Radioactive Decay: Calculating Half-Life Explained

by TextBrain Team 51 views

Hey guys! Let's dive into a fascinating problem involving radioactive decay and learn how to calculate the half-life of a substance. This is a classic physics problem, and understanding the concept of half-life is crucial in many scientific fields, from nuclear medicine to archaeology. So, let’s break it down step-by-step. We'll tackle the question: How do we determine the half-life of a radioactive substance if we know its initial mass and its mass after a certain period?

Understanding Half-Life

Before we jump into calculations, let’s make sure we're all on the same page about what half-life actually means. The half-life of a radioactive substance is the time it takes for half of the substance to decay. It's a fundamental property of radioactive isotopes and is constant for a given isotope. This means that if you start with a certain amount of a radioactive material, after one half-life, you'll have half of that amount remaining. After another half-life, you'll have half of that amount, and so on. This decay happens exponentially, which is a fancy way of saying it decreases rapidly at first and then slows down over time.

Imagine you have a room full of popcorn kernels, and every minute, half of them pop. The time it takes for half the kernels to pop is analogous to the half-life of a radioactive substance. The popping might be fast initially, but as you have fewer and fewer kernels, the rate of popping slows down. This is because the probability of a kernel popping depends on how many kernels are left. Similarly, with radioactive decay, the rate of decay is proportional to the amount of radioactive material present.

The half-life is a probabilistic concept. It doesn't mean that every atom will decay after exactly one half-life. Instead, it means that statistically, half of the atoms in a large sample will have decayed after that time. Some atoms will decay sooner, and some will decay later, but on average, half will be gone after one half-life. The half-life is a key characteristic of a radioactive isotope, helping scientists to identify and use these materials effectively in various applications. Understanding half-life is also vital for understanding the long-term effects of radioactive materials and managing nuclear waste.

Problem Setup: Radioactive Substance Decay

Okay, now let's get to the specific problem. We're told we have a sample of a radioactive substance that initially weighs 36 mg. That's our starting point. After one year, the sample weighs 27.5 mg. Our goal is to find the half-life of this substance. To do this, we'll use the concept of exponential decay and a little bit of algebra. Don't worry, it's not as scary as it sounds! We'll break it down into manageable steps. We need to figure out how long it takes for half of the substance to decay, given the information we have about its decay over one year.

The first thing to realize is that radioactive decay follows a predictable pattern. The amount of the substance remaining decreases exponentially with time. This means that the amount of substance at any time t can be described by the following formula:

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 time elapsed.
  • T is the half-life (what we want to find!).

This formula might look a bit intimidating at first, but it's really just saying that the amount remaining is equal to the initial amount multiplied by one-half raised to the power of the time elapsed divided by the half-life. The (1/2) term represents the halving of the substance with each half-life period. The exponent (t/T) tells us how many half-lives have passed in the given time t. So, the bigger the exponent, the more half-lives have passed, and the smaller the remaining amount will be.

In our specific problem, we know N₀ (the initial amount) is 36 mg, and we know N(1) (the amount after 1 year) is 27.5 mg. We also know that t is 1 year. The only unknown in our equation is T, the half-life, which is exactly what we want to find! This is great because now we can plug in the known values and solve for the unknown. The beauty of this formula is that it captures the fundamental process of exponential decay, where the decay rate is proportional to the amount of substance present. By using this formula, we can accurately predict the amount of radioactive material remaining after any given time, as long as we know the half-life and the initial amount.

Setting Up the Equation

Now, let's plug the values we know into our equation. We have:

  1. 5 mg = 36 mg * (1/2)^(1 year / T)

See? It's not so scary! We've just substituted the given information into the general decay formula. The next step is to isolate the term with the unknown, which is (1/2)^(1 year / T). We can do this by dividing both sides of the equation by 36 mg:

  1. 5 mg / 36 mg = (1/2)^(1 year / T)

This simplifies to:

  1. 7639 ≈ (1/2)^(1 year / T)

We've now got the equation in a form where the exponential term is isolated. This is a crucial step because it allows us to use logarithms to solve for the unknown exponent, which contains our half-life, T. Remember, the goal is to get T by itself, and logarithms are the mathematical tools that will help us unwind the exponential function. The left side of the equation, 0.7639, represents the fraction of the original substance that remains after one year. This value is between 0.5 and 1, which makes sense because one half-life hasn't yet passed (if it had, we'd have only half the substance remaining, corresponding to 0.5). Now, we are ready to take the next step and use logarithms to solve for T.

Solving for Half-Life Using Logarithms

To solve for T, we need to get rid of the exponent. This is where logarithms come in handy. We can take the logarithm of both sides of the equation. It doesn't matter which base of logarithm we use, but the natural logarithm (ln) and the common logarithm (log base 10) are the most common. Let's use the natural logarithm (ln) in this case:

ln(0.7639) = ln((1/2)^(1 year / T))

A key property of logarithms is that they allow us to bring down exponents as multipliers. So, we can rewrite the right side of the equation as:

ln(0.7639) = (1 year / T) * ln(1/2)

Now, we have T in the denominator of a fraction, but it's much easier to isolate. We can rearrange the equation to solve for T:

T = (1 year * ln(1/2)) / ln(0.7639)

This is a crucial step because we've successfully isolated the half-life, T, on one side of the equation. Now, it's just a matter of plugging the values into a calculator to get the numerical answer. Remember that ln(1/2) is a negative number because the logarithm of a number less than 1 is negative. Similarly, ln(0.7639) is also negative. The fact that both logarithms are negative means that the final result for T will be positive, which makes sense since half-life is a time period and must be a positive value. The logarithms have effectively helped us unravel the exponential relationship and transform it into a linear equation that we can easily solve for T.

Calculating the Final Answer

Now, let’s plug those values into a calculator. We find:

ln(0.7639) ≈ -0.2694

ln(1/2) ≈ -0.6931

So, our equation becomes:

T ≈ (1 year * -0.6931) / -0.2694

T ≈ 2.57 years

Remember, the problem asked us to round our answer to two decimal places. So, the half-life of this radioactive substance is approximately 2.57 years. This means that it takes about 2.57 years for half of the substance to decay. After another 2.57 years, half of the remaining substance will decay, and so on. The calculated half-life provides crucial information about the stability and decay rate of this particular radioactive isotope. It helps scientists to understand its behavior and predict its long-term presence in a sample or environment. We've successfully navigated through the problem, from understanding the concept of half-life to using the decay formula and logarithms to arrive at the final answer. Good job, guys!

Conclusion

So, there you have it! We've successfully calculated the half-life of the radioactive substance. The key takeaways here are understanding the exponential decay formula, knowing how to use logarithms to solve for unknowns in exponents, and applying these concepts to a real-world problem. Half-life calculations are essential in various fields, and mastering this skill opens up a world of understanding about radioactive materials and their behavior. If you ever encounter a similar problem, remember the steps we followed: set up the equation, isolate the exponential term, use logarithms, and solve for the half-life. You've got this! Keep practicing, and you'll become a pro at solving half-life problems in no time.