Problem 80 - Chemistry Central Science

Question

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

Step-by-Step Solution

Answer

The ratio of diffusion rates for gaseous \({ }^{235}\mathrm{U}\) and \({ }^{238}\mathrm{U}\) is approximately 1.0064, which is slightly higher than the given ratio for \(\mathrm{UF}_{6}\) (1.0043). This indicates that the separation between the uranium isotopes would be slightly more efficient in the hypothetical process involving gaseous uranium atoms compared to the actual process using \(\mathrm{UF}_{6}\).

Step 1: 1. Recall Graham's law of diffusion

Graham's law states that the rate of diffusion of a gas (r) is inversely proportional to the square root of its molar mass (M). Mathematically, it can be represented as: \(r \propto \frac{1}{\sqrt{M}}\)

Step 2: 2. Determine the molar masses of \({ }^{235} \mathrm{U}\) and \({ }^{238} \mathrm{U}\)

Since uranium has two isotopes with mass numbers 235 and 238, we can calculate the molar masses of these isotopes as follows: \({ }^{235} \mathrm{U}\): \(M_{235} = 235\, g/mol\) \({ }^{238} \mathrm{U}\): \(M_{238} = 238\, g/mol\)

Step 3: 3. Calculate the ratio of diffusion rates for \({ }^{235} \mathrm{U}\) and \({ }^{238} \mathrm{U}\)

Using Graham's law, we can find the ratio of the diffusion rates for the uranium isotopes. Since: \(r_1 / r_2 = \sqrt{M_2 / M_1}\) The ratio of diffusion rates for \({ }^{235} \mathrm{U}\) and \({ }^{238} \mathrm{U}\) is: \(r_{235} / r_{238} = \sqrt{M_{238} / M_{235}}\) Substitute the molar masses for each isotope: \(r_{235} / r_{238} = \sqrt{238 / 235} \approx 1.0064\)

Step 4: 4. Compare the calculated ratio to the given \(\mathrm{UF}_{6}\) ratio

The ratio of diffusion rates for \({ }^{235} \mathrm{U}\) and \({ }^{238} \mathrm{U}\) is approximately 1.0064, which is slightly higher than the given ratio for \(\mathrm{UF}_{6}\) in the essay (1.0043). This indicates that the separation between the uranium isotopes would be slightly more efficient in the hypothetical process involving gaseous uranium atoms compared to the actual process using \(\mathrm{UF}_{6}\).