Radioactive Decay Calculation How Much Isotope Remains After 3 Half-Lives

Hey guys! Ever wondered how radioactive stuff decays over time? It's a fascinating process governed by something called half-life. In this article, we'll break down a classic problem: figuring out how much of a radioactive isotope remains after a certain number of half-lives. We'll walk through the solution step-by-step, making sure you grasp the concept along the way. So, let's dive in!

Understanding Half-Life

Before we tackle the problem head-on, let's get a solid understanding of what half-life actually means. Imagine you have a bunch of radioactive atoms. These atoms are unstable, and they spontaneously transform into a different type of atom, a process we call radioactive decay. The half-life is the time it takes for half of the radioactive atoms in a sample to decay. It's a fixed property of each radioactive isotope, meaning each isotope has its own unique half-life, which can range from fractions of a second to billions of years!

Think of it like this: if you start with, say, 1000 radioactive atoms and the half-life of the isotope is 1 hour, then after 1 hour, you'll have about 500 radioactive atoms left. The other 500 have decayed into something else. After another hour (another half-life), half of the remaining 500 will decay, leaving you with approximately 250 atoms, and so on. This exponential decay is a key characteristic of radioactive materials. Understanding this core concept is crucial for tackling any radioactive decay problem. We'll use this knowledge to solve our sample problem, so make sure you've got it down!

The Problem: Radioactive Isotope Decay

Okay, now let's get to the heart of the matter. Our problem states: A sample contains 100 g of a radioactive isotope. How much radioactive isotope will remain in the sample after 3 half-lives? We are given the initial amount of the radioactive isotope which is 100g, and we need to determine how much of it will be left after 3 half-lives have passed. Remember, each half-life represents the time it takes for half of the radioactive material to decay. This is a classic radioactive decay problem, and it highlights the fundamental principle of half-life. The options given are:

A. 12.5 g B. 25 g C. 100 g D. 50 g

Before we jump into calculations, let's think conceptually. After one half-life, we know the amount will be halved. After two, it will be halved again, and so on. This repeated halving is the key to solving this problem. We'll explore two methods: a step-by-step approach and a more direct formula-based approach, to provide a comprehensive understanding of the decay process. So, let's get calculating and figure out the correct answer!

Method 1: Step-by-Step Calculation

The first method to solve this problem is the step-by-step calculation. This approach is intuitive and helps visualize the decay process. We simply track the amount of the radioactive isotope remaining after each half-life. We start with 100 g of the radioactive isotope. After the first half-life, half of the material will have decayed. So, we divide the initial amount by 2: 100 g / 2 = 50 g. This means that after one half-life, 50 g of the radioactive isotope remains.

Now, let's move on to the second half-life. Again, half of the remaining material will decay. We take the amount remaining after the first half-life, which is 50 g, and divide it by 2: 50 g / 2 = 25 g. So, after two half-lives, we have 25 g of the radioactive isotope left. Finally, we consider the third half-life. We repeat the process, halving the amount remaining after the second half-life: 25 g / 2 = 12.5 g. Therefore, after three half-lives, 12.5 g of the radioactive isotope will remain. This step-by-step method clearly illustrates the exponential decay, where the amount of radioactive material decreases by half with each passing half-life. This approach is perfect for solidifying your understanding of the concept.

Method 2: Using the Formula

For a more direct approach, we can use the radioactive decay formula. This formula allows us to calculate the remaining amount of a radioactive isotope after any number of half-lives in one single step. The formula is as follows:

N = N₀ * (1/2)ⁿ

Where:

• N is the amount of the radioactive isotope remaining after n half-lives.
• N₀ is the initial amount of the radioactive isotope.
• n is the number of half-lives.

Let's plug in the values from our problem. We have N₀ = 100 g (the initial amount), and n = 3 (the number of half-lives). So, the equation becomes:

N = 100 g * (1/2)³

First, we calculate (1/2)³ which is (1/2) * (1/2) * (1/2) = 1/8. Then, we multiply this by the initial amount:

N = 100 g * (1/8) N = 12.5 g

As you can see, using the formula gives us the same answer as the step-by-step method: 12.5 g. This formula is a powerful tool for solving radioactive decay problems, especially when dealing with a larger number of half-lives. It provides a concise and efficient way to calculate the remaining amount of radioactive material.

