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.

The gaseous diffusion process is a method used in enriching uranium. It involves reacting natural uranium with fluorine to form a mixture of \({ }^{238}\mathrm{UF}_{6}(g)\) and \({ }^{235}\mathrm{UF}_{6}(g)\). This uranium hexafluoride mixture is then passed through a series of diffusion barriers to separate the lighter isotope \({ }^{235}\mathrm{UF}_{6}(g)\) from the heavier isotope \({ }^{238}\mathrm{UF}_{6}(g)\). The lighter \({ }^{235}\mathrm{UF}_{6}(g)\) will pass through the barriers faster than the heavier \({ }^{238}\mathrm{UF}_{6}(g)\), thus increasing the concentration of \({ }^{235}\mathrm{U}\) to about \(3\%\).

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, this can be written as: \( \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} \) where: - \(r_1\) and \(r_2\) are the effusion rates of the two gases to be compared, and - \(M_1\) and \(M_2\) are their respective molar masses.

In the gaseous diffusion process, the major purpose is to separate \({ }^{235}\mathrm{UF}_{6}(g)\) from \({ }^{238}\mathrm{UF}_{6}(g)\). We can apply Graham's law of effusion to analyze this process since the effusion rate plays a crucial role in the separation of these isotopes. The effusion rate ratio between \({ }^{235}\mathrm{UF}_{6}(g)\) and \({ }^{238}\mathrm{UF}_{6}(g)\) can be given by: \( \frac{r_{235}}{r_{238}} = \sqrt{\frac{M_{238}}{M_{235}}} \) where: - \(r_{235}\) and \(r_{238}\) are the effusion rates of \({ }^{235}\mathrm{UF}_{6}(g)\) and \({ }^{238}\mathrm{UF}_{6}(g)\), respectively, and - \(M_{235}\) and \(M_{238}\) are their respective molar masses. Since the molar mass of \({ }^{235}\mathrm{U}\) is slightly less than the molar mass of \({ }^{238}\mathrm{U}\), \({ }^{235}\mathrm{UF}_{6}(g)\) will effuse faster than \({ }^{238}\mathrm{UF}_{6}(g)\). After passing through the series of diffusion barriers, the concentration of \({ }^{235}\mathrm{U}\) becomes enriched to reach the desired percentage of around \(3\%\). Therefore, Graham's law of effusion is critical in the uranium enrichment process by exploiting the slight mass difference between the two isotopes for separation and concentration enhancement.