Radioactive Decay Calculation: From 200g To 100 Days

Hey guys! Let's dive into a cool chemistry problem that involves radioactive decay. This is a super interesting topic, and it's all about how the mass of a radioactive substance decreases over time. In this case, we're starting with a 200-gram sample, and we know it loses 10% of its mass every 10 days. The question is: how much of the substance will be left after 100 days? Don't worry, we'll break it down step by step. It's like a countdown, but instead of a rocket launching, we're watching the mass of a substance shrink! Understanding this is key in fields like nuclear physics and even medicine, where radioactive isotopes are used for various purposes. So, grab your calculators, and let's get started. We will explore the concept of radioactive decay, its mathematical representation, and apply it to solve a practical problem. This process helps to illustrate how we can predict the behavior of radioactive substances over time, which is essential in various applications, from medical treatments to environmental monitoring. This article is intended to provide a clear understanding of the concepts and calculations involved in radioactive decay, ensuring that anyone can follow along and apply these principles to similar problems. We'll go through the problem, show you the formula, and then walk through the calculation. It’s gonna be fun, I promise!

Understanding the Problem: Radioactive Decay

Radioactive decay is a natural process where an unstable atomic nucleus loses energy by emitting radiation. This process transforms the original atom into a different atom. The rate at which this decay happens is constant and is usually expressed as a half-life, the time it takes for half of the substance to decay. In our problem, instead of using half-life, we're given the percentage of decay over a specific period. Imagine it like this: every 10 days, a certain portion of the substance vanishes, leaving us with less and less. This is a fundamental concept in nuclear chemistry. The rate of decay is constant, meaning a fixed percentage of the substance decays over a given time. The key here is to understand that the rate of decay is proportional to the amount of the substance present. The greater the amount of the substance, the greater the amount that will decay within a specific time. In real-world applications, this principle is used in various fields such as carbon dating, medical imaging, and nuclear power generation. Understanding radioactive decay helps us comprehend how these different fields function and allow us to predict the behavior of these substances.

Initial Conditions and Key Information

To solve this, let's clarify the given information:

• Initial Mass (M₀): 200 grams - This is our starting point, the amount of radioactive substance we have at the beginning.
• Time (t): 100 days - The total time duration over which we need to calculate the decay.
• Decay Percentage (p): 10% every 10 days - The percentage of the substance that decays every 10 days.
• Target Mass (Mt): ? - This is what we need to figure out, the remaining mass of the substance after 100 days.

We can visualize this as a series of 10-day intervals where the substance loses 10% of its mass. To get to our final answer, we must use our information so we can solve the problem!

The Formula for Radioactive Decay

Now that we know the problem's details, let's introduce the formula. This formula is our tool to solve the radioactive decay problem. It's like the secret recipe. The formula helps us to figure out how much of the substance will be left after a certain amount of time. It's a way to apply the known rate of decay over time. Let's unpack this formula piece by piece. The application of this formula is extremely valuable. Let's check it out.

The Decay Formula:

The formula we'll use to calculate the mass of the radioactive substance after a certain time is:

• Mt = M₀ (1 - p)^n

Where:

• Mt = Mass after time t (what we want to find)
• M₀ = Initial mass
• p = Decay rate (as a decimal)
• n = Number of decay periods

Breaking Down the Formula

1. Mt: This is the final mass of the radioactive substance after the decay process. This is what we are trying to find in our problem.
2. M₀: Represents the starting mass. This is the initial amount of the substance, and in our case, it's 200 grams.
3. (1 - p): This part of the formula represents the fraction of the substance that remains after each decay period. Since p is the decay rate, (1 - p) is the remaining proportion. The amount that isn't decaying.
4. n: This is the number of decay periods. In our case, since the decay happens every 10 days and we're looking at 100 days, we'll need to figure out how many 10-day periods are in 100 days.

Preparing to Calculate: Finding the number of decay periods.

Before we can plug our numbers into the formula, we need to figure out n, the number of decay periods. Since the substance decays by 10% every 10 days, and we want to know the mass after 100 days, we need to figure out how many 10-day periods are in 100 days. To calculate n, we do the following: Divide the total time by the decay period. In this scenario, we have the following:

• n = Total time / Decay period
• n = 100 days / 10 days per period = 10 periods

So, n = 10. Now we know that the substance goes through 10 cycles of decay over the 100 days.

Solving the Problem: Calculating the Remaining Mass

Alright, now that we've got the formula and all the pieces in place, it's time to do the math! Let's use the formula we discussed earlier: Mt = M₀ (1 - p)^n

Here's how to solve it, step by step. We will begin to find the mass after the substance decays over time. It will be a super easy step-by-step process. Let's go.

Step-by-step calculation

1. Identify the values:

   • M₀ = 200 grams
   • p = 10% = 0.1 (as a decimal)
   • n = 10 periods

2. Plug in the values into the formula:

   • Mt = 200 (1 - 0.1)^10*

3. Calculate (1 - p):

   • (1 - 0.1) = 0.9

4. Calculate (1 - p)^n:

   • 0.9^10 = 0.3486784401 (approximately)

5. Multiply M₀ by (1 - p)^n:

   • Mt = 200 * 0.3486784401
   • Mt = 69.73568802 grams (approximately)

Therefore, after 100 days, the mass of the radioactive substance will be approximately 69.74 grams (rounding to two decimal places). Congrats, you did it!

Conclusion: Understanding the Result

So, we've calculated that after 100 days, the mass of the radioactive substance has decreased from 200 grams to approximately 69.74 grams. This demonstrates how the concept of exponential decay works over time. The initial mass gradually decreases with each decay period. The formula helps us predict the mass of a radioactive substance over time. What started as a 200-gram sample has significantly reduced due to the radioactive decay process. Radioactive decay is essential in many areas. This includes medical treatments to nuclear energy and carbon dating. If we have to, we can apply this to other real-world scenarios. Remember, in real-world applications, this concept is critical for things like nuclear medicine, where you have to understand how radioactive isotopes decay in the body. Understanding this process helps us use these substances safely and efficiently. You can now predict the behavior of radioactive substances. This principle has broad applications. Keep it up!

Additional Notes and Considerations

• Units: Always make sure your units are consistent. In this case, we used grams for mass and days for time.
• Rounding: Rounding can affect the final answer. In our calculation, we rounded the final result to two decimal places for simplicity.
• Real-World Complexity: In reality, radioactive decay can be more complex, depending on the substance. Factors like temperature and pressure usually have a minimal effect, but can have an impact.
• Half-Life: While we used a decay percentage in this problem, the half-life is also a common way to describe the decay rate. The half-life is the time it takes for half of the substance to decay.

Understanding the concepts of radioactive decay is super important in many scientific fields. This helps us understand the behavior of radioactive materials. We can also use this information to make important decisions in real-life situations. Great job, guys! You now know how to calculate radioactive decay. Keep exploring and learning! You guys are amazing!