The Answer and Key Takeaways

So, after working through the problem using both the step-by-step method and the formula, we arrive at the same answer: 12.5 g. Therefore, the correct answer is A. 12.5 g. This means that after three half-lives, only 12.5 grams of the original 100 grams of the radioactive isotope will remain. The rest has decayed into other elements.

This problem beautifully illustrates the concept of half-life and how radioactive decay works. Remember, the key takeaway is that after each half-life, the amount of the radioactive isotope is reduced by half. We can calculate the remaining amount by either repeatedly dividing by 2 (the step-by-step method) or by using the radioactive decay formula. Understanding these methods equips you to tackle a wide range of radioactive decay problems. Furthermore, remember that the half-life is a constant for a given isotope. It does not depend on external factors like temperature or pressure. This predictable decay is what makes radioactive isotopes useful in various applications, from medical imaging to carbon dating.

Real-World Applications of Half-Life

The concept of half-life isn't just a theoretical exercise; it has tons of practical applications in various fields! For instance, in medicine, radioactive isotopes with short half-lives are used in imaging techniques like PET scans. The short half-life ensures that the patient isn't exposed to radiation for too long, while still allowing for clear images to be obtained. The rate at which these isotopes decay is crucial for dosage calculations and ensuring patient safety. In archaeological dating, carbon-14, a radioactive isotope of carbon, is used to determine the age of ancient artifacts and fossils. Carbon-14 has a half-life of about 5,730 years, which makes it suitable for dating materials up to around 50,000 years old. By measuring the amount of carbon-14 remaining in a sample, scientists can estimate when the organism died.

In nuclear power, the half-lives of radioactive materials are critical for managing nuclear waste. Some radioactive waste products have extremely long half-lives, meaning they will remain radioactive for thousands of years. This necessitates long-term storage solutions to prevent environmental contamination. Understanding half-life is also vital in radiation therapy for cancer treatment. Doctors use radioactive sources to target and destroy cancerous cells, and the half-life of the isotope used is a key factor in determining the duration and intensity of the treatment. These examples show that half-life is a fundamental concept with far-reaching implications, impacting our lives in numerous ways.

Practice Problems

To really master the concept of half-life, it's essential to practice solving different types of problems. So, let's try a couple more examples. These practice problems will help you solidify your understanding and build confidence in your problem-solving skills. Remember to use either the step-by-step method or the formula, whichever you find more comfortable. Don't be afraid to revisit the explanations and examples we've discussed earlier if you get stuck!

Practice Problem 1: A radioactive isotope has a half-life of 10 years. If you start with 200 grams of the isotope, how much will remain after 30 years?

Practice Problem 2: A sample initially contains 80 grams of a radioactive substance. After 24 days, only 5 grams of the substance remain. What is the half-life of this radioactive substance?

Working through these practice problems will not only reinforce your understanding of half-life but also improve your ability to apply this knowledge in different contexts. The key is to carefully identify the given information (initial amount, number of half-lives, or time elapsed) and then use the appropriate method to calculate the unknown quantity. Good luck, and happy problem-solving!

Conclusion

Alright guys, we've covered a lot in this article! We started with the basic definition of half-life, then walked through a step-by-step calculation and a formula-based approach to solve a radioactive decay problem. We even touched on some real-world applications of half-life, from medical imaging to carbon dating. Hopefully, you now have a solid understanding of how radioactive decay works and how to calculate the amount of radioactive material remaining after a certain number of half-lives. Remember, the concept of half-life is fundamental in nuclear physics and has practical implications in various fields. By understanding the principles we've discussed, you'll be well-equipped to tackle future problems and appreciate the fascinating world of radioactivity. Keep practicing, keep exploring, and you'll become a pro in no time!