Problem 136 - Chemistry The Science in Context

Question

Radon and helium are both by-products of the radioactive decay of uranium minerals. A fresh sample of carnotite, \(\mathrm{K}_{2}\left(\mathrm{UO}_{2}\right)_{2}\left(\mathrm{VO}_{4}\right)_{2}\) \(3 \mathrm{H}_{2} \mathrm{O},\) 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?

Step-by-Step Solution

Answer

Answer: Helium diffuses more rapidly through the display case, with a relative rate of diffusion of 7.45 times greater than radon.

Step 1: 1. Identify the Gases and their Molar Masses

The gases considered in the problem are helium and radon. First, we need to find their molar masses: Helium (He): 4 g/mol Radon (Rn): 222 g/mol

Step 2: 2. Graham's Law of Diffusion

Graham's law states that the rate of diffusion (r) of a gas is inversely proportional to the square root of its molar mass (M). Mathematically, this is expressed as: \(r \propto \frac{1}{\sqrt{M}}\) For this problem, we will compare the rates of diffusion for helium (r_He) and radon (r_Rn). So, we can write: \(\frac{r_{He}}{r_{Rn}}=\frac{\sqrt{M_{Rn}}}{\sqrt{M_{He}}}\)

Step 3: 3. Calculate the Relative Rates of Diffusion

Let's calculate the relative rates of diffusion for helium and radon by plugging in the molar masses: \(\frac{r_{He}}{r_{Rn}}=\frac{\sqrt{222}}{\sqrt{4}}=\frac{14.9}{2}=7.45\) The ratio 7.45 means that, under the same conditions of temperature and pressure, helium diffuses 7.45 times faster than radon.

Step 4: 4. Determine Which Gas Diffuses More Rapidly

Based on the calculation, helium diffuses more rapidly than radon through the display case. The relative rate of diffusion of helium is 7.45 times greater than that of radon. In conclusion, under fixed conditions of pressure and temperature, helium diffuses more rapidly than radon through the display case.

Key Concepts

Relative Rates of Diffusion

When it comes to gas diffusion, not all gases spread as quickly as others. This is where Graham's Law of Diffusion plays a crucial role. It helps us understand just how fast different gases can spread out into space. Take helium and radon, for instance. Graham's Law tells us that a gas's diffusion rate is inversely related to the square root of its molar mass.

In other words, lighter gases diffuse faster than heavier ones. By using the formula for Graham's Law:

• The diffusion rate (r) is proportional to the inverse square root of molar mass (M).
• The equation is: \( r \propto \frac{1}{\sqrt{M}} \)

When we want to compare two gases, like helium and radon, we use their molar masses in this formula:

• \( \frac{r_{He}}{r_{Rn}}=\frac{\sqrt{M_{Rn}}}{\sqrt{M_{He}}} \)

This equation allows us to calculate how much faster helium diffuses compared to radon, putting a number to the concept of relative diffusion rates.

Molar Mass

To understand why gases diffuse at different rates, we need to dive into the idea of molar mass. Every gas has a molar mass, which is essentially the weight of one mole of that gas. It's typically measured in grams per mole (g/mol).

• Helium, with a molar mass of 4 g/mol, is lighter than radon, which has a molar mass of 222 g/mol.
• Because of their different molar masses, helium and radon will have different diffusion rates.

The principle here is quite simple: a lighter gas has fewer particles per unit volume and moves more easily through available space compared to a heavier gas. Therefore, in gases, lower molar mass results in faster diffusion. When applying Graham's Law, the lighter helium moves rapidly through space and outpaces the heavier radon.

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by radiation. When uranium in minerals like carnotite undergoes radioactive decay, it releases particles that eventually form new elements such as radon and helium.

• Radon is a heavy, inert noble gas, while helium is a lighter, more agile noble gas.
• This decay process creates these gases as by-products over time.

The factors of radioactive decay tie into diffusion because they impact the environment's composition where diffusion occurs. Understanding the natural formation of these gases enriches our understanding of why gases diffuse differently. This knowledge allows us to better apply Graham's Law in practical situations, such as assessing how fast these gases escape from containers or natural reserves.