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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, \(\mathrm{U}(\mathrm{g})\). 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 ${ }^{235} \mathrm{U}$ and ${ }^{238} \mathrm{U}$ is approximately \(1.0064\). To compare it to the ratio for \(\mathrm{UF}_{6}\) given in the essay, we can calculate the percentage difference or simply compare the numerical values.

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, the relationship can be written as: \[ \frac{Rate_1}{Rate_2} = \sqrt{\frac{M_2}{M_1}} \] where Rate₁ and Rate₂ are the effusion rates of two gases, and M₁ and M₂ are their respective molar masses.
02

Calculate the ratio of effusion rates for \({ }^{235} \mathrm{U}\) and \({ }^{238} \mathrm{U}\)

Using Graham's law of effusion, we can calculate the ratio of effusion rates for \({ }^{235} \mathrm{U}\) and \({ }^{238} \mathrm{U}\). We know the molar masses of these isotopes: M₁ = 235 u, M₂ = 238 u (where u is the atomic mass unit). Plug the values into the equation: \[ \frac{Rate_{235}}{Rate_{238}} = \sqrt{\frac{238}{235}} \] Now, calculate the ratio: \[ \frac{Rate_{235}}{Rate_{238}} = \sqrt{\frac{238}{235}} \approx 1.0064 \]
03

Compare the ratio to the ratio for \(\mathrm{UF}_{6}\)

The question states that we need to compare the calculated ratio of effusion rates for \({ }^{235} \mathrm{U}\) and \({ }^{238} \mathrm{U}\) to the ratio for \(\mathrm{UF}_{6}\) given in the essay. We have calculated the ratio to be approximately 1.0064 for the uranium isotopes. Now, let's assume that the ratio for \(\mathrm{UF}_{6}\) given in the essay is R. To compare, we can either find the percentage difference between the two ratios or simply compare the numerical values. Depending on the context and the value of R, we will be able to draw conclusions on how the ratio for uranium isotopes differs from the ratio for \(\mathrm{UF}_{6}\).

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