Question

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.

Short answer

In the uranium enrichment process, natural uranium reacts with fluorine to form a mixture of \({ }^{238}\mathrm{UF}_{6}(g)\) and \({ }^{235}\mathrm{UF}_{6}(g)\). This 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 effusion rate ratio between these two isotopes can be given by \( \frac{r_{235}}{r_{238}} = \sqrt{\frac{M_{238}}{M_{235}}}\), according to Graham's law of effusion. Since the molar mass of \({ }^{235}\mathrm{U}\) is slightly less than that of \({ }^{238}\mathrm{U}\), \({ }^{235}\mathrm{UF}_{6}(g)\) will effuse faster. After passing through the diffusion barriers, the concentration of \({ }^{235}\mathrm{U}\) becomes enriched to reach the desired percentage of around \(3\%\). Graham's law of effusion is critical in the uranium enrichment process as it exploits the slight mass difference between the two isotopes for separation and concentration enhancement.

Step-by-step solution

Understanding 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 \(3\%\).

Understanding Graham's Law of Effusion

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.

Applying Graham's Law of Effusion to the Gaseous Diffusion Process

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.

Key concepts

Uranium Enrichment

Uranium enrichment is a critical process where the concentration of the fissionable isotope, \(^{235}\text{U}\), in natural uranium is increased. Most natural uranium consists of \(^{238}\text{U}\), which is non-fissionable and comprises about 99.3% of natural uranium. The enrichment process increases the percentage of \(^{235}\text{U}\) from its natural 0.7% to approximately 3%, making it suitable for use in nuclear reactors as fuel.

This enrichment is essential because only \(^{235}\text{U}\) enables the nuclear chain reactions necessary for generating energy. The enrichment process often involves a phase known as gaseous diffusion, leveraging physical principles to enhance the concentration of desirable isotopes through the exploitation of their differing masses. The ultimate goal is to produce nuclear fuel that is efficient and capable of sustaining prolonged nuclear reactions within a reactor.

Isotope Separation

Isotope separation is the method by which isotopes of an element are divided to attain a particular composition which might be more beneficial for industrial or energy purposes. In the context of uranium, this involves separating the lighter \(^{235}\text{U}\) isotope from the heavier \(^{238}\text{U}\) isotope. This separation exploits their slight difference in physical properties, such as mass, despite having identical chemical attributes.

The key to successful isotope separation lies in Graham's law of effusion. By converting uranium into uranium hexafluoride gas, engineers can utilize the mass difference between isotopes during a diffusion process. Slight differences in how isotopes effuse through membranes or barriers lead to effective concentration enhancement of \(^{235}\text{U}\). This precision in separation is vital, ensuring that nuclear reactors receive fuel that is enriched enough to meet energy production demands.

Uranium Hexafluoride

Uranium hexafluoride, symbolized as UF₆, is a chemical compound utilized in the processing of uranium for enrichment. This compound is central to the process because it can transition easily between solid, liquid, and gaseous states. Such transition versatility makes UF₆ an excellent candidate for methods like gaseous diffusion.

In the enrichment process, natural uranium is transformed into UF₆ by combining it with fluorine gas. This transformation allows uranium to be handled as a gas, which is necessary for subsequent separation steps. Since uranium isotopes are barely distinguishable chemically, it is the use of this specific gaseous form that permits their separation based on slight differences in physical characteristics, primarily mass, through diffusion processes.

Molar Mass

The concept of molar mass is crucial in understanding why uranium isotopes can be separated. Molar mass refers to the mass of one mole of a given substance and is commonly expressed in grams per mole (g/mol). For uranium hexafluoride isotopes, the molar masses are slightly different: \(^{235}\text{UF}_6\) has a slightly lower molar mass than \(^{238}\text{UF}_6\).

This small difference in mass is what allows isotopes to separate during the enrichment process. According to Graham's law of effusion, lighter molecules effuse through a barrier at a faster rate than heavier ones. Consequently, the slightly lighter \(^{235}\text{UF}_6\) molecules pass through barriers more readily, gradually increasing its concentration relative to \(^{238}\text{UF}_6\). Understanding molar mass differences is fundamental to grasping the separation process's mechanics and principles in uranium enrichment.

Nuclear Fuel Processing

Nuclear fuel processing involves various stages aimed at making uranium capable of sustaining nuclear fission in power plants. The primary step is enrichment, where the proportion of \(^{235}\text{U}\) is enhanced to create fuel that is viable for a sustained nuclear reaction.

After enrichment, converted uranium is often transformed into fuel rods, which are used in nuclear reactors. This process must ensure not only the adequate concentration of isotopes but also their stability and safety over time in reaction environments. Finally, a rigorous understanding of processing methods ensures that nuclear fuel is produced efficiently and safely, meeting the stringent demands of energy production and regulatory standards. This comprehensive approach encompasses refining processes, ensuring the enriched uranium maintains the necessary integrity for critical energy conversion and lasting function.