Calculating Remaining Radioactive Sample After N Half-Lives

Hey everyone! Today, let's dive into a fascinating topic in physics: radioactive decay. Specifically, we're going to tackle the question of how to calculate the amount of a radioactive sample remaining after a certain number of half-lives. This is a fundamental concept in nuclear physics and has applications in various fields, from medicine to archaeology. So, let's get started!

What is Radioactive Decay?

Before we jump into the calculations, let's quickly recap what radioactive decay is all about. Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This radiation can take the form of alpha particles, beta particles, or gamma rays. The rate at which a radioactive substance decays is characterized by its half-life. The half-life is the time it takes for half of the radioactive atoms in a sample to decay. This decay happens in an exponential manner, meaning that the amount of the radioactive substance decreases rapidly at first and then more slowly as time goes on. Understanding the concept of half-life is essential for figuring out how much of a radioactive sample remains after a certain period.

Key Concepts to Remember

• Radioactive decay: The process where unstable atomic nuclei lose energy by emitting radiation.
• Half-life: The time required for half the radioactive atoms in a sample to decay.
• Exponential Decay: The decay rate decreases over time, starting rapidly and slowing down.

The Formula for Calculating Remaining Radioactive Material

Now, let's get to the heart of the matter: how do we calculate the amount of a radioactive sample that remains after n half-lives? The key is understanding the exponential nature of radioactive decay. After one half-life, half of the original sample remains. After two half-lives, half of that half remains, which is one-quarter of the original sample. This pattern continues, with the amount remaining being halved with each successive half-life. This leads us to a very handy formula.

To figure out how much of the original radioactive sample is left after n half-lives, we use the following expression:

(1 / 2)^n

This formula tells us that for each half-life (n) that passes, the original amount of the radioactive sample is multiplied by 1/2 raised to the power of n. Let's break this down a bit more to make sure we understand why this formula works so well. The term (1/2) represents the fraction of the sample that remains after one half-life. When we raise this fraction to the power of n, we are essentially multiplying (1/2) by itself n times. Each multiplication represents another half-life passing. For instance, after one half-life (n = 1), the fraction remaining is (1/2)^1 = 1/2. After two half-lives (n = 2), the fraction remaining is (1/2)^2 = 1/4, and so on. This exponential decay is a hallmark of radioactive processes, and this formula neatly captures that behavior. It's a powerful tool for anyone working with radioactive materials or studying nuclear physics.

Why This Formula Works

• (1/2): Represents the fraction remaining after one half-life.
• n: The number of half-lives that have passed.
• (1/2)^n: The fraction of the original sample remaining after n half-lives.

Applying the Formula

Let’s illustrate this with an example. Suppose we start with a radioactive sample of 100 grams, and we want to know how much will be left after 3 half-lives. Using our formula, we calculate (1/2)^3, which equals 1/8. This means that after 3 half-lives, 1/8 of the original sample remains. So, 1/8 of 100 grams is 12.5 grams. That’s how much of the radioactive substance will be left. This calculation shows the practical application of the formula and how it helps us predict the decay of radioactive materials over time.

Let's walk through a few more examples to solidify our understanding. Imagine we have a different radioactive isotope with a half-life of 5 years, and we start with 200 grams of it. We want to know how much will remain after 15 years. First, we need to figure out how many half-lives have passed. Since each half-life is 5 years, 15 years represents 15 / 5 = 3 half-lives. Now we can use our formula: (1/2)^3 = 1/8. So, after 15 years, 1/8 of the original 200 grams will remain. That's 200 grams * 1/8 = 25 grams. These examples highlight the power of the (1/2)^n formula in predicting radioactive decay and are essential for various applications, such as determining the age of ancient artifacts through carbon dating or calculating the dosage of radioactive isotopes in medical treatments. Understanding these principles helps us work safely and effectively with radioactive materials.

Example Calculations

• 100 grams initial sample, 3 half-lives: (1/2)^3 = 1/8 remaining, so 12.5 grams remain.
• 200 grams initial sample, 3 half-lives: (1/2)^3 = 1/8 remaining, so 25 grams remain.

Why Other Options Are Incorrect

Now, let's address why the other options given in the original question are incorrect. This will not only help us confirm our correct answer but also deepen our understanding of radioactive decay. The original question presented four options:

• A. (1 / 2) × n
• B. (1 / n)^2
• C. (1 / 2)^n
• D. 1 / (2 n)

We’ve already established that option C, (1 / 2)^n, is the correct expression. Let’s break down why the others don’t work. Option A, (1 / 2) × n, represents a linear decrease rather than an exponential one. This means the amount remaining would decrease by a constant amount for each half-life, which isn’t what happens in radioactive decay. The amount halves each time, not decreases by a fixed quantity. Option B, (1 / n)^2, is also incorrect because it doesn’t accurately reflect the halving process. This formula would imply a different rate of decay that isn't consistent with the exponential nature of radioactive decay. Finally, option D, 1 / (2 n), again suggests a linear relationship and doesn’t capture the fundamental concept of halving the sample size with each half-life. By understanding why these options are wrong, we reinforce our grasp on the correct principles governing radioactive decay.

• Option A: (1 / 2) × n Incorrect because it represents linear decay, not exponential decay.
• Option B: (1 / n)^2 Incorrect because it does not accurately model the halving process of radioactive decay.
• Option D: 1 / (2 n) Incorrect because it suggests a linear relationship, not exponential decay.

Conclusion: Mastering Radioactive Decay Calculations

So, to wrap things up, the original amount of a radioactive sample should be multiplied by the expression (1 / 2)^n to calculate the amount of the sample that remains after n half-lives have passed. Understanding this formula is crucial for anyone studying physics, chemistry, or any field that involves radioactive materials. It's not just about memorizing a formula, though; it's about understanding the underlying principle of exponential decay and how it governs the behavior of radioactive substances.

By grasping this concept, you're well-equipped to tackle a variety of problems related to radioactive decay, whether it's determining the age of artifacts using carbon dating or calculating the dosage of radioactive isotopes in medical applications. Keep practicing, and you'll become a pro at radioactive decay calculations in no time! Remember, the key to mastering physics is understanding the concepts, not just memorizing the formulas. Keep exploring, keep questioning, and you'll unlock the secrets of the universe one step at a time. Happy calculating, everyone!