Natural uranium is mostly nonfissionable \(^{238} \mathrm{U} ;\) it contains only about \(0.7 \%\) of fissionable \(^{235}\) U. For uranium to be useful as a nuclear fuel, the relative amount of \(^{235}\) U must be increased to about \(3 \% .\) This is accomplished through a gas diffusion process. In the diffusion process, natural uranium reacts with fluorine to form a mixture of \(^{238} \mathrm{UF}_{6}(g)\) and \(^{235} \mathrm{UF}_{6}(g) .\) The fluoride mixture is then enriched through a multistage diffusion process to produce a \(3 \%^{235} \mathrm{U}\) nuclear fuel. The diffusion process utilizes Graham's law of effusion (see Chapter \(8,\) Section \(8-7\) ). Explain how Graham's law of effusion allows natural uranium to be enriched by the gaseous diffusion process.

Graham's Law of Effusion is a principle in chemistry that describes the relationship between the effusion rates of gases and their molar masses. Specifically, it states that the rate at which a gas escapes through a small hole into a vacuum is inversely proportional to the square root of the molar mass of its particles. This fundamental law can be represented mathematically as \[\frac{Rate_{1}}{Rate_{2}} = \sqrt{\frac{M_{2}}{M_{1}}}\] where \(Rate_1\) and \(Rate_2\) are the effusion rates of two different gases, and \(M_1\) and \(M_2\) are their respective molar masses.
When applying this law to the process of uranium enrichment, it becomes clear why different isotopes of uranium hexafluoride effuse at different rates. The lighter \(^{235}\mathrm{UF}_6\) will effuse more rapidly than the heavier \(^{238}\mathrm{UF}_6\), allowing for a separation based on mass and ultimately leading to the enrichment of the fissionable \(^{235}\mathrm{U}\).

Uranium Hexafluoride (\(\mathrm{UF}_6\)) is a key compound used in the nuclear fuel preparation process. This chemical is the form in which uranium is typically processed for enrichment, due to its unique properties. It is a volatile substance, which allows it to be in a gaseous state at relatively low temperatures; this is crucial for the enrichment process.
Uranium in its natural state is composed mainly of the isotope \(^{238}\mathrm{U}\), with only about 0.7% of the more reactive \(^{235}\mathrm{U}\) isotope, which is capable of sustaining a nuclear chain reaction. To become viable as nuclear fuel, the proportion of \(^{235}\mathrm{U}\) must be increased. The first step in this process is combining natural uranium with fluorine to create \(\mathrm{UF}_6\), consisting of both \(^{238}\mathrm{UF}_6\) and \(^{235}\mathrm{UF}_6\). The uranium hexafluoride is then subjected to gaseous diffusion to separate these isotopes, leveraging the principles outlined in Graham's law of effusion.

Isotopic separation is the cornerstone of creating usable nuclear fuel. It involves increasing the percentage of the fissionable \(^{235}\mathrm{U}\) isotope within uranium to make it suitable for use in nuclear reactors. This process hinges on the slight differences in physical properties between isotopes, most notably their mass.
The gaseous diffusion method is one example of isotopic separation. It takes advantage of the different effusion rates of \(^{238}\mathrm{UF}_6\) and \(^{235}\mathrm{UF}_6\) as predicted by Graham's Law. Another method is centrifugation, where isotopes are separated due to their different masses under the force of a rapidly spinning rotor. These techniques aim to increase the proportion of \(^{235}\mathrm{U}\) from 0.7% to approximately 3-5% for reactor-grade nuclear fuel, an essential step for powering nuclear energy facilities.

Nuclear fuel is a material that can be 'burned' by nuclear fission to derive energy. It primarily consists of the isotope \(^{235}\mathrm{U}\), which is capable of sustaining a controlled chain reaction. To harness this energy, nuclear fuel must first be processed through enrichment to increase the concentration of \(^{235}\mathrm{U}\).
Once enriched to the proper levels, the nuclear fuel can be fabricated into fuel rods or assemblies and placed into a nuclear reactor. Inside the reactor, atoms of \(^{235}\mathrm{U}\) are bombarded with neutrons, causing them to split and release a tremendous amount of energy in the form of heat. This heat can then be used to produce steam, which in turn drives turbines to generate electricity. The careful and detailed process of creating this fuel, starting from mining uranium ore to the isotopic separation and enrichment, illustrates the intricate work behind the power that many take for granted when they flip on a light switch.