Natural uranium is mostly nonfissionable \({ }^{238} \mathrm{U} ;\) it contains only about \(0.7 \%\) of fissionable \({ }^{235} \mathrm{U}\). For uranium to be useful as a nuclear fuel, the relative amount of \({ }^{235} \mathrm{U}\) 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 5, Section 5.7). Explain how Graham's law of effusion allows natural uranium to be enriched by the gaseous diffusion process.

Graham's law of effusion states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. Mathematically, it can be represented as: \( \frac{Rate_{1}}{Rate_{2}} = \sqrt{\frac{M_{2}}{M_{1}}} \) where \(Rate_{1}\) and \(Rate_{2}\) are the effusion rates of gas 1 and gas 2, respectively, while \(M_{1}\) and \(M_{2}\) are their molar masses.

In the given problem, natural uranium reacts with fluorine gas to form a gas mixture of uranium hexafluoride, specifically: - \({ }^{238} \mathrm{U} + 6\mathrm{F} \rightarrow { }^{238}\mathrm{UF_6}(g)\) - \({ }^{235} \mathrm{U} + 6\mathrm{F} \rightarrow { }^{235}\mathrm{UF_6}(g)\) These two gaseous reactants, uranium-238 hexafluoride, and uranium-235 hexafluoride, are then enriched through a multistage diffusion process.

Using Graham's law of effusion, we can determine the ratio of effusion rates between the two uranium hexafluorides: \(\frac{Rate_{^{235}\mathrm{UF}_6}}{Rate_{^{238}\mathrm{UF}_6}} = \sqrt{\frac{M_{^{238}\mathrm{UF}_6}}{M_{^{235}\mathrm{UF}_6}}}\) Here, we can see that the rate of effusion of the two uranium hexafluoride gaseous compounds depends on the ratio of their molar masses.

Since \({ }^{235}\mathrm{U}\) is lighter than \({ }^{238}\mathrm{U}\), the \({ }^{235}\mathrm{UF_6}\) will effuse faster than the \({ }^{238}\mathrm{UF_6}\). This difference in effusion rates is used in the gaseous diffusion process. The \({ }^{235}\mathrm{UF_6}\) will move faster through a porous barrier, while the heavier \({ }^{238}\mathrm{UF_6}\) will move slower. Through multiple stages of this diffusion process, the concentration of the lighter \({ }^{235}\mathrm{UF_6}\) increases relative to \({ }^{238}\mathrm{UF_6}\). This allows the uranium to become enriched with a higher percentage of \({ }^{235}\mathrm{U}\), reaching the desired level of about \(3\%\) for nuclear fuel. So, Graham's law of effusion is crucial for the enrichment process of natural uranium, as it helps to separate and increase the concentration of the fissionable \({ }^{235}\mathrm{U}\) isotopes in the nuclear fuel.