Problem 73 - General Chemistry: Principles and Modern Applications

Question

For medical uses, radon-222 formed in the radioactive decay of radium-226 is allowed to collect over the radium metal. Then, the gas is withdrawn and sealed into a glass vial. Following this, the radium is allowed to disintegrate for another period, when a new sample of radon- 222 can be withdrawn. The procedure can be continued indefinitely. The process is somewhat complicated by the fact that radon-222 itself undergoes radioactive decay to polonium- 218 , and so on. The half-lives of radium-226 and radon-222 are \(1.60 \times 10^{3}\) years and 3.82 days, respectively.(a) Beginning with pure radium- \(226,\) the number of radon-222 atoms present starts at zero, increases for a time, and then falls off again. Explain this behavior. That is, because the half-life of radon-222 is so much shorter than that of radium- \(226,\) why doesn't the radon-222 simply decay as fast as it is produced, without ever building up to a maximum concentration?(b) Write an expression for the rate of change \((d \mathrm{D} / d t)\) in the number of atoms (D) of the radon- 222 daughter in terms of the number of radium- 226 atoms present initially ( \(\mathrm{P}_{0}\) ) and the decay constants of the parent \(\left(\lambda_{\mathrm{p}}\right)\) and daughter \(\left(\lambda_{\mathrm{d}}\right)\) (c) Integration of the expression obtained in part (b) yields the following expression for the number of atoms of the radon-222 daughter (D) present at a time \(t\).$$\mathrm{D}=\frac{\mathrm{P}_{0} \lambda_{\mathrm{p}}\left(\mathrm{e}^{-\lambda_{\mathrm{p}} \times t}-\mathrm{e}^{-\lambda_{\mathrm{d}} \times t}\right)}{\lambda_{\mathrm{d}}-\lambda_{\mathrm{p}}}$$,Starting with \(1.00 \mathrm{g}\) of pure radium- \(226,\) approximately how long will it take for the amount of radon222 to reach its maximum value: one day, one week, one year, one century, or one millennium?

Step-by-Step Solution

Answer

It takes approximately 4.8 days for the amount of radon-222 to reach its maximum value when starting with 1.00 g of pure radium-226.

Step 1: Conceptual Understanding

When radium-226 decays, it forms radon-222. At first, the number of radon-222 atoms increases because radon-222 starts accumulating as radium-226 decays. However, since radon-222 itself is radioactive and decays faster than radium-226 due to its shorter half-life, its concentration starts to decrease once it has reached a saturation point where the rate of its formation equals the rate of its decay.

Step 2: Derive Rate Expression

The rate of change of radon-222 (dD/dt) can be expressed as the difference between its rate of production and its rate of decay. The rate of production is directly proportional to number of radium-226 atoms and their decay constant \(\lambda_{\mathrm{p}}\), and the rate of decay is directly proportional to the number of radon-222 atoms and its decay constant \(\lambda_{\mathrm{d}}\). This leads to the differential equation: \(dD/dt = \lambda_{p}(P - D) - \lambda_{d} D\).

Step 3: Apply Calculus

To find the time when the radon-222 atom concentration is at its maximum, we need to find when the rate of change is zero (dD/dt = 0). Substituting the values for the decay constants \(\lambda_{p}\) and \(\lambda_{d}\) and solving the differential equation, we can obtain the time it takes for radon-222 to reach maximum concentration. Solving this, we get t = (\(ln(\lambda_{\mathrm{d}} / \lambda_{\mathrm{p}}) / (\lambda_{\mathrm{d}} - \lambda_{\mathrm{p}})\))

Step 4: Compute Maximum Time

Substituting the values of \(\lambda_{\mathrm{p}}\) and \(\lambda_{\mathrm{d}}\) as 1.37 x \(-14\) yr\(-1\) and 0.181 day\(-1\) respectively (derived from their half-lifes), we gets that it takes approximately 4.8 days for the amount of radon222 to reach its maximum value

Key Concepts

Radium-226

Radium-226 is a naturally occurring radioactive isotope of radium. It is known for its role in the decay chains of heavy elements.
Radium-226 undergoes radioactive decay to form radon-222, releasing alpha particles in the process.
This decay is part of a longer decay series that eventually leads to the formation of stable lead. In practical applications, the decay of radium-226 has a significant impact on the environment and human health due to the radon-222 it produces.
The half-life of radium-226 is very long, approximately 1600 years, which means it remains active for many generations. As a result, radium-226 provides a continuous supply of radon-222 for extended periods.

Radon-222

Radon-222 is an inert, colorless, and tasteless radioactive gas that is produced as a decay product of radium-226.
Being a noble gas, radon-222 doesn't chemically react with much, but its presence is noteworthy in enclosed spaces due to its radioactive properties.
Radon-222 undergoes radioactive decay relatively quickly compared to radium-226, with a half-life of about 3.82 days.
This short half-life means it decays faster, forming polonium-218, which continues down the decay chain. Despite its fleeting nature, radon-222 is crucial in understanding radioactive decay processes because it demonstrates how a radioactive substance can build up to a maximum concentration before decaying away.

Half-Life

The concept of half-life is a fundamental aspect of radioactive decay. It represents the time required for half of a given sample of a radioactive substance to decay.
Radium-226 has a half-life of 1600 years, while radon-222 has a much shorter half-life of 3.82 days.
Why does half-life matter? It helps us predict how long a radioactive element will be active and how it will behave over time.

• In the case of radium-226 and radon-222, understanding their half-lives allows us to gauge their effects over different time scales.
• Longer half-lives imply prolonged radioactivity, while shorter half-lives mean rapid decay and transformation to other elements.

This knowledge is crucial when considering the implications of radioactive materials in various applications, from medical uses to safety protocols.

Decay Constant

The decay constant is a key parameter in the study of radioactive decay, indicating the probability per unit time of a nucleus decaying.
It is denoted by the Greek letter lambda (\(\lambda\)) and is related to the half-life of an isotope.

• For radium-226, the decay constant \(\lambda_{p}\) is around 1.37 x 10\({-14}\) per year.
• For radon-222, \(\lambda_{d}\) is approximately 0.181 per day.

Understanding the decay constants of radium-226 and radon-222 helps us understand their decay rates and behavior.
The decay constant provides insight into how quickly an isotope will decay relative to its half-life.
It’s important in fields like nuclear physics and environmental science, where predicting the behavior of radioactive materials is crucial.