Problem 31 - University Physics with Modern Physics

Question

Gaseous Diffusion of Uranium. (a) A process called gaseous diffusion is often used to separate isotopes of uranium that is, atoms of the elements that have different masses, such as 235 \(\mathrm{U}\) and 238 \(\mathrm{U} .\) The only gaseous compound of uranium at ordinary temperatures is uranium hexafluoride, UF \(_{6}\) . Speculate on how 235 \(\mathrm{UF}_{6}\) and \(^{238} \mathrm{UF}_{6}\) molecules might be separated by diffusion. (b) The molar masses for \(^{235} \mathrm{UF}_{6}\) and 238 \(\mathrm{UF}_{6}\) molecules are 0.349 \(\mathrm{kg} / \mathrm{mol}\) and \(0.352 \mathrm{kg} / \mathrm{mol},\) respectively. If uranium hexafluoride acts as an ideal gas, what is the ratio of the root-meansquare speed of \(^{235} \mathrm{UF}_{6}\) molecules to that of \(^{238} \mathrm{UF}_{6}\) molecules if the temperature is uniform?

Step-by-Step Solution

Answer

235UF₆ diffuses faster than 238UF₆, with a speed ratio of about 1.004.

Step 1: Understand the Concept of Gaseous Diffusion

Gaseous diffusion relies on the principle that lighter gas molecules will diffuse faster than heavier ones. In this context, this means that the lighter 235UF₆ molecules will diffuse faster than the heavier 238UF₆ molecules due to their respective masses.

Step 2: Consider the Molecular Masses

The molar masses are given as 0.349 kg/mol for 235UF₆ and 0.352 kg/mol for 238UF₆. The small difference in mass will affect how fast each type of molecule diffuses.

Step 3: Apply the Formula for Root-Mean-Square Speed

The root-mean-square (RMS) speed of gas molecules is given by the formula: \[ v_{rms} = \sqrt{\frac{3kT}{m}} \]where \( v_{rms} \) is the root-mean-square speed, \( k \) is the Boltzmann constant, \( T \) is the temperature, and \( m \) is the molar mass of the gas molecules in kg/mol. Since the temperature \( T \) is uniform, we can focus on the relationship between \( v_{rms} \) and \( m \).

Step 4: Calculate the Ratio of RMS Speeds

Using the relationship from Step 3: \[\frac{v_{rms, 235UF_6}}{v_{rms, 238UF_6}} = \sqrt{\frac{m_{238UF_6}}{m_{235UF_6}}}\]Substitute the molar masses: \[\frac{v_{rms, 235UF_6}}{v_{rms, 238UF_6}} = \sqrt{\frac{0.352}{0.349}}\]Calculate this expression to find the ratio.

Step 5: Perform the Calculation

Calculate the numerical value:\[\frac{v_{rms, 235UF_6}}{v_{rms, 238UF_6}} = \sqrt{\frac{0.352}{0.349}} \approx 1.004\]Therefore, the 235UF₆ molecules have an RMS speed approximately 1.004 times that of the 238UF₆ molecules.

Key Concepts

Uranium Isotopes

Isotopes are variants of a particular chemical element that differ in neutron number. They have the same number of protons (hence belong to the same element) but differ in the number of neutrons, and consequently, their atomic masses. In the case of uranium, the two important isotopes are uranium-235 ( 235 U) and uranium-238 ( 238 U).
Both isotopes are crucial, especially in nuclear energy and weaponry, due to their differing nuclear properties. In industry, separating uranium isotopes is vital for enriching uranium, often for use in nuclear reactors or weapons. Gaseous diffusion is one method employed, relying on the slight mass difference between isotopes.
Uranium hexafluoride (UF$_6$) is used in this process because it is the only gaseous compound of uranium at usable temperatures. Diffusion exploits the principle that lighter molecules ( 235 UF$_6$) diffuse faster than heavier molecules ( 238 UF$_6$). This difference allows for the physical separation of isotopes.

Root-Mean-Square Speed

The root-mean-square (RMS) speed is a statistical measure used to describe the speed of particles in a gas. It is derived from kinetic theory and provides an average measure of molecular velocity, taking into account the kinetic energy distribution.
The RMS speed, \(v_{rms}\), for gas molecules is calculated using the formula:\[v_{rms} = \sqrt{\frac{3kT}{m}}\]where \(k\) is the Boltzmann constant, \(T\) is the temperature in Kelvin, and \(m\) is the molar mass of the gas molecules in kilograms per mole.In the context of uranium isotope separation, because temperature \(T\) is uniform, \(v_{rms}\) is determined by the molar mass. Thus, lighter molecules (235UF\(_6\)) will have a slightly higher RMS speed compared to heavier molecules (238UF\(_6\)).
It is this principle that allows gaseous diffusion to be an effective method for isotope separation.

Molecular Mass

Molecular mass, synonymous with molar mass in this context, is a critical factor in gaseous diffusion. It refers to the mass of a given molecule, often expressed in grams per mole or kilograms per mole.
The molecular mass determines how quickly a molecule can move under set conditions, like temperature and pressure, due to inertia. For uranium hexafluoride, $^{235}$ UF$_6$ has a marginally lower molar mass (0.349 kg/mol) compared to $^{238}$ UF$_6$ (0.352 kg/mol). This small difference plays a significant role in diffusion processes. Since lighter molecules have less inertia, they diffuse quicker. Therefore, even a minor variation in molecular mass can effectively separate isotopes through gaseous diffusion, as observed in the operation scale of uranium enrichment. Understanding and calculating molecular masses are crucial steps in predicting and exploiting gas behavior—which is essential for practical applications like isotope separation.