Problem — Pressure and molecular collisions

Problem

Pressure and molecular collisions. At constant volume, the pressure of a gas rises if the temperature increases. Translate this macroscopic observation into the language of molecular collisions on the walls: what changes about each individual collision and what changes in their frequency?

Solution

At constant volume the pressure is the average force per unit area due to the molecules colliding with the walls. Raising the temperature increases the mean kinetic energy, and hence the root-mean-square speed: $v_\text{rms}=\sqrt{\frac{3k_B T}{m}}\propto \sqrt{T}$ This affects collisions in two ways that combine together:

• Each individual collision is more energetic. Faster molecules arrive with greater momentum and, bouncing back, transfer a larger impulse $\Delta p \propto v$ to the wall.
• Collisions are more frequent. Moving faster, each molecule reaches the wall more often: the number of collisions per second also grows $\propto v$.

The product of the two effects (impulse per collision $\times$ collision frequency) gives a mean force $\propto v^2 \propto T$, i.e. the pressure grows proportionally to the absolute temperature, consistent with $p = n k_B T$ at constant volume. $\ev{p\propto T:\ \text{more energetic collisions (}\Delta p\propto v\text{) and more frequent (}\propto v\text{)}\Rightarrow p\propto v^2\propto T}$

Links

Topics: Kinetic theory of gases Concepts: Pressure · Temperature Skills: Micro-macro interpretation