The rate of diffusion is inversely proportional to the square root of the density of a gas at constant temperature and pressure due to the relationship between molecular collisions and diffusion.
Diffusion is the process by which molecules move from an area of higher concentration to an area of lower concentration. It occurs due to the random motion of molecules and the collisions they undergo with one another. In gases, diffusion is primarily driven by the kinetic energy of the molecules.
The rate of diffusion is affected by two factors: the speed at which molecules move and the frequency of molecular collisions. The speed of molecules is directly related to their kinetic energy, which is determined by temperature. Higher temperatures lead to faster molecular motion and, consequently, faster diffusion.
The frequency of molecular collisions, on the other hand, is influenced by the density of the gas. A higher gas density implies a higher number of molecules in a given volume, resulting in an increased likelihood of collisions. More collisions facilitate the mixing of molecules and, hence, faster diffusion.
When considering the relationship between diffusion rate and density, we observe that as the density increases, the frequency of collisions rises, which promotes faster diffusion. However, higher densities also mean a larger number of molecules per unit volume, which reduces the average distance traveled by each molecule before colliding with another. As a result, individual molecules have less distance to diffuse before encountering other molecules.
The net effect of these factors is that the increase in collision frequency due to higher density is counterbalanced by the decreased distance traveled by each molecule before collision. This inverse relationship between diffusion rate and the square root of density is described by Graham's law of diffusion, which states that the rate of diffusion of a gas is inversely proportional to the square root of its molar mass or density at constant temperature and pressure.
Mathematically, Graham's law can be expressed as: Rate of diffusion ∝ 1/√(density)
Therefore, at constant temperature and pressure, gases with higher densities diffuse more slowly compared to gases with lower densities.