Question

Radon \((\mathrm{Rn})\) is the heaviest, and only radioactive, member of Group \(8 \mathrm{~A}(18)\) (noble gases). It is a product of the disintegration of heavier radioactive nuclei found in minute concentrations in many common rocks used for building and construction. In recent years, there has been growing concern about the cancers caused from inhaled residential radon. If \(1.0 \times 10^{15}\) atoms of radium (Ra) produce an average of \(1.373 \times 10^{4}\) atoms of Rn per second, how many liters of Rn, measured at STP, are produced per day by \(1.0 \mathrm{~g}\) of Ra?

Short answer

Approximately 1.17~mL of \(\text{Rn}\) gas is produced per day by \(1.0~g\) of \(\text{Ra}\).

Step-by-step solution

- Determine the number of moles of Ra

First, find the molar mass of radium \text {Radium has an atomic mass of } \text {226 g/mol.}\text {Compute the moles of Ra in 1.0 g of Ra using the formula:}\[ \text{moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \]\[ \text{moles of Ra} = \frac{1.0~\text{g}}{226~\text{g/mol}} \]

- Calculate the number of atoms of Ra in 1.0 g

Now, use Avogadro's number to convert moles of Ra to atoms of Ra:\[ \text{Number of atoms} = \text{moles} \times 6.022 \times 10^{23} \]Substitute the value from Step 1.

- Determine the total number of Rn atoms produced per day

Given that 1.0 \times 10^{15} atoms of Ra produce 1.373 \times 10^{4} atoms of Rn per second, establish a conversion factor:\[ \frac{1.373 \times 10^{4} \text{ atoms Rn}}{1.0 \times 10^{15} \text{ atoms Ra/sec}} \]Using this factor, calculate the number of Rn atoms produced by the total number of Ra atoms found in Step 2 over one day (86400 seconds):\[ \text{Rn atoms/day} = \text{Ra atoms} \times \frac{1.373 \times 10^{4}}{1.0 \times 10^{15}} \times 86400 \]

- Calculate moles of Rn produced per day

Convert the number of Rn atoms produced per day into moles using Avogadro's number:\[ \text{moles of Rn} = \frac{\text{Rn atoms/day}}{6.022 \times 10^{23}} \]Substitute the value obtained in Step 3 to get the moles of Rn.

- Find the volume of Rn gas produced at STP

At STP (Standard Temperature and Pressure), 1 mole of any gas occupies 22.4 liters.Calculate the volume of Rn gas in liters using the moles of Rn from Step 4:\[ \text{Volume at STP} = \text{moles of Rn} \times 22.4~\text{L/mol} \]

Key concepts

radioactive decay

Radioactive decay is a process where unstable atomic nuclei lose energy by emitting radiation. In this problem, radium (Ra) decays into radon (Rn) by releasing particles and energy. This decay follows an exponential law, meaning that the quantity of Ra atoms decreases exponentially over time, producing Rn atoms.
Understanding the decay process helps in calculating how much radon is generated over a specific period. It’s critical for applications in fields like nuclear physics, environmental science, and health physics.
In our specific exercise, we focus on the decay of radium to radon and how many radon atoms are produced per day from a given amount of radium.

noble gases

Noble gases are a group of chemical elements with similar properties. They are odorless, colorless, monatomic gases with very low chemical reactivity. Radon (Rn) is one of these gases and is unique for being radioactive.
Noble gases are found in group 18 of the periodic table and include helium, neon, argon, krypton, xenon, and radon. They are often used in lighting, welding, and space technologies due to their stability. However, radon, due to its radioactivity, poses health risks if inhaled in large quantities.
Understanding the characteristics of radon is important in this exercise, as its noble gas properties influence how we measure and handle it, including the volume it occupies at standard temperature and pressure (STP).

Avogadro's number

Avogadro's number is a key concept in chemistry and physics, representing the number of atoms, ions, or molecules in one mole of a substance, which is approximately \(6.022 \times 10^{23}\). This constant allows us to count particles by weighing them.
In our exercise, we use Avogadro's number to convert the amount of radium (in moles) into the actual number of radium atoms. This is crucial for determining how many radon atoms are produced since the decay rate is given per radium atom.
By knowing the number of radium atoms, we can then find the number of radon atoms produced per day and eventually convert that into the volume of gas produced.

molar mass calculation

Molar mass is the mass of one mole of a substance (in grams). It is calculated by summing the atomic masses of all the atoms in a molecule. For radium, the molar mass is 226 grams per mole.
Calculating the molar mass is the first step in determining how many moles of radium are present in a given sample. Using the formula:

• \(\text{moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}}\)

we can convert the mass of radium into moles. This conversion is foundational in our exercise, as it bridges the mass of the substance to the number of atoms, enabling further calculations involving radioactive decay and gas volume.