15.3 Kinetic theory of gases

The ideal-gas kinetic model assumes particles are point-like compared with container volume, move randomly, collide elastically and exert negligible forces except during collisions.

These assumptions explain pressure and temperature while marking the limits of the model for dense or strongly interacting gases.

At lower density, the average spacing increases and intermolecular attractions become less important, so ideal behaviour improves.

“No forces” applies between collisions in the ideal model; collisions still transfer momentum and create pressure.

Molecules colliding with a wall change momentum; the rate of momentum transfer per unit area produces macroscopic pressure.

More frequent or harder collisions raise pressure. Increasing temperature raises average molecular kinetic energy and therefore collision impulses and rates.

Compressing a gas at fixed temperature increases wall-collision frequency per unit area and raises pressure.

Pressure is not caused by molecules “pushing continuously” between collisions; it is the aggregate effect of impacts.

The root-mean-square speed c_rms is defined by c_rms=√⟨c²⟩ and for an ideal gas c_rms=√(3RT/M), where M is molar mass.

Use absolute temperature and molar mass in kg mol⁻¹; lighter gases move faster at the same temperature.

At fixed T, helium has a larger rms speed than oxygen because its molar mass is smaller.

c_rms is not the arithmetic mean speed and does not mean every molecule travels at that speed.

For N molecules of mass m, pV=⅓Nm⟨c²⟩. Combining with pV=NkT links pressure to mean translational kinetic energy per particle.

Use ⟨c²⟩ as the mean square speed, not the square of the mean speed, and keep N as particle count.

At fixed volume, raising T raises mean molecular kinetic energy and therefore pressure in proportion.

The equation describes a statistical average; not every molecule has the same speed or energy.