Given That The Half-life Of Iron-52 Is Approximately 8.3 Hours, How Much Of A 13-gram Sample Of Iron-52 Would Remain After 2 Hours? Round The Answer To Three Decimal Places.
In the realm of nuclear chemistry, understanding radioactive decay is paramount. Radioactive isotopes, like iron-52, decay at a specific rate, characterized by their half-life. The half-life represents the time it takes for half of the radioactive material to decay. This concept is crucial in various applications, from medical imaging to archaeological dating. In this article, we will delve into the half-life of iron-52, which is approximately 8.3 hours, and calculate how much of a 13-gram sample would remain after 2 hours. We will explore the underlying principles of radioactive decay, the formula used for calculations, and the step-by-step process of determining the remaining amount of iron-52. This comprehensive guide aims to provide a clear understanding of this essential concept in nuclear chemistry.
Understanding Half-Life
The half-life of a radioactive isotope is a fundamental concept in nuclear chemistry. It is defined as the time required for one-half of the atoms in a radioactive sample to decay. This decay process follows first-order kinetics, meaning the rate of decay is proportional to the amount of radioactive material present. Each radioactive isotope has a unique half-life, ranging from fractions of a second to billions of years. For instance, iron-52 has a half-life of approximately 8.3 hours, while uranium-238 has a half-life of 4.5 billion years. Understanding half-life is crucial for determining the age of ancient artifacts, the dosage of radioactive drugs in medical treatments, and the safe storage of nuclear waste. The concept of half-life is not just a theoretical construct; it has practical implications in various scientific and industrial applications. For example, in nuclear medicine, isotopes with short half-lives are preferred to minimize patient exposure to radiation. In contrast, in geological dating, isotopes with long half-lives are used to determine the age of rocks and minerals. Therefore, a thorough understanding of half-life is essential for anyone working with radioactive materials.
The Formula for Radioactive Decay
The mathematical formula that governs radioactive decay is derived from the principles of first-order kinetics. This formula allows us to calculate the amount of a radioactive substance remaining after a certain period, given its initial amount and half-life. The formula is expressed as follows:
N(t) = N₀ * (1/2)^(t/T)
Where:
- N(t) is the amount of the substance remaining after time t.
- N₀ is the initial amount of the substance.
- t is the elapsed time.
- T is the half-life of the substance.
This formula is a powerful tool for predicting the decay of radioactive isotopes. It highlights the exponential nature of radioactive decay, where the amount of substance decreases by half for every half-life that passes. The term (1/2)^(t/T) represents the fraction of the original substance remaining after time t. This fraction decreases exponentially as time increases. The formula is widely used in various fields, including nuclear medicine, environmental science, and archaeology. For instance, in carbon dating, the formula is used to determine the age of organic materials by measuring the remaining amount of carbon-14, a radioactive isotope with a half-life of 5,730 years. Understanding and applying this formula is crucial for accurately calculating radioactive decay and its implications.
Step-by-Step Calculation for Iron-52 Decay
To calculate the remaining amount of iron-52 after 2 hours, we will use the radioactive decay formula mentioned above. Here’s a step-by-step guide:
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Identify the Given Values:
- Initial amount (N₀) = 13 grams
- Half-life (T) = 8.3 hours
- Elapsed time (t) = 2 hours
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Plug the Values into the Formula:
N(t) = N₀ * (1/2)^(t/T)
N(2) = 13 grams * (1/2)^(2/8.3)
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Calculate the Exponent:
2 / 8.3 ≈ 0.241
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Calculate the Fraction Remaining:
(1/2)^0.241 ≈ 0.845
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Calculate the Remaining Amount:
N(2) = 13 grams * 0.845
N(2) ≈ 10.985 grams
Therefore, after 2 hours, approximately 10.985 grams of the 13-gram sample of iron-52 would remain. This calculation demonstrates the practical application of the radioactive decay formula in determining the amount of a radioactive substance remaining after a specific period. The step-by-step approach ensures clarity and accuracy in the calculation process. Understanding each step is crucial for applying this method to other radioactive isotopes and scenarios. The result highlights the gradual decay of iron-52, with a significant portion still remaining after 2 hours due to its relatively longer half-life.
Detailed Calculation and Rounding
To provide a more detailed breakdown of the calculation, let's revisit the steps involved in determining the remaining amount of iron-52 after 2 hours. We start with the radioactive decay formula: N(t) = N₀ * (1/2)^(t/T). We have N₀ = 13 grams, T = 8.3 hours, and t = 2 hours. Plugging these values into the formula, we get:
N(2) = 13 * (1/2)^(2/8.3)
The first step is to calculate the exponent, 2/8.3. This yields approximately 0.240963855. Next, we calculate (1/2) raised to this power: (1/2)^0.240963855. This is equivalent to 2^(-0.240963855), which is approximately 0.844987. Now, we multiply this fraction by the initial amount, 13 grams:
13 * 0.844987 ≈ 10.984831
The question requires us to round the answer to three decimal places. Therefore, we round 10.984831 to 10.985 grams. This detailed calculation ensures accuracy and precision in our result. The rounding step is crucial for providing an answer that meets the specified requirements. The final answer, 10.985 grams, represents the amount of iron-52 remaining after 2 hours, considering its half-life of 8.3 hours. This thorough approach highlights the importance of careful calculation and attention to detail in nuclear chemistry.
Practical Implications of Iron-52 Half-Life
The half-life of iron-52, approximately 8.3 hours, has significant practical implications, particularly in the field of nuclear medicine. Iron-52 is a radioactive isotope used in Positron Emission Tomography (PET) scans, a medical imaging technique that visualizes the metabolic processes in the body. Its relatively short half-life offers several advantages in this context. Firstly, it minimizes the patient's exposure to radiation. Since iron-52 decays relatively quickly, the total radiation dose received by the patient is limited compared to isotopes with longer half-lives. Secondly, the short half-life allows for imaging studies to be conducted and completed within a reasonable timeframe. The diagnostic information can be obtained relatively quickly, which is crucial for timely medical decisions.
However, the short half-life also presents logistical challenges. Iron-52 must be produced close to the location where it will be used, as it decays rapidly. This necessitates the availability of cyclotrons or other particle accelerators in or near the medical facilities that perform PET scans using iron-52. The production, transportation, and administration of iron-52 require careful coordination and adherence to strict safety protocols. Despite these challenges, the benefits of using iron-52 in PET imaging, such as reduced radiation exposure and timely diagnostic information, make it a valuable tool in nuclear medicine. The understanding of iron-52's half-life is, therefore, crucial for optimizing its use in medical applications and ensuring patient safety.
Conclusion
In conclusion, understanding the half-life of radioactive isotopes is crucial in various scientific and medical applications. In the case of iron-52, with a half-life of approximately 8.3 hours, we have demonstrated how to calculate the remaining amount of a sample after a given time using the radioactive decay formula. Starting with a 13-gram sample, we calculated that approximately 10.985 grams would remain after 2 hours. This calculation underscores the practical application of the half-life concept and the importance of the radioactive decay formula.
Furthermore, we discussed the practical implications of iron-52's half-life, particularly in nuclear medicine and PET scans. Its relatively short half-life offers advantages in terms of minimizing patient radiation exposure and enabling timely diagnostic imaging. However, it also presents logistical challenges related to production and transportation. Overall, the study of iron-52’s half-life provides valuable insights into the behavior of radioactive isotopes and their applications in various fields. A thorough understanding of these principles is essential for scientists, medical professionals, and anyone working with radioactive materials.