Diffusion and Effusion

Gaseous diffusion is the process whereby a gas spreads out through another gas to occupy a space with uniform partial pressure. A gas or vapor having a relatively high partial pressure will spread out toward regions of lower partial pressure of that gas until the partial pressure becomes equal everywhere in the space. Thus, if we release an odorous substance at one corner of a room, minutes later it will be detected at the opposite corner. (It is well known, for example, how quickly the smell of baking apple pie draws people to the kitchen.)

We might consider the spread of an odor in minutes to be fast yet when we think about our kinetic-theory calculations of molecular speed, we might ask why diffusion is not even much faster than it is. Why does it take minutes for the gas to diffuse throughout a room when the molecules are moving at perhaps a thousand miles per hour? This was in fact one of the first criticisms of kinetic theory. The answer is simply that a molecule never travels very far in one direction (at ordinary pressures) before it collides with another molecules and moves off in another direction. If we could trace out the path of an individual molecule, it would be a zigzagging trail reminiscent of Brownian motion. For a molecule to cross a room, it has to travel many times the straight-line distance.

Although the rate of diffusion certainly depends in part on the average molecular speed, the effect of molecular collisions makes the theoretical Picture a bit complicated. Effusion, like diffusion, is a process involving the flow of a gas, but is theoretically much simpler.

If we place a container of gas in a vacuum and then make a very small hole in the container, the gas escapes through the hole at the same speed it had in the container. The proses is called effusion. It was first studied by Thomas Graham who, in 1846, discovered the law that bears his name. Graham’s law of effusion states: The rate of effusion of gas molecules from a partivular hole is inversely proportional to the square root of the molecular weight of the gas at constant temperature and pressure.

Let us consider a kinetic-theory analysis of an effusion experiment. Suppose the hole in the container is made small enough so that the gas molecules continue to move randomly (rather than moving together as they would in a wind). When a molecule happens to encounter the hole, it leaves the container. The collection of molecules leaving the container by chance encounters with the hole constitutes the effusing gas. All we have to consider for effusion is the rate at which molecules encounter the hole in the container.

The rate of effusion of molecules from a container depends on three factors: (1) the cross-sectional area of the hole (the larger this is, the more likely molecules are to escape); (2) the number of molecules per unit volume (the more crowded the molecules are, the more likely they are to encounter the hole); and (3) the average molecular speed (the faster the molecules are moving, the sooner they will escape).

If we compare the effusion of different gases from the same container, at the same temperature and pressure, factors (1) and (2) will be the same. The average molecular speeds will be different, however.

Since the average molecular speed essentially equals , where is the molar mass, we see that the rate of effusion is proportional to 1/ . That is, the rate of effusion is inversely proportional to the square root of the molar mass (or molecular weight), as Graham’s law states. The derivation of Graham’s law from kinetic theory was considered a triumph of the theory, and greatly strengthened confidence in its validity.

 

Graham’s law effusion:

Rate of effusion of molecules α 1/ (for the same container at constant T and P)

Grahams law has practical application in the preparation of fuel rods for nuclear fission reactors. Such reactors depend on the fact that the uranium-235 nucleus undergoes fission (splits) when bombarded with neutrons. When the nucleus splits, several neutrons are emitted and a large amount of energy is liberated. These neutrons bombard more urenium-235 nuclei, and the process continues with the evolution of more energy. However, natural uranium consist of 99.27% uranium-238 (which does not undergo fission) and only 0.72% uranium-235 (which does undergo fission). A uranium fuel rod must contain about 3% uranium-235 to sustain the nuclear reaction.

To increase the percentage of uranium-235 in a sample of uranium (a process called enrichment), one first prepares uranium hexafluoride, U , a white, crystalline solid that is easily vaporized.

Uranium hexafluride vapor is allowed to pass through a series of porous membranes. Each membrabe has many small holes through which the vapor can effuse. Since the U molecules with the lighter isotope of uranium travel about 0.4% faster than the U molecules with the heavier isotope, the gas that passes through first will be somewhat richer in uranium-235 vapor becomes further concentrated. It takes many effusion stages to reach the necessary enrichment; for complete sepetaration of the isotopes, as required for bomb-grade uranium, about two million effusion stages are needed.