Radioactive Decay Rate: Solving Half-Life Equations

Have you ever wondered how scientists determine how quickly a radioactive substance decays? It all comes down to understanding the concept of half-life and using a bit of math. In this article, we'll break down a classic problem involving radioactive decay and walk you through the steps to calculate the annual decay rate. Let's dive in!

The Radioactive Decay Problem

Let's consider this problem: A radioactive element has a half-life of five years. A scientist starts with 18 grams of this element. The amount of the element remaining after ${x}$ years can be modeled by the equation {g = 18(0.5)^{\frac{x}{5}}\, where \(g} represents the number of grams. The question we need to answer is: What is the annual rate of decay for this element?

To solve this, we'll explore the fundamentals of radioactive decay, understand the given equation, and then calculate the annual decay rate step by step. This process involves converting the half-life information into an annual decay percentage, which is crucial in fields like nuclear medicine, environmental science, and geology. Understanding decay rates helps in safely handling radioactive materials, dating geological samples, and using radioisotopes in medical treatments.

Delving into Radioactive Decay

Radioactive decay is a fundamental process in nuclear physics where an unstable atomic nucleus loses energy by emitting radiation in the form of particles or electromagnetic waves. This process transforms the original nuclide, or specific type of atomic nucleus, into a different nuclide or a different energy state of the same nuclide. Radioactive decay is a spontaneous process, meaning it occurs randomly and does not require external factors like temperature or pressure to initiate. The rate at which a radioactive substance decays is constant and is described by its half-life, which we'll explore further.

Understanding Half-Life

The half-life of a radioactive element is the time it takes for half of the substance to decay. It’s a crucial concept for understanding how long a radioactive material will remain hazardous and how quickly it will transform into a more stable form. Different radioactive isotopes have vastly different half-lives, ranging from fractions of a second to billions of years. For example, some isotopes used in medical imaging have short half-lives to minimize radiation exposure to patients, while isotopes used in dating ancient artifacts have extremely long half-lives.

The Decay Equation

The equation provided, (g = 18(0.5)^{\frac{x}{5}}, is a mathematical representation of exponential decay. Here’s a breakdown:

• ${g}$ represents the amount of the radioactive element remaining after ${x}$ years.
• 18 is the initial amount of the element in grams.
• 0. 5 (or ${\frac{1}{2}}$) signifies that the substance halves with each half-life period.
• ${\frac{x}{5}}$ indicates the number of half-lives that have passed after ${x}$ years, given that the half-life is 5 years.

The exponential nature of this equation shows that the decay starts rapidly and then slows down as the amount of the radioactive substance decreases. This is because the rate of decay is proportional to the amount of the substance present. Initially, when there is a large amount of the substance, the decay rate is high. As the substance decays, there is less of it left, and the rate of decay decreases proportionally.

Calculating the Annual Decay Rate

Now, let's calculate the annual decay rate. The key here is to transform the half-life information into a yearly decay percentage. This involves understanding how the decay occurs over one year rather than over the entire half-life period.

Step 1: Focus on One Year

To find the annual decay rate, we need to determine what fraction of the substance remains after one year. We can do this by plugging ${x = 1}$ into the given equation:

${g = 18(0.5)^{\frac{1}{5}}}$

This calculation will give us the amount of the substance remaining after one year, relative to the initial amount.

Step 2: Calculate the Remaining Amount

Let’s calculate the value of ((0.5)^{\frac{1}{5}}, which represents the fraction of the substance remaining after one year. Using a calculator, we find:

${(0.5)^{\frac{1}{5}} ≈ 0.87055}$

This means that after one year, approximately 87.055% of the substance remains. This value is crucial because it tells us the proportion of the original substance that has not decayed after one year. It reflects the stability of the radioactive element over a shorter time frame than the half-life, giving us a more granular view of the decay process.

