Question

The \({ }^{235} \mathrm{U}\) isotope undergoes fission when bombarded with neutrons. However, its natural abundance is only 0.72 percent. To separate it from the more abundant \({ }^{238} \mathrm{U}\) isotope, uranium is first converted to \(\mathrm{UF}_{6},\) which is easily vaporized above room temperature. The mixture of the \({ }^{235} \mathrm{UF}_{6}\) and \({ }^{238} \mathrm{UF}_{6}\) gases is then subjected to many stages of effusion. Calculate how much faster \({ }^{235} \mathrm{UF}_{6}\) effuses than \({ }^{238} \mathrm{UF}_{6}\)

Short answer

\(^{235}\mathrm{UF}_{6}\) effuses approximately 1.0043 times faster than \(^{238}\mathrm{UF}_{6}\).

Step-by-step solution

Understand the Effusion Concept

Effusion refers to the process by which gas molecules escape through a small hole into a vacuum. According to Graham's law of effusion, the rate of effusion of a gas is inversely proportional to the square root of its molar mass. This means lighter gases effuse faster than heavier gases.

Apply Graham's Law of Effusion

Graham's law states that the rate of effusion of two gases is given by: \(\frac{\text{Rate of effusion of } ^{235}\mathrm{UF}_{6}}{\text{Rate of effusion of } ^{238}\mathrm{UF}_{6}} = \sqrt{\frac{M_{^{238}\mathrm{UF}_{6}}}{M_{^{235}\mathrm{UF}_{6}}}} \)where \(M\) is the molar mass of each respective molecule.

Calculate the Molar Masses

- The molar mass of \(^{235}\mathrm{UF}_{6}\) is \(235 + 6(19.00) + (9.00) \approx 349.0 \text{ g/mol}\).- The molar mass of \(^{238}\mathrm{UF}_{6}\) is \(238 + 6(19.00) + (9.00) \approx 352.0 \text{ g/mol}\).

Calculate the Effusion Rate Ratio

Substitute the molar masses into Graham's Law formula:\[\frac{\text{Rate of effusion of } ^{235}\mathrm{UF}_{6}}{\text{Rate of effusion of } ^{238}\mathrm{UF}_{6}} = \sqrt{\frac{352.0}{349.0}} \approx 1.0043 \]

Interpret the Result

The calculated ratio \(1.0043\) indicates that \(^{235}\mathrm{UF}_{6}\) effuses approximately 1.0043 times faster than \(^{238}\mathrm{UF}_{6}\). This means that \(^{235}\mathrm{UF}_{6}\) effuses slightly faster than \(^{238}\mathrm{UF}_{6}\) during the separation process.

Key concepts

uranium isotopes

Uranium isotopes are different forms of the element uranium, varying in the number of neutrons within their nuclei. The most commonly discussed isotopes are

• Uranium-235
• Uranium-238

Uranium-235 ( ²³⁵ U) is notable for its ability to sustain a chain reaction, which is crucial for both nuclear reactors and weapons. It is less abundant in nature, comprising roughly 0.72% of natural uranium samples. Meanwhile, Uranium-238 ( ²³⁸ U) makes up about 99.28% of natural uranium.

The separation of these isotopes is essential in nuclear chemistry, especially for enriching ²³⁵ U which is needed in nuclear applications. Chemical and physical processes are employed to isolate ²³⁵ U from ²³⁸ U, given their minute differences in mass. These processes often involve converting uranium into a gaseous form like Uranium Hexafluoride ( UF₆ ), enabling methods like effusion to efficiently separate isotopes.

Graham's law of effusion

Graham's Law of Effusion is a fundamental principle in gas chemistry, first articulated by scientist Thomas Graham. It predicts how gas molecules escape through small openings. The law states that the rate at which a gas effuses is entirely dependent on its molar mass.

According to Graham's Law, smaller or lighter gas molecules effuse more rapidly than larger or heavier ones. This is depicted mathematically as:

• \(r_1 \propto \frac{1}{\sqrt{M_1}}\)
• \(r_2 \propto \frac{1}{\sqrt{M_2}}\)

Where \(r\) is the effusion rate and \(M\) represents the molar mass of the gases. By applying this law, you can compare the effusion rates of two different gases with the formula:\[\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}\]This formula is invaluable for separating isotopes, as in the case of ²³⁵UF₆ and ²³⁸UF₆.

molar mass calculation

Calculating the molar mass of a compound involves adding up the atomic masses of all atoms in a molecule. The process is straightforward and essential for understanding how substances behave in reactions and separations.

For uranium hexafluoride ( UF₆ ), let's calculate the molar mass of two isotopic variants:

• First, for ²³⁵ UF₆:
• ²³⁵ U = 235 g/mol
• F = 19 g/mol, but since there are six fluorine atoms, it totals 6 × 19 = 114 g/mol
• Adding these, the molar mass is 235 + 114 = 349 g/mol.

Similarly, with ²³⁸ UF₆, the calculation is:

• ²³⁸ U = 238 g/mol
• Therefore, 238 + 114 = 352 g/mol

These calculations are pivotal, as slight differences in molar mass influence the behavior in effusion processes, as illustrated by Graham's Law.

effusion process

Effusion is an important process involving the movement of gas molecules through a small opening. During this process, gases escape into a vacuum. The rate at which effusion occurs is determined by the molecular weight of the gas.

Lighter gases effuse faster than heavier ones. This concept is key in processes like isotope separation, where uranium isotopes are refined using differences in effusion rates.

In nuclear chemistry, the ability to separate isotopes such as ²³⁵ UF₆ from ²³⁸ UF₆ is vital for enriching uranium necessary for nuclear power and weapons. By understanding effusion, scientists can efficiently utilize physical properties, like differences in isotopic mass, to achieve separation.