B.3.6—Molecular pressure model

Pressure from collisions

Gas particles collide with a wall and change momentum. The wall exerts a force on the particles; by Newton’s third law, the particles exert an equal and opposite force on the wall. Pressure is this normal force per unit area.

Kinetic-theory relation

For an ideal gas,

$$P=\frac13\rho\overline{v^2}$$

where ρ is gas density and $$\overline{v^2}$$ is the mean square molecular speed. The speed in this equation is not simply the square of the average speed.

Read the trends

Greater molecular speed increases momentum change per collision and collision rate, increasing pressure. At fixed speed, greater density means more mass per unit volume and therefore greater pressure.

Worked example from the mapped local textbook

Nitrogen at pressure $1.0\times10^5\,\mathrm{Pa}$ has density $1.17\,\mathrm{kg\,m^{-3}}$. Rearranging the kinetic-theory relation gives

$$v_{\mathrm{rms}}=\sqrt{\overline{v^2}}=\sqrt{\frac{3P}{\rho}}=\sqrt{\frac{3(1.0\times10^5)}{1.17}}=5.1\times10^2\,\mathrm{m\,s^{-1}}$$

This is the root-mean-square speed, not the ordinary arithmetic mean speed.

Common trap

A single particle’s collision force is not the total gas force. Pressure is a statistical average over many collisions on the surface.