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Chapter 10: Problem 86

As discussed in the "Chemistry Put to Work" box in Section 10.8 , enriched uranium can be produced by effusion of gaseous \(\mathrm{UF}_{6}\) across a porous membrane. Suppose a process were developed to allow effusion of gaseous uranium atoms, U(g). Calculate the ratio of effusion rates for \(^{235} \mathrm{U}\) and \(^{238} \mathrm{U},\) and compare it to the ratio for \(\mathrm{UF}_{6}\) given in the essay.

Short Answer

Expert verified
The ratio of effusion rates for gaseous uranium isotopes \(^{235}\mathrm{U}\) and \(^{238}\mathrm{U}\) is approximately 1.0063, which is slightly higher but still quite comparable to the given ratio of 1.0043 for \(\mathrm{UF_{6}}(g)\) containing these isotopes. This suggests that the effusion process would be more efficient for separating uranium isotopes in the gaseous form, although the difference is not significant.

Step by step solution

01

Recall 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 is expressed as: $$\frac{Rate_A}{Rate_B} = \sqrt{\frac{M_B}{M_A}}$$ where Rate_A and Rate_B are the effusion rates of gas A and B respectively, and M_A and M_B are their molar masses.
02

Identify the given information

We are given the two isotopes of uranium: 1. \(^{235}\mathrm{U}\) 2. \(^{238}\mathrm{U}\) We need to find the ratio of effusion rates for these isotopes.
03

Apply Graham's law of effusion

Applying the formula from Step 1: $$\frac{Rate_{^{235}\mathrm{U}}}{Rate_{^{238}\mathrm{U}}} = \sqrt{\frac{M_{^{238}\mathrm{U}}}{M_{^{235}\mathrm{U}}}$$
04

Calculate the effusion rates ratio

Now we can plug in the molar masses of the two uranium isotopes: $$\frac{Rate_{^{235}\mathrm{U}}}{Rate_{^{238}\mathrm{U}}} = \sqrt{\frac{238 \mathrm{g/mol}}{235 \mathrm{g/mol}}}$$ Solving for the ratio: $$\frac{Rate_{^{235}\mathrm{U}}}{Rate_{^{238}\mathrm{U}}} = \sqrt{\frac{238}{235}} = 1.0063$$ The ratio of effusion rates for \(^{235}\mathrm{U}\) and \(^{238}\mathrm{U}\) is approximately 1.0063.
05

Compare the ratio for uranium isotopes to 𝑈𝐹6 ratios

The ratio of effusion rates for \(\mathrm{UF_{6}}(g)\) containing \(^{235}\mathrm{U}\) and \(^{238}\mathrm{U}\) is given in the essay as 1.0043. The ratio for gaseous uranium isotopes, 1.0063, is slightly higher but still quite comparable to the ratio for \(\mathrm{UF_{6}}(g)\). This indicates that the effusion process would be more efficient for separating the uranium isotopes in the gaseous form, but the difference is not significant.

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