As indicated by the gas laws, all gases show similar physical characteristics and behavior. A theoretical model to explain the behavior of gases was developed during the second half of the 19th century. The combined efforts of Boltzmann, Maxwell, and others led to a simple explanation of gaseous molecular behavior based on the motion of individual molecules. This model is called the kinetic molecular theory of gases. Like the gas laws, this theory was developed in reference to ideal gases, but it can be applied with reasonable accuracy to real gases as well.
According to the kinetic molecular theory of gases, the average kinetic energy (and therefore average velocity) of a gas particle is proportional to the absolute temperature of the gas:
where k is the Boltzmann constant. However, because of the large number of rapidly and randomly moving gas particles, the speed of an individual gas molecule is nearly impossible to define. Instead, it is the average speed of all the gas particles that can be related exactly to the temperature. Some particles will be moving at higher speeds and some at lower speeds.
A Maxwell-Boltzmann distribution curve shows the distribution of speeds of gas particles at a given temperature. The curve below shows a distribution curve of molecular speeds at two temperatures, T1 and T2, where T2 > T1. Notice that the bell-shaped curve flattens and shifts to the right as the temperature increases, indicating that, at higher temperatures, more molecules are moving at high speeds.
Diffusion of gases can provide a demonstration of random motion when the molecules of these gases mix with one another by virtue of their individual kinetic properties. Diffusion occurs when gas molecules move through a mixture. Diffusion accounts for the fact that an open bottle of perfume can quickly be smelled across a room. The kinetic molecular theory of gases predicts that heavier gas molecules diffuse more slowly than lighter ones because of their differing average speeds. In 1832, Thomas Graham showed mathematically that, under isothermal and isobaric conditions, the rates at which two gases diffuse are inversely proportional to the square root of their molar masses. Thus:
where r1 and MM1 represent the diffusion rate and molar mass of gas 1, respectively, and r2 and MM2 represent the diffusion rate and molar mass of gas 2.
Effusion is the flow of gas particles under pressure from one compartment to another through a small opening. Graham used the kinetic molecular theory of gases to show that for two gases at the same temperature, the rates of effusion are proportional to the average speeds. He then expressed the rates of effusion in terms of molar mass and found that the relationship is the same as that for diffusion.