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

[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.

Thanks in advance for your help.

 

[SteamKing]

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?

 

[Andrej.N]

Hi! Thanks a lot! 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?