Question

The element krypton has several radioactive isotopes. Below are listed several of these isotopes along with their half-lives. Which of the isotopes is most stable? Which of the isotopes is "hottest"? If we were to begin a half-life experiment with separate \(125-\mu g\) samples of each isotope, approximately how much of each isotope would remain after 24 hours? \(\begin{array}{ll}\text { Isotope } & \text { Half-life } \\ \mathrm{Kr}-73 & 27 \mathrm{s} \\ \mathrm{Kr}-74 & 11.5 \mathrm{min} \\ \mathrm{Kr}-76 & 14.8 \mathrm{h} \\ \mathrm{Kr}-81 & 2.1 \times 10^{5}\mathrm{yr}\end{array}\)

Short answer

The most stable isotope is Kr-81 with a half-life of \(2.1 \times 10^5\) years, and the "hottest" isotope is Kr-73 with a half-life of 27 seconds. After 24 hours, approximately 0 \(\mu g\) of Kr-73, 0.0026 \(\mu g\) of Kr-74, 94.2537 \(\mu g\) of Kr-76, and 125 \(\mu g\) of Kr-81 would remain.

Step-by-step solution

Determine the most stable and "hottest" isotopes

We need to compare the half-lives of the given isotopes to determine which one is the most stable and which one is the "hottest". The most stable isotope is the one with the longest half-life, and the "hottest" isotope is the one with the shortest half-life. For the provided isotopes, we have: - Kr-73 has a half-life of 27 seconds. - Kr-74 has a half-life of 11.5 minutes. - Kr-76 has a half-life of 14.8 hours. - Kr-81 has a half-life of \(2.1 \times 10^5\) years. From the given data, Kr-81 has the longest half-life, making it the most stable isotope, and Kr-73 has the shortest half-life, making it the "hottest" isotope.

Calculate the remaining amounts of the isotopes after 24 hours

We will use the formula for radioactive decay to determine the remaining amounts of each isotope after 24 hours: \(N_t = N_0 e^{-\lambda t}\) where: - \(N_t\) is the remaining amount of the isotope after time \(t\) - \(N_0\) is the initial amount of the isotope (\(125-\mu g\)) - \(e\) is the base of the natural logarithm (approximately 2.718) - \(\lambda\) is the decay constant (calculated as \(\lambda = \frac{\ln{2}}{T_{1/2}}\), where \(T_{1/2}\) is the half-life) - \(t\) is the time that has passed (24 hours = 86400 seconds) We will first convert the half-lives of the isotopes into seconds, then calculate the decay constants and, finally, use the radioactive decay formula to find the remaining amounts of each isotope after 24 hours. Kr-73: \(T_{1/2} = 27 s \Rightarrow \lambda = \frac{\ln{2}}{27}\) Kr-74: \(T_{1/2} = 11.5 \text{ min} = 690 s \Rightarrow \lambda = \frac{\ln{2}}{690}\) Kr-76: \(T_{1/2} = 14.8 \text{ h} = 53280 s \Rightarrow \lambda = \frac{\ln{2}}{53280}\) Kr-81: \(T_{1/2} = 2.1 \times 10^5 \text{ yr} = 6.629 \times 10^9 s \Rightarrow \lambda = \frac{\ln{2}}{6.629 \times 10^9}\) Now, we calculate the remaining amounts of each isotope after 24 hours (\(t = 86400 s\)): Kr-73: \(N_t = 125e^{-\frac{\ln{2}}{27} \times 86400} \approx 0\, \mu g\) Kr-74: \(N_t = 125e^{-\frac{\ln{2}}{690} \times 86400} \approx 0.0026\, \mu g\) Kr-76: \(N_t = 125e^{-\frac{\ln{2}}{53280} \times 86400} \approx 94.2537 \, \mu g\) Kr-81: \(N_t = 125e^{-\frac{\ln{2}}{6.629 \times 10^9} \times 86400} \approx 125 \, \mu g\) After 24 hours, approximately 0 \(\mu g\) of Kr-73, 0.0026 \(\mu g\) of Kr-74, 94.2537 \(\mu g\) of Kr-76, and 125 \(\mu g\) of Kr-81 would remain.

Key concepts

Half-Life

Half-life is an important concept in radioactive decay, determining how quickly a radioactive isotope breaks down. It's the time required for half of the isotope's nuclei to decay. Each half-life reduces the amount by half, repeatedly unfolding this cycle.
This is key when considering stability. Longer half-life indicates a more stable isotope since it decays more slowly.
- Kr-73, with a 27-second half-life, decays rapidly, while Kr-81, with a half-life of 2.1 million years, is highly stable. When experimenting, understanding half-life allows predictions of remaining isotope quantities over time, crucial in fields like medicine, archaeology, and nuclear physics.

Decay Constant

The decay constant, symbolized as \(\lambda\), helps quantify the rate of radioactive decay. It's directly related to the half-life by the formula \(\lambda = \frac{\ln{2}}{T_{1/2}}\). This constant helps calculate the amount of a radioactive substance remaining at any given time.
- For Kr-73 with \(T_{1/2} = 27\) seconds, \(\lambda\) is relatively high, indicating a rapid decay.- Conversely, Kr-81's \(\lambda\) is very small due to its lengthy half-life.The decay constant plays a critical role in predicting how much and how quickly an isotope will decay, which is essential in managing radioactive materials safely and efficiently.

Isotopes

Isotopes are different forms of the same chemical element, having the same number of protons but a different number of neutrons. This variance in neutron numbers gives isotopes distinct atomic masses.
Commonly, isotopes are identified by their mass number, like Kr-73 or Kr-81. They often display different physical properties like stability, which can influence their applications and found uses in various domains.
While some isotopes are stable, those that are unstable undergo radioactive decay, transforming into other elements over time. Understanding isotopes is crucial for grasping basic and applied scientific principles, especially in fields like chemistry and physics.

Radioactive Isotopes

Radioactive isotopes, or radioisotopes, are isotopes with unstable nuclei that release energy by radiation in the form of particles or electromagnetic waves. Over time, these isotopes transform into more stable forms often by emitting alpha, beta, or gamma rays.
In the example, isotopes like Kr-73 and Kr-81 are radioactive and have distinct half-lives, impacting their surrounding environments differently. Radioactive isotopes are used broadly, from medical applications like cancer treatment to archaeological methods, where they help date ancient artifacts.
Understanding their behavior, decay patterns, and interactions allows for safer handling and diverse applications, maximizing benefits while minimizing risks.