Step 3: Determine the Decay Rate

The decay rate is the percentage of the substance that has decayed in one year. To find this, we subtract the remaining amount from 1 (or 100%):

${1 - 0.87055 = 0.12945}$

Converting this to a percentage, we get approximately 12.945%. This value represents the fraction of the substance that has undergone radioactive decay in a single year. It is a key metric for assessing the rate at which the radioactive material is losing its radioactivity and transforming into a more stable state.

Step 4: Round to the Nearest Percentage

Rounding 12.945% to the nearest whole percentage, we get approximately 13%. Therefore, the annual decay rate is about 13%.

Analyzing the Answer Choices

Given the answer choices:

• A. 3%
• B. 13%
• C. 87%
• D. 97%

The correct answer is B. 13%. This result indicates that each year, approximately 13% of the radioactive element decays. This annual decay rate is essential for predicting the long-term behavior of the radioactive substance and for managing the risks associated with its radioactivity.

Understanding the Incorrect Options

It's also important to understand why the other options are incorrect:

• A. 3% is far too low, suggesting a much slower decay rate than what the half-life indicates.
• C. 87% represents the amount remaining after one year, not the amount that has decayed.
• D. 97% is an unrealistically high decay rate for an element with a half-life of five years.

Real-World Implications

Understanding the annual decay rate is crucial in various real-world applications. Here are a few examples:

Nuclear Medicine

In nuclear medicine, radioactive isotopes are used for diagnostic imaging and therapeutic treatments. The decay rate of these isotopes is a critical factor in determining the dosage and timing of treatments. Isotopes with shorter half-lives and faster decay rates minimize the patient's exposure to radiation, while providing effective imaging or therapy. Understanding the annual decay rate helps medical professionals ensure that the right amount of radiation is administered for the required duration, optimizing treatment outcomes while minimizing risks.

Environmental Science

Radioactive materials can be environmental pollutants. Knowing the decay rate is essential for assessing the long-term impact of radioactive contamination and for developing remediation strategies. For instance, understanding how quickly radioactive substances in soil or water decay helps in estimating the duration of contamination and the effectiveness of cleanup efforts. Environmental scientists use decay rates to model the spread and persistence of radioactive pollutants, ensuring that appropriate measures are taken to protect public health and the environment.

Geological Dating

Radioactive decay is a fundamental principle in radiometric dating techniques, which are used to determine the age of rocks and fossils. Isotopes with long half-lives, such as uranium-238 and carbon-14, are used to date very old geological samples and archaeological artifacts, respectively. By measuring the ratio of the parent isotope to the daughter isotope (the product of decay) in a sample, geologists can calculate how many half-lives have passed since the sample was formed, thus determining its age. The precision of these dating methods relies on the accurate knowledge of the decay rates of the isotopes involved.

Safety and Regulation

Radioactive materials are used in various industries, including energy, manufacturing, and research. Regulations governing the handling, storage, and disposal of radioactive substances are based on their decay rates. Understanding how quickly a material will decay is essential for ensuring safe practices and minimizing the risk of radiation exposure. Regulatory agencies set limits on the amount of radioactive materials that can be released into the environment and require specific safety protocols based on the decay characteristics of the substances. This ensures that radioactive materials are managed responsibly, safeguarding workers, the public, and the environment.

Conclusion

Calculating the annual decay rate from a half-life equation is a vital skill in understanding radioactive decay. By breaking down the problem into manageable steps, we've shown how to determine the percentage of a substance that decays in a year. This knowledge is not only useful for solving mathematical problems but also for understanding real-world applications in medicine, environmental science, geology, and safety regulations.

In summary, the annual decay rate of the radioactive element in our example is approximately 13%. This rate helps scientists and professionals in various fields predict the behavior of radioactive materials and manage their use and disposal safely and effectively. The principles discussed here highlight the importance of mathematical modeling in understanding natural phenomena and in developing practical solutions to real-world problems.

For further information on radioactive decay and its applications, you can visit the World Nuclear Association.