Radioactive Decay: Solving The Half-Life Problem

Hey guys, let's dive into a classic chemistry problem involving radioactive decay! This scenario throws us a curveball by asking us to calculate the mass of a radioactive element that decays over a specific time, instead of the mass remaining. This kind of problem often pops up in exams, so understanding the fundamentals is super important. Don't worry, we'll break it down step by step to make sure everything is crystal clear. We will explore the concept of half-life and its relationship to exponential decay. This knowledge isn't just for passing tests, it's also crucial for understanding how scientists use radioactive isotopes in fields like medicine, archaeology, and environmental science. Let's get started and become experts on nuclear chemistry!

Understanding the Problem and Key Concepts

So, the problem says a radioactive element decays to half its mass in 45 minutes. This period is known as the half-life of the element. It's a fundamental concept in nuclear physics, defining the time required for half of the radioactive atoms in a sample to undergo radioactive decay. In this case, our element has a half-life of 45 minutes. This means that after every 45-minute interval, the amount of the radioactive substance is halved. If we begin with 50 grams, after 45 minutes, we'll have 25 grams. After another 45 minutes (90 minutes total), we'll have 12.5 grams, and so on. This follows an exponential decay pattern.

Now, the problem wants to know how much of the element decays over a 3-hour period. Notice the tricky wording? Instead of asking for the remaining mass, it's asking about the decayed mass. This subtle difference is the key to correctly solving the problem. To tackle this, first, we'll figure out how many half-lives occur within the 3-hour timeframe. Once we know this, we can determine the remaining mass and, finally, calculate the mass that has decayed. This will involve using the formula related to half-life. Let's break this down.

To work through this problem effectively, remember these essential terms: radioactive decay, half-life, exponential decay, and the distinction between the mass that remains and the mass that decays. Grasping these concepts will make the calculations much easier. The problem also indicates that the initial mass of the substance is 50 grams. The question is, how much of the element has decayed after 3 hours? This is where we need to put our knowledge of half-life to work.

We'll go through the solution carefully, calculating the mass that remains after 3 hours and then figuring out the mass that has decayed. This approach ensures we don't get confused by the wording of the question. It's always a good idea to understand the core principles and not just memorize formulas; it will boost your problem-solving abilities.

Step-by-Step Solution

Alright, let's crunch some numbers! First, we need to convert the time into the same units. Our half-life is in minutes (45 minutes), so let's convert the total time (3 hours) into minutes. There are 60 minutes in an hour, so 3 hours is 3 * 60 = 180 minutes. Now that we have the total time in minutes, we can calculate the number of half-lives. The number of half-lives is the total time divided by the half-life duration: 180 minutes / 45 minutes = 4 half-lives.

This means that in the 3-hour period, the element will go through 4 half-lives. Now let's calculate the remaining mass after each half-life. Initially, we have 50 grams. After the 1st half-life (45 minutes), the mass remaining is 50 grams / 2 = 25 grams. After the 2nd half-life (90 minutes total), the mass remaining is 25 grams / 2 = 12.5 grams. After the 3rd half-life (135 minutes total), the mass remaining is 12.5 grams / 2 = 6.25 grams. Finally, after the 4th half-life (180 minutes or 3 hours total), the mass remaining is 6.25 grams / 2 = 3.125 grams.

So, after 3 hours, 3.125 grams of the radioactive element is left. But remember, the problem asks us for the mass that decayed. To find the mass decayed, we subtract the remaining mass from the initial mass: 50 grams (initial mass) - 3.125 grams (remaining mass) = 46.875 grams. Therefore, the mass of the element that decayed during the 3-hour period is 46.875 grams. Thus, we've successfully solved the problem.

In this scenario, the initial mass plays a crucial role, and the half-life is the constant that determines how fast the element decays. The problem's wording wants the mass that has decayed, and we use the formula to get the answer.

Checking the Answer and Understanding the Decay Process

Let's take a moment to verify our answer and ensure it makes sense. We started with 50 grams, and after 3 hours (or 4 half-lives), 3.125 grams remain. This means that a significant portion of the element has decayed. The mass that decayed is 46.875 grams. Considering the exponential nature of radioactive decay, this result aligns with our expectations. It's essential to check your answer, especially in chemistry problems, to catch any calculation errors or misunderstandings. A quick review of the steps involved can also reinforce your understanding of the concept.

In radioactive decay, the rate of decay is directly proportional to the amount of the radioactive substance present. This is why it follows an exponential pattern. During each half-life, half of the remaining substance decays. This process continues until the amount of the substance becomes negligible. In nuclear physics, half-life is used as a key parameter to determine the stability of an element. Elements with short half-lives are highly unstable and decay rapidly, while elements with long half-lives are relatively stable and decay slowly. Understanding the process is essential. This involves mastering the concept of half-life. Now that you've worked through this problem, you're ready to tackle similar challenges with confidence.

So, the answer is E. 46.875 gram. Congratulations, you've successfully navigated a radioactive decay problem! Keep practicing, and you'll become a pro at these types of calculations. Remember that radioactive decay is not a one-size-fits-all process; it depends on the element's identity, which determines its specific half-life. Keep practicing, and you'll be acing these problems in no time.