Uranium has two naturally occurring isotopes. ²³⁸U has a natural abundance of 99.3% and ²³⁵U has an abundance of 0.7%. It is the rarer ²³⁵U that is needed for nuclear reactors. The isotopes are separated by forming uranium hexafluoride, UF₆, which is a gas, then allowing it to diffuse through a series of porous membranes. ²³⁵UF₆ has a slightly larger rms speed than ²³⁸UF₆ and diffuses slightly faster. Many repetitions of this procedure gradually separate the two isotopes. What is the ratio of the rms speed of ²³⁵UF₆ to that of ²³⁸UF₆?

Verified step by step guidance

1. Identify the molecular masses of the isotopes ²³⁵UF₆ and ²³⁸UF₆. The molecular mass of UF₆ can be calculated by adding the atomic mass of uranium (either 235 or 238) to six times the atomic mass of fluorine (approximately 19).
2. Recall the formula for root mean square (rms) speed, which is given by \(v_{rms} = \sqrt{\frac{3kT}{m}}\), where \(k\) is the Boltzmann constant, \(T\) is the temperature, and \(m\) is the mass of the gas molecule.
3. Note that the rms speed of a gas is inversely proportional to the square root of its molecular mass. This means that the lighter molecule (²³⁵UF₆) will have a higher rms speed compared to the heavier molecule (²³⁸UF₆).
4. Set up the ratio of the rms speeds of ²³⁵UF₆ to ²³⁸UF₆. This can be expressed as \(\frac{v_{rms, 235}}{v_{rms, 238}} = \sqrt{\frac{m_{238}}{m_{235}}}\), where \(m_{235}\) and \(m_{238}\) are the molecular masses of ²³⁵UF₆ and ²³⁸UF₆, respectively.
5. Calculate the ratio using the molecular masses obtained in step 1. This will give you the ratio of the rms speeds of the two isotopes.

Key Concepts

Root Mean Square Speed

The root mean square (rms) speed is a measure of the average speed of particles in a gas, calculated as the square root of the average of the squares of the speeds of the particles. It is directly related to the temperature and molar mass of the gas, with lighter gases having higher rms speeds. This concept is crucial for understanding how different isotopes of uranium behave when diffusing through membranes.

Diffusion

Diffusion is the process by which particles spread from areas of high concentration to areas of low concentration. In the context of gases, lighter molecules diffuse faster than heavier ones due to their higher average speeds. This principle is essential for separating isotopes, as it explains why ²³⁵UF₆, being lighter, diffuses more quickly than ²³⁸UF₆.

Isotopes and Their Properties

Isotopes are variants of a chemical element that have the same number of protons but different numbers of neutrons, resulting in different atomic masses. The properties of isotopes, such as their mass and stability, influence their behavior in physical processes like diffusion. Understanding the differences between ²³⁵U and ²³⁸U is key to solving the problem of their separation in nuclear applications.