How Is Gas Pressure Related to Molecular Speed?

For both SL and HL, gas pressure is proportional to the mean square molecular speed when the gas density is constant. The kinetic theory relationship is $P=\frac{1}{3}\rho\langle v^2\rangle=\frac{1}{3}\rho v_{\text{rms}}^2$.

The Reasoning

Gas pressure results from gas molecules colliding with the walls of their container. During each collision, a molecule's momentum component perpendicular to the wall changes, so the wall experiences a force according to the rate of change of momentum.

Faster molecules undergo larger momentum changes and make collisions more frequently. Both effects increase the average force on the wall. Since pressure is force per unit area, the pressure increases.

For molecules moving randomly in three dimensions, kinetic theory gives:

$$P=\frac{1}{3}\rho\langle v^2\rangle$$

Here, $\rho$ is the gas density and $\langle v^2\rangle$ is the mean square speed. The factor $\frac{1}{3}$ arises because random molecular motion is distributed equally among three perpendicular directions.

The root-mean-square speed is defined by $v_{\text{rms}}=\sqrt{\langle v^2\rangle}$, so the equation can also be written as:

$$P=\frac{1}{3}\rho v_{\text{rms}}^2$$

At constant density, doubling $v_{\text{rms}}$ produces four times the pressure because $P\propto v_{\text{rms}}^2$. For example, increasing from to increases the pressure by .

A numerical application shows how the equation is used. If a gas has density $1.2\text{ kg m}^{-3}$ and root-mean-square speed $500\text{ m s}^{-1}$, then

$$P=\frac{1}{3}(1.2)(500)^2=1.0\times10^5\text{ Pa}.$$

This calculation assumes random molecular motion and the stated density. The result is close to atmospheric pressure, showing how molecular speeds produce substantial macroscopic forces through enormous numbers of collisions. Root-mean-square speed, not ordinary average speed, must be substituted.

This also links molecular speed to absolute temperature because $\frac{1}{2}m\langle v^2\rangle=\frac{3}{2}k_{\text{B}}T$. Therefore, for a fixed number of molecules in a fixed volume, higher temperature means greater molecular speed and greater pressure.

Exam Technique

For subtopic B.3 Gas laws, explain pressure using molecular collisions and momentum change before stating the equation. A common misconception is that pressure is proportional to mean speed; it is proportional to mean square speed, provided density remains constant.