# How many effusion stages required to increase the 235U to 4%

• Andrej.N
In summary, the task is to determine the minimum number of effusion stages needed to increase the 235U abundance from 0.7% to 4%. The rate of effusion is proportional to the vrms and follows Graham's Law. The ratio of the vrms for the two isotopes is 1.0043, suggesting an enrichment of 0.43% at each stage. This leads to an expression for the number of stages required, which is approximately 407. Another approach is to use a recursive equation, which yields a similar result.
Andrej.N

## Homework Statement

Uranium has two common isotopes with atomic masses of 238 and 235. One way to separate these isotopes is to combine the uranium with fluorine to make uranium hexafluoride gas, UF6, and then to let it effuse through a small hole. The natural abundance of the 235U isotope is 0.7%. Light‐water reactors, the world’s most common type of nuclear reactor, require a 235U abundance of approximately 4%. How many effusion stages are at least required to increase the 235U abundance to this level? The rate of effusion is proportional to v_rms.

## Homework Equations

I'm not sure how to approach this question since we haven't covered rates of effusion and I'm unfamiliar with the concept of effusions stage. What defines an effusion stage?

## The Attempt at a Solution

Following Schroder's book on Thermodynamics, the rate of effusion I assume reduces to $v_{rms-238U F6}/v_{rms-235UF6}$, which reduces to Graham's Law, which is: $\sqrt {m_{(238)UF6}/m_{(235)UF6)}\approx1.0043$
From here I don't know how to proceed to get the minimal number of stages requiered.

A ratio of the vrms for the two isotopes is 1.0043, which suggests an enrichment of 0.43% at each stage. Since you want to enrich from 0.7% to 4.0%, this suggests that starting with 0.7%, the next stage would have 0.7*1.0043, then 0.7*1.0043^2, and so on.

Care to try now?

Hi! Thanks alot! Then the expression for number of stages ##n## is: $$n=\frac{\log{\frac{4.0}{0.7}}}{\log{1.0043}}=407$$
My original solution to the problem was equivalent: ##y_{n+1}=1.0043y_{n}##. Where ##y_n## is the portion of ##^{235}U## at each iterration. I couldn't solve this analyically. :((

Is this the right way to model the effusion process?

## 1. How many stages are typically required to increase 235U to 4%?

The number of stages required to increase 235U to 4% depends on the initial enrichment level of the uranium. However, on average, it takes about 5-6 stages to reach this enrichment level.

## 2. What is the purpose of increasing the 235U to 4%?

The purpose of increasing the 235U to 4% is to produce enriched uranium that can be used as fuel in nuclear reactors. This level of enrichment is necessary to sustain a chain reaction and generate energy.

## 3. How does the enrichment process work?

The enrichment process involves separating the uranium isotopes, specifically increasing the concentration of 235U through a series of physical and chemical processes. This is typically done using centrifuges, which spin at high speeds to separate the isotopes based on their weight.

## 4. Are there any risks associated with increasing the 235U to 4%?

There are potential risks associated with the enrichment process, as it involves handling radioactive materials. Proper safety measures and regulations are in place to minimize these risks and ensure the safe production of enriched uranium.

## 5. What are the benefits of increasing the 235U to 4%?

The main benefit of increasing the 235U to 4% is the production of enriched uranium, which is used as fuel for nuclear reactors. This allows for the generation of electricity, as well as the production of medical isotopes and other nuclear materials for various industries.

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