Question

Decay Products of Uranium Minerals Radon and helium are both by-products of the radioactive decay of uranium minerals. A fresh sample of carnotite, \(\mathrm{K}_{2}\left(\mathrm{UO}_{2}\right)_{2}\left(\mathrm{VO}_{4}\right)_{2}\) \(\cdot 3 \mathrm{H}_{2} \mathrm{O},\) is put on display in a museum. Calculate the relative rates of diffusion of helium and radon under fixed conditions of pressure and temperature. Which gas diffuses more rapidly through the display case?

Short answer

Short Answer: Under fixed conditions of pressure and temperature, helium diffuses more rapidly compared to radon in the display case, as the ratio of their diffusion rates is greater than 1 (\(Rate_{He} / Rate_{Rn} = \sqrt{55.5}\)).

Step-by-step solution

Recall Graham's Law of Diffusion formula

Graham's law of diffusion states that the rate of diffusion of a gas is inversely proportional to the square root of its molar mass. Mathematically, it is written as: \(Rate_{1} / Rate_{2} = \sqrt{M_{2} / M_{1}}\) Where \(Rate_{1}\) and \(Rate_{2}\) are the diffusion rates of the gases, and \(M_{1}\) and \(M_{2}\) are their molar masses respectively.

Find the Molar Masses of Helium and Radon

We need to find the molar masses of helium and radon in order to apply Graham's law. For helium: \(\mathrm{He}\), molecular weight = 4 g/mol. For radon: \(\mathrm{Rn}\), molecular weight = 222 g/mol.

Calculate the Relative Rates of Diffusion

Now, we will use Graham's law to find the ratio of their diffusion rates. \(Rate_{He} / Rate_{Rn} = \sqrt{M_{Rn} / M_{He}}\) \(Rate_{He} / Rate_{Rn} = \sqrt{(222\ g/mol) / (4\ g/mol)}\) \(Rate_{He} / Rate_{Rn} = \sqrt{55.5}\)

Determine which gas diffuses more rapidly

Since the ratio \(Rate_{He} / Rate_{Rn}\) is greater than 1, it means that helium diffuses more rapidly than radon through the display case. Therefore, under the fixed conditions of pressure and temperature, helium diffuses more rapidly compared to radon in the display case.

Key concepts

Rates of Diffusion

The rates of diffusion are an essential concept in understanding how gases spread through different mediums. You can visualize diffusion like the way a drop of ink spreads in water. The rate at which this spreading occurs depends on several factors, one of them being Graham's Law of Diffusion.
Graham's Law states that the rate of diffusion of a gas is inversely proportional to the square root of its molar mass. This relationship allows us to compare how fast two gases will diffuse under the same conditions. By knowing the molar masses of these gases, we use the formula:

• \( Rate_{1} / Rate_{2} = \sqrt{M_{2} / M_{1}} \)

Where \( Rate_{1} \) and \( Rate_{2} \) are the diffusion rates and \( M_{1} \) and \( M_{2} \) are the respective molar masses. A higher rate indicates faster diffusion, aiding in predicting how quickly a gas will penetrate through materials, which is crucial for chemical reactions and industrial applications.

Molar Mass

Molar mass is the weight of one mole of a substance, measured in grams per mole (g/mol). It's like a heavier element is made up of heavier 'building blocks,' which affects how it moves. In the context of diffusion, gases with lower molar mass diffuse more quickly because lighter particles move faster.
Helium and radon in the given problem have very different molar masses. Helium has a molar mass of 4 g/mol, making it very light. On the other hand, radon has a molar mass of 222 g/mol. This significant difference directly impacts their rates of diffusion.
Using the concept of molar mass, students can determine which gas will diffuse faster by applying Graham’s Law. Simply, the lighter the gas (lower the molar mass), the faster it will diffuse through a medium.

Uranium Decay Products

Understanding uranium decay products brings us into the world of radioactivity and isotopes. Uranium, a heavy metal, undergoes radioactive decay, producing several by-products including radon and helium.
Radon is a noble gas, heavier than many other gases due to its high molar mass (222 g/mol). It is a significant by-product because it is radioactive, posing health risks if not properly managed. Helium, in this context, is lighter and non-radioactive. It's produced when alpha particles (which are helium nuclei) are emitted during radioactive decay.
The diffusion properties of these decay products are significant for safe storage and management in environments like museums. Because helium diffuses more quickly than radon, understanding these properties ensures effective strategies to monitor and manage these gases, protecting both artifacts and people from potential harm.