessel. “No intermolecular forces” applies between the model's brief collisions, which change direction and transfer momentum. Elastic collisions conserve total kinetic energy for the colliding pair, not each particle's individual speed.

Each time a particle strikes a wall it reverses direction, which means the wall exerted a force on it and it exerted an equal force on the wall. Pressure is the sum of those tiny forces divided by the wall area. Anything that raises the collision rate or the force per collision raises the pressure.

A particle striking a wall changes its momentum, and the wall receives an equal opposite impulse. Many impulses per second per unit area produce the macroscopic pressure. To explain a pressure change, identify whether particle number, speed, collision frequency or momentum transfer changes, and what is held fixed. A heavier particle transfers more momentum per collision at a given speed, but at the same temperature heavier particles move more slowly. Those effects combine so equal numbers of ideal particles at the same T and V give the same pressure regardless of molar mass.

The average translational kinetic energy of a gas particle is (3/2)kT, and for one mole it is (3/2)RT. Because this depends only on temperature, two different gases at the same temperature have the same average kinetic energy, even though the heavier gas moves more slowly.

The per-particle expression uses the Boltzmann constant k_B; the per-mole expression uses R because R = N_A k_B. With R = 8.314 J mol⁻¹ K⁻¹, at 300 K the mean translational energy is (3/2)(8.314)(300) = 3.74 × 10³ J per mole. This is translational energy, not necessarily all rotational and vibrational energy in a polyatomic gas. Individual particles span a distribution of energies; equal temperatures establish equal averages, not identical energies for every particle.

Shrink the volume at fixed temperature and the particles hit the walls more often, so pressure rises. Raise the temperature at fixed volume and the particles hit harder and more often, so pressure rises. Add more particles and there are simply more collisions. Every gas law you will meet is this model in algebraic form.

Doubling T does not double average speed: characteristic speeds scale as √T. A gas heated from 25 °C to 50 °C rises from about 298 K to 323 K, so its pressure increases by about 8.4% at fixed n and V, rather than doubling.

The ideal model ignores molecular attractions and the volume occupied by particles. At low temperature attractions matter more because collisions have less kinetic energy; at high pressure the particles are crowded enough that their volume matters. The direction of a deviation depends on conditions, so identify which neglected feature dominates before claiming the measured pressure is above or below ideal.

When attractions dominate, particles approaching a wall are pulled inward by their neighbors and the measured pressure can lie below nRT/V. At very high density, finite particle size and repulsive effects can make the measured pressure exceed the ideal prediction. High temperature and low density generally make the ideal approximation more useful, but the degree of deviation also depends on the gas. A single rule that real gases always give lower pressure is therefore unreliable.