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Chapter 14: Problem 19

If a radioactive isotope has a 6 -month half-life, what fraction will remain after 5 years?

Short Answer

Expert verified
After 5 years, \( \frac{1}{1024} \) of the original isotope remains.

Step by step solution

01

Determine the Number of Half-Lives

First, we need to determine how many half-lives pass in 5 years. Since the half-life of the isotope is 6 months, there are 2 half-lives in one year. Therefore, in 5 years, there will be:\[ 5 \text{ years} \times 2 \frac{\text{half-lives}}{1 \text{year}} = 10 \text{ half-lives} \]
02

Understand Half-Life Decay

Each half-life reduces the remaining amount of the substance to half its previous amount. This means if you start with an amount \( A_0 \), after one half-life you have \( \frac{A_0}{2} \), after two half-lives you have \( \frac{A_0}{2^2} \), and so on.
03

Calculate Fraction Remaining

Using the formula for half-life decay, the fraction of the isotopes that remain after \( n \) half-lives is \( \frac{1}{2^n} \). In this case, \( n = 10 \):\[ \text{Remaining fraction} = \frac{1}{2^{10}} = \frac{1}{1024} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Half-Life
The concept of "half-life" is essential in understanding radioactive decay. Half-life is the time it takes for half of the radioactive atoms in a sample to decay. This period can vary from a few milliseconds to billions of years, depending on the isotope. During each half-life:
  • 50% of the isotopes decay, leaving 50% remaining.
  • The next half-life reduces this remainder by half again, continuing this pattern exponentially.
Understanding the half-life allows scientists and students to predict how quickly a substance will decay.
This process is crucial in fields like archaeology, geology, and medicine, where dating techniques and radiation treatments rely on this principle.
Exponential Decay
Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value. In radioactive decay, this concept manifests as isotopes losing half their quantity over consistent time intervals, known as half-lives. Let's break it down:
  • The initial quantity decays, decreasing exponentially over time.
  • The formula used involves exponents to represent this decay process as the number of half-lives increase.
Given enough time, even a large sample will be reduced to a minute fraction of its original quantity. Mathematically, this is represented as:\[N(t) = N_0 \left(\frac{1}{2}\right)^n\]
Where:

- \(N(t)\) is the amount remaining.- \(N_0\) is the initial amount.- \(n\) is the number of half-lives.
This formula shows how radioactive substances decrease predictably over time.
Isotopes
Isotopes are variations of the same chemical element with differing numbers of neutrons but the same number of protons. This difference in neutron count does not affect the chemical properties significantly but can affect the stability of the nucleus. Radioactive isotopes are unstable and will decay over time, releasing radiation until they reach a more stable form.
  • Each element can have multiple isotopes with varying stability.
  • Stable isotopes do not undergo radioactive decay.
  • Unstable or radioactive isotopes are key in nuclear reactions and radiometric dating.
These isotopes have immense value in scientific research, medical applications like cancer radiotherapy, and even power generation through nuclear reactors. Understanding them allows us to harness nuclear energy and date ancient objects.

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Most popular questions from this chapter

If a radioactive isotope has a 1 -year half-life, what fraction will remain after 5 years?

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