Graham's Law

The Intuitive Picture

Imagine two identical rooms, each filled with a different gas at the same temperature and pressure. One room contains helium, a very light gas, and the other contains xenon, a very heavy gas. If you were to open a small door in each room leading to a vacuum, you would observe that the helium escapes much faster than the xenon. Why does this happen? It is not because the helium molecules are "pushed" harder, but because they are moving significantly faster. Temperature is a measure of the average kinetic energy of the molecules. Since kinetic energy is defined as one-half mass times velocity squared, for two different gases to have the same average kinetic energy (same temperature), the lighter molecules must have a higher velocity. Therefore, in any given second, a helium molecule will strike the wall of the container, and specifically the area around the tiny hole, many more times than a xenon molecule. Because the rate of effusion depends on how often molecules hit the hole, the faster-moving, lighter gas effuses at a higher rate. This inverse relationship between mass and speed is the core of Graham's law. It tells us that if you double the molar mass of a gas, its effusion rate decreases by a factor of the square root of two. This simple square-root relationship allows us to compare the behaviour of any two gases without needing to know their specific temperatures or pressures, provided they are identical for both gases.

Mathematical Formulation

The quantitative statement of Graham's law arises from the kinetic theory of gases. The rate of effusion, \(\Phi\), is proportional to the number density of molecules, \(n\), and their mean speed, \(\langle v \rangle\). From the Maxwell-Boltzmann distribution, we know that the mean speed of a gas molecule is inversely proportional to the square root of its molar mass, \(M\). Therefore, at a fixed temperature, the rate of effusion is proportional to \(1/\sqrt{M}\). When comparing two different gases, gas 1 and gas 2, under identical conditions of temperature and pressure, the ratio of their effusion rates is equal to the square root of the inverse ratio of their molar masses. This can be expressed as:

It is crucial to note the inversion of the molar masses. The lighter gas (smaller \(M\)) will have the larger rate. For example, if gas 1 is hydrogen (\(M_1 = 2.016\ \mathrm{g\,mol^{-1}}\)) and gas 2 is oxygen (\(M_2 = 32.00\ \mathrm{g\,mol^{-1}}\)), the ratio of their rates is \(\sqrt{32.00 / 2.016}\), which is approximately 4. This means hydrogen effuses four times faster than oxygen. This mathematical form allows for precise calculations in laboratory settings where the time taken for a certain volume of gas to effuse is measured. The units of molar mass must be consistent, but since it is a ratio, \(\mathrm{g\,mol^{-1}}\) and \(\mathrm{kg\,mol^{-1}}\) yield the same result.

Effusion vs. Diffusion

While often used interchangeably in casual conversation, effusion and diffusion are distinct physical processes. Effusion occurs when gas molecules pass through a tiny aperture into a vacuum or a region of significantly lower pressure, such that intermolecular collisions near the hole are negligible. The rate of effusion is determined solely by the molecular speed and the size of the hole. Diffusion, on the other hand, is the spreading of molecules from a region of high concentration to a region of low concentration through a medium, involving frequent intermolecular collisions. In diffusion, the rate is more complex and depends on the collision cross-section and the mean free path, not just the speed. However, for many practical purposes, especially when dealing with gases in open spaces or through porous materials, Graham's law provides a good approximation for the relative rates of diffusion as well. The key distinction is that in effusion molecules pass through the hole one at a time without colliding, whereas diffusion is governed by frequent collisions. In the context of this lecture, we focus strictly on effusion, as it is the process that allows for the clean application of the \(1/\sqrt{M}\) dependence.

Practical Applications

The most significant application of Graham's law is in the separation of isotopes. Isotopes of the same element have the same chemical properties but different masses. For example, uranium-235 and uranium-238 have very similar molar masses, but uranium-235 is slightly lighter. When uranium hexafluoride, \(\mathrm{UF_6}\), is allowed to effuse through a porous barrier, the lighter \(\mathrm{^{235}UF_6}\) molecules effuse slightly faster than the heavier \(\mathrm{^{238}UF_6}\) molecules. Although the difference is small, repeated stages of effusion can enrich the gas in the lighter isotope. This process was historically crucial for the production of nuclear fuel and weapons. Another application is in the determination of molar masses of unknown gases. By measuring the rate of effusion of an unknown gas relative to a known gas, one can calculate the molar mass of the unknown. This technique is particularly useful for gases that are difficult to condense or that decompose upon heating. The precision of this method depends on the accuracy of the rate measurement and the assumption that the gas behaves ideally.