Question

How much of the parent isotope would be remaining after 7 half-lives have passed? a. \(6.25 \%\) b. \(1.56 \%\) c. \(0.78 \%\) d. \(0.39 \%\)

Short answer

After 7 half-lives, 0.78% of the parent isotope remains (option c).

Step-by-step solution

Understand Half-life Concept

The concept of half-life refers to the time it takes for half of the radioactive isotope to decay. After each half-life, the amount of the parent isotope is reduced by half.

Calculate Remaining Fraction After 7 Half-Lives

The remaining fraction of the parent isotope after \( n \) half-lives can be calculated using the formula \( \left( \frac{1}{2} \right)^n \). In this case, \( n = 7 \), so the remaining fraction is \( \left( \frac{1}{2} \right)^7 = \frac{1}{128} \).

Convert Fraction to Percentage

To convert the remaining fraction into a percentage, multiply by 100. Therefore, \( \frac{1}{128} \times 100 \% \approx 0.78 \% \).

Select the Correct Answer

Based on the previous calculations, we find that after 7 half-lives, approximately \(0.78\%\) of the parent isotope remains, which corresponds to option c.

Key concepts

Understanding Half-Life

Half-life is a crucial concept in radioactive decay processes. Imagine you have a certain amount of a radioactive substance. The half-life is the time it takes for exactly half of that substance to decay. For instance, if the half-life of a substance is 10 years, then in 10 years, only half of the original substance remains. This process continues successively. So, after one half-life, 50% remains, after two half-lives, 25% remains, and so on. Understanding half-lives helps predict how long a substance will remain active and to plan for safe disposal times.
By comprehending the concept, you can predict the reduction of radioactive material over time, which is especially useful in fields like archaeology, geology, and medicine.

Parent Isotope Explained

In radioactive decay, we often talk about the parent isotope. This is the original radioactive substance that undergoes decay to transform into a different atom, known as the daughter isotope.

• The parent isotope is crucial because its decay rate is predictable based on its half-life.
• Knowing the amount of parent isotope helps scientists estimate age or duration in processes like radiometric dating.

Imagine a block of uranium. Uranium is the parent isotope, and over time, it decays into lead, the daughter isotope. Tracking how much of the parent isotope remains over time enables scientists to understand geological and archaeological time scales.

Percentage Calculation in Decay Problems

Calculating percentages in decay problems is essential. It helps translate mathematical fractions into easily understandable numbers. After a series of half-lives, you often need to know what percentage of the initial substance remains.

• To calculate remaining percentages, use the formula: Remaining Fraction = \( \left( \frac{1}{2} \right)^n \) where \( n \) is the number of half-lives.
• For clearer understanding, convert this fraction into a percentage by multiplying by 100.

For example, if \( \left( \frac{1}{2} \right)^7 = \frac{1}{128} \), then \( \frac{1}{128} \times 100 = 0.78\% \). This means that after 7 half-lives, only 0.78% of the parent isotope remains. Converting these fractions into percentages aids in grasping how much of the material lasts over time, making predictions straightforward and clear for practical applications.