Argon-39 Decay Calculation Determining Remaining Fraction After 1076 Years

Understanding Radioactive Decay and Half-Life

In the realm of nuclear chemistry, radioactive decay is a fundamental process where unstable atomic nuclei lose energy by emitting radiation. This process transforms one nuclide into another, often involving the release of alpha particles, beta particles, or gamma rays. A crucial concept in understanding radioactive decay is half-life, which is the time required for one-half of the radioactive material to decay. This decay occurs exponentially, meaning that the amount of the substance decreases by half during each half-life period. The half-life is a constant for a given radioactive isotope and is a key characteristic used in various applications, including radioactive dating, medical treatments, and industrial processes. For instance, carbon-14, with a half-life of 5,730 years, is widely used in archaeology to date organic materials. Understanding half-life allows scientists to predict the rate of decay and the amount of radioactive material remaining after a certain period. The process is governed by first-order kinetics, and the decay constant, λ, is inversely related to the half-life (T₁/₂), expressed by the equation λ = ln(2)/T₁/₂. This relationship enables precise calculations of radioactive decay rates and the determination of the age of various materials using methods like radiometric dating. Radioactive isotopes find uses in medical imaging, cancer therapy, and industrial radiography, underscoring the significance of understanding their decay properties. Different isotopes have vastly different half-lives, ranging from fractions of a second to billions of years, making some suitable for short-term applications and others for long-term tracking and dating. In environmental science, understanding half-lives helps in managing radioactive waste and assessing the impact of nuclear accidents.

Problem Statement: Argon-39 Decay

In this specific problem, we delve into the radioactive decay of argon-39 (Ar-39), an isotope with a half-life of 269 years. Argon-39 decays into krypton-39 through a process known as beta decay. Our objective is to determine the fraction of the original amount of argon-39 that will remain after 1,076 years. This is a classic application of the half-life concept, where we need to calculate how many half-life periods have elapsed within the given time frame and then use that information to find the remaining fraction. To solve this, we first need to find out how many half-lives of argon-39 fit into the 1,076-year period. This involves dividing the total time by the half-life of the isotope. Once we know the number of half-lives, we can calculate the remaining fraction by repeatedly halving the initial amount. For each half-life that passes, the amount of argon-39 is reduced by half. This means after one half-life, 1/2 of the original amount remains; after two half-lives, 1/4 remains; after three half-lives, 1/8 remains, and so on. This exponential decay is a characteristic feature of radioactive isotopes, making half-life a convenient measure for quantifying the rate of decay. Understanding this decay process is crucial not only in chemistry and physics but also in fields like geology, where radioactive dating is used to determine the age of rocks and minerals. The predictability of radioactive decay based on half-lives allows for accurate measurements and estimations in various scientific applications.

Step-by-Step Solution

1. Calculate the Number of Half-Lives: The first step in solving this problem is to determine how many half-lives of argon-39 occur within the given time frame of 1,076 years. We can calculate this by dividing the total time by the half-life of argon-39. Mathematically, this is represented as: Number of half-lives = Total time / Half-life = 1,076 years / 269 years. Performing this division, we find that 1,076 divided by 269 equals 4. This means that four half-life periods of argon-39 have passed in 1,076 years. This calculation is crucial because it sets the foundation for understanding how much of the original argon-39 has decayed over this time. Each half-life represents a halving of the substance, so knowing the number of half-lives allows us to determine the fraction of the original substance remaining. This step is fundamental in radioactive decay problems, as it quantifies the extent of decay and provides a direct link to calculating the final amount of the radioactive isotope. The precision of this calculation is important because it directly affects the accuracy of the final result. In practical applications, such as radioactive dating, accurate determination of the number of half-lives is essential for reliable age estimations.
2. Determine the Remaining Fraction: Now that we know there are four half-lives within the 1,076-year period, we can calculate the fraction of the original argon-39 remaining. After each half-life, the amount of the substance is halved. So, after one half-life, 1/2 of the original amount remains. After two half-lives, 1/2 of the remaining 1/2 remains, which is (1/2) * (1/2) = 1/4. Following this pattern, after three half-lives, the remaining fraction is (1/2) * (1/2) * (1/2) = 1/8. And finally, after four half-lives, the fraction remaining is (1/2) * (1/2) * (1/2) * (1/2) = 1/16. Mathematically, this can be represented as (1/2)^n, where n is the number of half-lives. In our case, n = 4, so the remaining fraction is (1/2)^4 = 1/16. This calculation demonstrates the exponential nature of radioactive decay, where the amount of the radioactive substance decreases by half with each passing half-life. Understanding this exponential decay is crucial for predicting the behavior of radioactive isotopes over time. The final fraction, 1/16, indicates that only a small portion of the original argon-39 remains after 1,076 years, highlighting the significant decay that occurs over time for isotopes with relatively short half-lives.

Final Answer

Therefore, after 1,076 years, the fraction of the original amount of argon-39 remaining is 1/16. This corresponds to option A. This calculation illustrates the power of the half-life concept in predicting the decay of radioactive materials. By understanding half-lives, we can accurately determine the amount of a radioactive substance that will remain after a certain period. This is particularly useful in various applications, such as nuclear medicine, where radioactive isotopes are used for diagnostic and therapeutic purposes, and in environmental monitoring, where the decay of radioactive contaminants needs to be assessed. The half-life is a fundamental property of radioactive isotopes, and its application extends across multiple scientific and technological domains. In the context of this problem, the argon-39 isotope, with its 269-year half-life, serves as a clear example of how radioactive decay reduces the quantity of a substance over time. The result, 1/16, emphasizes the significant reduction in the amount of argon-39 after four half-lives, reinforcing the concept of exponential decay. Understanding and applying these principles are essential for anyone working with radioactive materials or studying nuclear processes.