Calculating Half-Life Of Polonium-209 A Step-by-Step Guide

by Scholario Team 59 views

Hey guys! Ever wondered how long it takes for radioactive stuff to decay? Let's dive into the fascinating world of nuclear chemistry and figure out how to calculate the half-life of Polonium-209. We’re going to break down a problem where we start with 200g of Polonium-209, and after 306 years, we’re left with just 25g. Sounds like a puzzle? Let’s solve it together!

Understanding Half-Life: The Basics

Before we jump into the calculations, let’s make sure we’re all on the same page about what half-life actually means. In the simplest terms, half-life is the time it takes for half of a radioactive substance to decay. Think of it like this: if you have a room full of popcorn kernels, the half-life is the time it takes for half of those kernels to pop. For radioactive elements, it’s about how long it takes for half of the atoms in a sample to undergo radioactive decay.

Why is this important? Well, half-life helps us understand how stable a radioactive isotope is. Isotopes with short half-lives decay quickly, while those with long half-lives stick around for quite a while. Knowing the half-life is crucial in various fields, from medicine (like understanding how long a radioactive tracer will be active in a patient) to archaeology (like carbon-14 dating).

Key Concepts to Grasp

  1. Radioactive Decay: This is the process where an unstable atomic nucleus loses energy by emitting radiation. It’s like the atom is shaking off extra weight to become more stable.
  2. Isotopes: These are variants of a chemical element which have the same number of protons but different numbers of neutrons. Polonium-209, for instance, is an isotope of Polonium.
  3. Exponential Decay: Radioactive decay follows an exponential pattern. This means the amount of substance decreases by half in each half-life period, not linearly. It’s like cutting a cake in half, then cutting one of the halves in half again, and so on.

So, with the basics down, we can appreciate how the concept of half-life helps us measure and predict the behavior of radioactive materials. Now, let’s get to the fun part: solving our problem!

Problem Setup: Polonium-209 Decay

Alright, let’s tackle the problem at hand. We’re dealing with Polonium-209 ( P o-209), and we've got some key information to start with. Initially, we have a sample mass of 200 grams. After 306 years, only 25 grams of Polonium-209 remain. Our mission is to find the half-life of this radioactive isotope. Think of it as a detective game where we’re piecing together clues to uncover the hidden half-life!

Identifying the Given Values

First, let’s break down what we know:

  • Initial mass ( Mâ‚€ ): 200 grams
  • Final mass ( M t ): 25 grams
  • Total time ( t ): 306 years

What we’re looking for is the half-life ( T 1/2

). This is the time it takes for half of the original sample to decay. To find this, we'll use the formula that describes radioactive decay. This formula connects the initial and final amounts of the substance with the number of half-lives that have passed.

The Radioactive Decay Formula

The formula we’re going to use is:

N t = Nâ‚€ * (1/2)^(t / T 1/2

)

Where:

N t is the amount of the substance remaining after time t . *
Nâ‚€ is the initial amount of the substance. *
t is the total time that has passed. *
T 1/2

is the half-life of the substance.

This formula might look a bit intimidating, but don't worry! We're going to take it step by step. The key here is understanding that the fraction of the substance remaining is related to how many half-lives have occurred during the given time. The (1/2)^(t / T 1/2

) part of the formula tells us exactly that.

With our problem set up and the formula in hand, we're ready to plug in the numbers and start crunching. So, let’s roll up our sleeves and dive into the mathematical part of this journey!

Step-by-Step Calculation

Now comes the exciting part where we actually use our formula and the given values to calculate the half-life of Polonium-209. Don't worry, we’ll take it one step at a time to make sure everything’s clear. Grab your calculators, folks! Let's get started.

Plugging in the Values

Remember our formula?

N t = Nâ‚€ * (1/2)^(t / T 1/2

)

Let’s plug in the values we identified earlier:

N t = 25 grams *
Nâ‚€ = 200 grams *
t = 306 years

So, our equation now looks like this:

25 = 200 * (1/2)^(306 / T 1/2

)

Simplifying the Equation

The first thing we want to do is simplify the equation to isolate the exponential term. To do this, we'll divide both sides by 200:

25 / 200 = (1/2)^(306 / T 1/2

)

This simplifies to:

1/8 = (1/2)^(306 / T 1/2

)

Recognizing the Exponential Relationship

Here's a cool trick: we can express 1/8 as a power of 1/2. Think about it – how many times do you need to multiply 1/2 by itself to get 1/8? The answer is 3, because (1/2) * (1/2) * (1/2) = 1/8. So, we can rewrite our equation as:

(1/2)³ = (1/2)^(306 / T 1/2

)

Now, we have the same base (1/2) on both sides of the equation. This means we can equate the exponents:

3 = 306 / T 1/2

Solving for Half-Life

We’re in the home stretch now! To solve for the half-life ( T 1/2

), we need to isolate it. We can do this by multiplying both sides of the equation by T 1/2

and then dividing by 3:

3 * T 1/2

= 306

T 1/2

= 306 / 3

T 1/2

= 102 years

And there we have it! The half-life of Polonium-209 is 102 years. That means it takes 102 years for half of a sample of Polonium-209 to decay. Pretty neat, huh?

Conclusion: The Half-Life Unveiled

So, guys, we did it! We successfully calculated the half-life of Polonium-209, which turned out to be 102 years. We took a problem that might have seemed tricky at first and broke it down into manageable steps. We started with the basics of half-life, set up our problem with the given values, used the radioactive decay formula, and then solved for the half-life.

Recap of Our Journey

  1. Understood Half-Life: We clarified what half-life means and why it’s important in understanding radioactive decay.

  2. Set Up the Problem: We identified the initial mass, final mass, and total time, and recognized that we needed to find the half-life.

  3. Used the Formula: We applied the radioactive decay formula:

    N t = Nâ‚€ * (1/2)^(t / T 1/2

) 4. Simplified and Solved: We plugged in the values, simplified the equation, recognized the exponential relationship, and solved for T 1/2

.

Why This Matters

Understanding how to calculate half-life isn’t just an academic exercise. It’s crucial in many real-world applications. For instance:

  • Medicine: Radioactive isotopes are used in medical imaging and cancer treatment. Knowing their half-lives helps doctors determine the right dosage and timing.
  • Archaeology: Carbon-14 dating uses the half-life of carbon-14 to estimate the age of ancient artifacts and fossils.
  • Environmental Science: Understanding the decay rates of radioactive pollutants helps in assessing environmental risks and planning cleanup efforts.

By mastering this concept, you've gained a valuable tool for understanding the world around you. Whether you’re interested in nuclear chemistry, environmental science, or just curious about how things work, knowing how to calculate half-life is a fantastic skill to have.

Keep exploring, keep questioning, and keep learning! Who knows what other fascinating concepts you’ll uncover next? And remember, even complex problems become manageable when you break them down step by step. Until next time, happy calculating!