Problem 78 - A Graphical Approach to College Algebra

Question

Half-Life of a Radioactive Substance The half-life of a radioactive substance is the time it takes for half the substance to decay. Suppose the half-life of a substance is 3 years and \(10^{15}\) molecules of the substance are present initially. How many molecules will be present after 15 years?

Step-by-Step Solution

Answer

Approximately \(3.125 \times 10^{13}\) molecules will be present after 15 years.

Step 1: Understanding Half-Life Concept

The half-life of a radioactive substance is the time required for half of its quantity to decay. In this case, the half-life is 3 years.

Step 2: Initial Quantities

Initially, there are \(10^{15}\) molecules of the substance present.

Step 3: Determine Number of Half-Lives

Calculate how many half-lives occur in 15 years. Since each half-life is 3 years, in 15 years there are \(\frac{15}{3} = 5\) half-lives.

Step 4: Apply Half-Life Decay Formula

After each half-life, the quantity of the substance is halved. The formula to find the remaining amount after \(n\) half-lives is:\[N = N_0 \left( \frac{1}{2} \right)^n\]where \(N_0\) is the initial amount, \(N\) is the remaining amount, and \(n\) is the number of half-lives.

Step 5: Calculate Remaining Molecules

Substitute the values into the formula:\[N = 10^{15} \times \left( \frac{1}{2} \right)^5 = 10^{15} \times \frac{1}{32}\]Calculate \(10^{15} \div 32\) to find \(N\).

Step 6: Compute the Final Answer

\(10^{15} \div 32 \approx 3.125 \times 10^{13}\). Therefore, approximately \(3.125 \times 10^{13}\) molecules will remain after 15 years.

Key Concepts

Half-Life Calculation

Half-life is a fascinating concept in radioactive decay. It's the period it takes for half of a radioactive substance to decay into another element or isotope. This proces is predictable and allows us to calculate how much of a substance remains after a certain time. For instance, if a substance has a half-life of 3 years, in 3 years half of it will decay. If you start with 100 grams, after 3 years, you'll have 50 grams left, and after 6 years, just 25 grams.

An important formula used in half-life calculations is:

• \[N = N_0 \left( \frac{1}{2} \right)^n\]

Here:

• \(N_0\) is the initial quantity of the substance,
• \(N\) is the remaining quantity after time \(t\),
• \(n\) is the number of half-lives elapsed.

If you break down the math, it is simple: divide the total time by the half-life to find \(n\). Then plug \(n\) into the formula with your initial quantity, and you've solved how much remains!

Exponential Decay

Exponential decay is a process where quantities decrease rapidly at first and then slow over time. It's like watching a fast sand hourglass where at first, the sand seems to pour quickly, but as time progresses, the falling sand appears to take longer due to the decreasing amount.In radioactive decay, we often witness this process. The key mathematical representation of exponential decay in this context is:

• \[N = N_0 e^{-\lambda t}\]

Here, \(\lambda\) is the decay constant, related to the half-life, and \(e\) is the base of the natural logarithm. For many practical purposes, half-life discussions use the simpler half-life formula, but this equation helps understand more continuous decay.This continuous aspect means that after a certain period, not only half but a fixed percentage of the remaining sample will always decay, demonstrating the relentless and predictable nature of exponential processes.

Radioactive Substances

Radioactive substances are materials that contain unstable nuclei. These nuclei lose energy abundantly by emitting radiation in the form of particles or electromagnetic waves. This process is spontaneous and eventually leads to the formation of more stable elements or isotopes. Understanding radioactive substances is crucial because they have varied applications and implications:

• Used in medical treatments like cancer radiation therapy.
• Essential for power generation in nuclear reactors.
• Utilized in dating methods, such as carbon dating, to determine the age of archaeological finds.

However, it's important to handle them carefully as they can be hazardous to health. The radiation emitted can cause damage to living tissues which is why safety precautions are paramount. By calculating half-lives and understanding decay, scientists can handle these substances safely and effectively.