1.6: The Kinetic Model of Gases

The kinetic model of gases explains the properties of a perfect gas using three main assumptions: molecules move in ceaseless random motion, their size is negligible compared to the distances between them, and they do not interact except during perfectly elastic collisions. The total energy of a gas is the sum of the kinetic energies of all its constituent molecules. The pressure exerted by the gas arises from the continual bombardment of the container walls by billions of colliding molecules. Moreover, the molecular speeds in a gas are not uniform but follow a specific distribution.

The kinetic model of gases describes the distribution of molecular speeds in a sample, with some molecules moving slower or faster than the average. Collisions continually redistribute molecular speeds. The probability distribution governing these speeds is known as the Maxwell–Boltzmann distribution. It shows that the probability of molecules having a certain speed increases proportionally to the range of speeds. The probability of very high speeds is small, decreasing more rapidly for heavier molecules and more slowly for higher temperatures.

Conversely, the probability of very low speeds is also low. The distribution also indicates that there's a greater chance of molecules having high speeds at higher temperatures. At a fixed temperature, molecules with different molar masses exhibit distinct distributions. Heavier molecules tend to have lower average speeds and a narrower range of speeds. In contrast, lighter molecules, like H₂, have higher average speeds and a broader range of speeds, meaning many molecules will be moving significantly slower or faster than the average.

The mean free path (λ) represents the average distance a molecule travels before colliding with another molecule. This distance is shorter in liquids but can span several hundred molecular diameters in gases. The collision frequency (z) is the average number of collisions a molecule experiences within a specific time frame. As pressure increases, the mean free path decreases due to the increased density of molecules. Molecules with larger collision cross-sections typically have shorter mean free paths. Similarly, the collision frequency escalates with increasing gas pressure. Heavy molecules, given the identical collision cross-sections, exhibit lower collision frequencies compared to their lighter counterparts.