Problem 68 - College Algebra and Calculus: An Applied Approach

Question

Radioactive Decay Neptunium-237 has a half-life of 2.1 million years. The amount \(N\) of 200 kilograms of neptunium- 237 present after \(t\) years is given by \(N=200 e^{-0.00000033007 t}\) How much of the 200 kilograms will remain after 20,000 years?

Step-by-Step Solution

Answer

The amount of Neptunium-237 remaining after 20,000 years will be approximately 167.54kg. This is calculated by evaluating the expression \(200 e^{-0.00000033007 * 20000}\).

Step 1: Understand Radioactive Decay of Neptunium-237

The amount \(N\) of 200 kg of Neptunium-237 after \(t\) years is given by the equation \(N=200 e^{-0.00000033007 t}\), which represents the decay over time. Here, \(N\) is the mass of Neptunium-237 left after \(t\) years, and \(e^{-0.00000033007 t}\) is the exponential decay law.

Step 2: Substitute the given time

Substitute \(t=20000\) into the given equation to find out the amount remaining after 20,000 years. It becomes, \(N=200 e^{-0.00000033007 * 20000}\)

Step 3: Evaluate

Evaluate the expression to find the mass of Neptunium-237 remaining. Here, it's essential to use an accurate calculator.

Key Concepts

Exponential Decay

Exponential decay is a fundamental concept in understanding various natural processes, including radioactive decay. It's characterized by a decrease in quantity at a rate proportional to its current value. Think about it like a cake that gets smaller each time you take a bite - the bigger the bite, the faster it's gone. In mathematical terms, exponential decay can be represented by the formula:
\[ N(t) = N_0 e^{-kt} \]
where:

• \( N(t) \) is the quantity remaining after time \( t \).
• \( N_0 \) is the initial quantity.
• \( e \) is the base of natural logarithms.
• \( k \) is the decay constant, unique to each decaying substance.

What sets exponential decay apart from other types is the 'half-life', a constant time it takes for half the substance to decay - no matter how much you start with. So, if you're dealing with a pesky radioactive element or your savings account interest, exponential decay gives you the tools to predict how much will be left after a certain period.

Half-life Calculation

Calculating the half-life of a radioactive isotope is crucial for understanding how long it will stick around in the environment or a certain space. The half-life is the time required for half of the radioactive nuclei in a sample to decay. Akin to setting a timer for your favorite tea to steep - it's all about that perfect timing to know when it's done.
For example, Neptunium-237 has a half-life of 2.1 million years. This means that every 2.1 million years, the amount of Neptunium-237 will be reduced to half. If we start with 100%, after the first half-life, we'll have 50%, then 25% and so on. To calculate how much remains after a certain period, we use the initial amount and the number of half-lives passed, which we can express as:
\[ N(t) = N_0 \times \frac{1}{2}^{\frac{t}{t_{1/2}}} \]
where:

• \( N(t) \) is the remaining amount after time \( t \).
• \( N_0 \) is the initial amount.
• \( t_{1/2} \) is the half-life of the substance.

Understanding half-lives not only helps in labs but also in real-life scenarios like medical treatments with radioactive elements or figuring out the age of an archaeological find.

Neptunium-237

Dive into the world of radioisotopes and you'll meet Neptunium-237 (Np-237), a heavyweight from the periodic table's actinide series. With a lengthy half-life of 2.1 million years, this guy takes his time to bow out. Found as a by-product in nuclear reactors and in the making of plutonium, Np-237 is one element you'd track for the long haul.
Its significant half-life calls for careful lab and environmental management, as it quietly hangs around, unassumingly contributing to the nuclear waste scenario. The long half-life also makes Np-237 a good candidate for studying the mechanics of radioactive decay and for testing our understanding of long-term nuclear waste storage solutions. A neat fact about Np-237 is its potential use in advanced nuclear reactors and deep-space batteries - hands down, it's a multi-talented actinide.

Algebraic Modeling

Ever tried fitting the pieces of a puzzle together? That's what algebraic modeling is like—creating equations to represent real-world scenarios, like finding how much a piece of radioactive rock will weigh after a while or maxing out your high score in a video game.
In the case of radioactive decay, we stitch the initial mass, decay constant, and time together to form a nifty algebraic model that depicts the decay process. By plugging in values for time or mass, we can predict how much a radioactive isotope will be left at any given point. This magic of algebra makes things less of a guessing game and more of a calculated forecast. For instance, the algebraic model for Neptunium-237, based on its decay process, might look complex, but it hands you the power to predict its behavior over millennia with just a few numbers and an equation.
So, the next time you work through an algebraic model, remember that you're piecing together a puzzle that can unravel everything from ancient mysteries to futuristic technologies.