Question

If a radiometric element has a half-life of 425 years, how old would a rock be that only had \(3.125 \%\) of the parent isotope remaining? a. 2125 years b. 1700 years c. 2550 years d. 3400 years

Short answer

The rock is 2125 years old.

Step-by-step solution

Understand Half-life of a Radioactive Isotope

The half-life of an isotope is the time it takes for half of the isotope to decay. In this case, we know that the isotope has a half-life of 425 years.

Calculate the Number of Half-Lives

Since the rock has 3.125% of the parent isotope remaining, we need to find out how many half-lives are required to reach this percentage. Recognize that 3.125% is equivalent to \(\frac{3.125}{100} = \frac{1}{32}\) of the original quantity.

Determine Number of Half-Lives

By observing powers of 2, we know that \(\frac{1}{32} = \left(\frac{1}{2}\right)^5\). This means that 5 half-lives are needed to reduce the isotope to \(\frac{1}{32}\) of its original amount, or 3.125% remaining.

Calculate the Age of the Rock

Multiply the number of half-lives (5) by the duration of one half-life (425 years). This gives the age of the rock as: \[5 \times 425 = 2125 \text{ years}\].

Key concepts

Half-Life Calculation

Understanding the half-life of a radioactive isotope is crucial for determining the age of certain geological materials. The half-life is the time it takes for half of the radioactive atoms in a sample to decay. This concept helps us measure the decay process and calculate how long it has been happening.
For instance, if you start with a 100% amount of a radioactive element, after one half-life you will have 50% of it left. After two half-lives, it reduces to 25%, three half-lives bring it to 12.5%, and so on. This exponential decay process continues until the material virtually disappears.
When calculating the age of a rock, you determine how many half-lives it took to reach the current state. Multiply the number of half-lives by the length of one half-life to find the total duration. This method allows scientists to make accurate estimates of the age of ancient materials, enhancing our understanding of Earth's history.

Radioactive Isotopes

Radioactive isotopes are unstable atoms that emit radiation as they decay into stable forms. These isotopes are integral to the process of radiometric dating, which estimates the age of materials by gauging the ratio of remaining unstable isotopes to stable products.
Different isotopes decay at different rates, characterized by their distinct half-lives. Some decay quickly, within days or months, like certain medical tracers, while others take millions or even billions of years, like the isotopes used for dating rocks.
The choice of isotope for dating depends on the material's age and the type of rock or mineral it's contained in. Uranium-lead dating is useful for the oldest materials, while carbon-14 dating applies to much younger archaeological finds.
Radioactive isotopes offer a window into the past, revealing not just the age of materials but also insights into the chemical and physical conditions of the time they formed.

Geological Time Scale

The geological time scale is a system that organizes Earth's history into different time periods based on geological and paleontological events. It helps geologists and paleontologists understand the chronological order of past events and the evolution of life on Earth.
This scale is divided into eons, eras, periods, epochs, and ages. Each segment represents significant geological or biological changes, like mass extinctions or major climate shifts.
Radiometric dating, using radioactive isotopes, plays an essential role in defining these time units. As isotopes decay at known rates, they provide clocks that help place geological events in chronological order.
By understanding the geological time scale and using radiometric dating, scientists can piece together the history of Earth, offering insights into its dynamic nature and the evolutionary traject of life.