- The half-life \(t_{1/2}\) represents the duration of one decay cycle.
- The total age is the product of the number of cycles (\(n\)) and the duration of each cycle (\(t_{1/2}\)).
- The relationship is defined by \(\text{Age} = n \times t_{1/2}\).
- The correct formula is \(\boxed{\text{Age of object} = n \times t_{\frac{1}{2}}}\).
Explanation
- Defining the Variables and Relationship
To determine the age of a radioactive sample using its half-life, we first define the core components involved. The half-life, denoted as \(t_{1/2}\), is defined as the time required for half of a radioactive substance to decay. The number of half-lives that have elapsed, denoted by \(n\), represents how many such decay cycles have passed since the sample was initially created or isolated.
- Formulating the Equation
The total age of the object is the elapsed time since its formation. Since each individual half-life period has a duration of \(t_{1/2}\) and a total of \(n\) such periods have passed, the total time (or age) is the product of the number of periods and the duration of each period. This is expressed mathematically as:
\[\text{Age} = n \times t_{1/2}\]
This formula calculates the total duration by summing up all the elapsed half-life intervals.
- Selecting the Correct Formula
Comparing our derived formula, \(\text{Age} = n \times t_{1/2}\), with the provided options:
- \(\text{Age} = \frac{t_{1/2}}{n}\)
- \(\text{Age} = \frac{n}{\frac{t_{1/2}}{t^2}}\)
- \(\text{Age} = n \times t_{1/2}\)
- \(\text{Age} = n + t_{1/2}\)
The third option matches our derived relationship perfectly.
- Conclusion
The age of a sample is calculated by multiplying the number of elapsed half-lives by the duration of one half-life. Therefore, the correct formula is \(\boxed{\text{Age of object} = n \times t_{\frac{1}{2}}}\).
Examples
Radioactive dating is a vital tool in geology and archaeology for determining the age of ancient materials. For example, if a bone sample is found to have undergone 3 half-lives of Carbon-14, and the half-life of Carbon-14 is approximately 5,730 years, we can calculate the age of the bone using the derived formula: \(\text{Age} = 3 \times 5730 = 17,190 \text{ years}\). This method allows scientists to place historical artifacts and geological formations into a chronological timeline accurately.