Question

41\. 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 \(125-\mu \mathrm{g}\) 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 due to its longest half-life of \(2.1 * 10^5\) years, while the "hottest" isotope is Kr-73 with the shortest half-life of 27 seconds. After 24 hours, approximately 0 μg of Kr-73 and Kr-74, 7.3 μg of Kr-76, and 125 μg of Kr-81 would be left.

Step-by-step solution

Rank isotopes based on stability

The stability of an isotope is directly related to its half-life. Longer half-lives indicate that the isotope is less likely to decay, thus increasing its stability. Comparing the half-lives, we can determine that Kr-81 has the longest half-life (2.1 * 10^5 years), making it the most stable. The shortest half-life belongs to Kr-73 (27 seconds), making it the least stable.

Determine the "hottest" isotope

In terms of radioactivity, a "hotter" isotope can be interpreted as an isotope that has a higher rate of decay. In this context, isotopes with shorter half-lives are considered hotter. Based on this interpretation, the "hottest" isotope is Kr-73, as it has the shortest half-life (27 seconds).

Calculate the amount of each isotope remaining after 24 hours

To find the amount of each isotope remaining after 24 hours, we will use the half-life formula: \(FinalAmount = StartingAmount * (0.5) ^ {\frac{ElapsedTime}{Half-Life}}\) Each isotope initially has a 125 μg sample. We will now find the amount left for each after 24 hours. For Kr-73: \(RemainingAmount_{73} = 125 * (0.5)^{\frac{24 * 3600}{27}} ≈ 0\) Approximately 0 μg of Kr-73 is left after 24 hours. For Kr-74: \(RemainingAmount_{74} = 125 * (0.5)^{\frac{24 * 60}{11.5}} ≈ 0\) Approximately 0 μg of Kr-74 is left after 24 hours. For Kr-76: \(RemainingAmount_{76} = 125 * (0.5)^{\frac{24}{14.8}} ≈ 7.3\) Approximately 7.3 μg of Kr-76 is left after 24 hours. For Kr-81: \(RemainingAmount_{81} = 125 * (0.5)^{\frac{24}{2.1 * 10^5 * 365.25 * 24}} ≈ 125\) Approximately 125 μg of Kr-81 is left after 24 hours. Therefore, the remaining amounts after 24 hours are approximately 0 μg for Kr-73, 0 μg for Kr-74, 7.3 μg for Kr-76, and 125 μg for Kr-81.

Key concepts

half-life

The concept of half-life is a fundamental principle in the study of radioactive isotopes. It describes the time needed for half of a sample of a radioactive substance to decay into a more stable form. When a radioactive isotope undergoes decay, it loses particles and emits radiation, steadily altering its atomic structure.

Half-life is particularly useful in radioactivity measurements because it is constant for any given isotope. This means that no matter how much of a substance you have, the time it takes for half of it to decay will always be the same. For example, if a substance has a half-life of 2 hours, starting with 100 grams, after 2 hours, you will have 50 grams left. After another 2 hours, you’ll have 25 grams, and so forth.

This property aids scientists in predicting the quantity of an isotope remaining after a specific period, which can have applications in dating archaeological finds, medical diagnostics, and nuclear power management.

isotope stability

Isotope stability is closely tied to an element's half-life. Essentially, isotopes with longer half-lives are more stable. Stability, in this context, relates to how resistant an isotope is to decaying. A stable isotope will have a minimal amount of radioactivity or decay very slowly.

There are several factors that contribute to isotope stability including:

• Number of protons and neutrons: A balanced proportion often leads to greater stability.
• Nuclear forces: Strong nuclear forces within the nucleus also contribute to stability.
• Energy states: Low energy levels typically correspond with stable isotopes.

For instance, krypton-81 with a half-life of 2.1 x 10^5 years is considered extremely stable because it decays at a very slow rate, making it the most stable of the krypton isotopes mentioned.

radioactivity decay rate

Radioactivity decay rate refers to how quickly a radioactive isotope transforms into a different element through the emission of radiation. This rate is what makes an isotope "hot" or "cold" in terms of radioactivity.

In simple terms, isotopes with shorter half-lives decay more rapidly, making them more radioactive or "hotter" because their atoms are transitioning at a higher frequency. This can be contrasted to isotopes with longer half-lives that decay over extended periods, meaning they are less active or "colder."

If we consider krypton isotopes, Kr-73 with a half-life of 27 seconds has the highest decay rate, thus it's the hottest isotope of those listed. Meanwhile, Kr-81, with its lengthy half-life, has a slower, less frequent decay process. Understanding decay rates is critical for applications in fields like nuclear medicine, environmental monitoring, and radiometric dating.