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 short, Graham's law of effusion allows natural uranium to be enriched through the gaseous diffusion process by exploiting the different effusion rates between the two uranium hexafluorides, \({ }^{235}\mathrm{UF_6}\) and \({ }^{238}\mathrm{UF_6}\), which arise due to their difference in molar mass. This difference enables the lighter \({ }^{235}\mathrm{UF_6}\) to pass through a porous barrier faster than the heavier \({ }^{238}\mathrm{UF_6}\), leading to an increased concentration of the fissionable \({ }^{235}\mathrm{U}\) isotope in the nuclear fuel after multiple stages of diffusion.

Step-by-step solution

State 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, 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.

Identify the Gaseous Reactants

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.

Determine the ratio of effusion rates

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.

Explain the enrichment process

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.

Key concepts

Uranium enrichment

Uranium enrichment is a crucial process in preparing uranium for use in nuclear reactors. Natural uranium consists mostly of the isotope ^{238}U, which is not suitable for sustaining a nuclear reaction. Only about 0.7% of natural uranium is ^{235}U, the isotope capable of sustaining chain reactions to release energy. However, to be effective as nuclear fuel, the amount of ^{235}U must be increased to approximately 3%.
This is where uranium enrichment comes into play. By enriching uranium, the percentage of the fissionable ^{235}U isotope is increased, making it viable for nuclear reactors and weapons. The enrichment process is typically done using various methods, but one of the most common is the gaseous diffusion process.

• Uses physical properties to separate isotopes.
• Relies on molecular diffusion techniques.

Nuclear fuel

Nuclear fuel is a material that can undergo fission, releasing energy in the form of heat. This heat is used in nuclear reactors to produce steam, driving turbines to generate electricity. Uranium, specifically the isotope ^{235}U, is one of the most common nuclear fuels due to its ability to maintain a fission chain reaction.
Before uranium can be used as a nuclear fuel, it has to be enriched from its natural state. Enriched uranium, with a higher concentration of ^{235}U, is necessary because it provides the necessary conditions for sustained and controlled nuclear reactions.

• ^{235}U enriched to about 3% is typically used in power plants.
• Acts as a core component in nuclear reactors.
• Requires careful handling due to its radioactive nature.

Uranium isotopes

Isotopes of uranium are variations of the same element, each with different numbers of neutrons. The most common isotopes are ^{238}U and ^{235}U. While both have the same number of protons in their nuclei (92 protons), ^{238}U has 146 neutrons, and ^{235}U has 143 neutrons.
The difference in neutron count affects their physical and nuclear properties, with ^{235}U being the only naturally occurring isotope capable of undergoing a sustained fission reaction. No other uranium isotopes are capable of achieving the chain reaction needed for nuclear energy production.

• ^{235}U is rare, comprising only about 0.7% of natural uranium.
• ^{238}U does not support a chain reaction despite being more abundant.

Gas diffusion process

The gas diffusion process is one of the earliest methods used to enrich uranium and relies on the physical principle explained by Graham's Law of Effusion. This method involves converting uranium into uranium hexafluoride gas (UF6), which can then be processed to separate isotopes based on their molecular weight.
The essence of the diffusion process is that the lighter ^{235}UF_6 will effuse through a porous material more quickly than the heavier ^{238}UF_6. Over multiple stages of diffusion, uranium becomes gradually enriched with ^{235}U until the desired concentration for nuclear fuel is reached.

• Initially used in the Manhattan Project.
• Simplicity is outweighed by high energy requirements.
• Gradually replaced by more efficient techniques, like centrifuge enrichment.