Question

A hypothetical radioactive isotope has a half-life of 10,000 years. If the ratio of radioactive parent to stable daughter product is \(1: 3,\) how old is the rock containing the radioactive material?

Short answer

The rock is 20,000 years old.

Step-by-step solution

Understanding the Ratio

The ratio of radioactive parent ( P ) to stable daughter ( D ) product is given as 1:3. This means for every part of parent isotope, there are 3 parts of daughter product. The total parts (parent + daughter) are 1 + 3 = 4 parts.

Using the Decay Formula

The decay of radioactive isotopes is exponential and can be described using the decay formula: \( \frac{N}{N_0} = \left(\frac{1}{2}\right)^n \) where \( N \) is the amount of radioactive substance at time \( t \), \( N_0 \) is the initial amount, and \( n \) is the number of half-lives that have passed. In this scenario, \( \frac{N}{N_0} = \frac{1}{4} \) because 1 out of 4 parts of the sample is radioactive parent.

Solving for Number of Half-lives

To find the number of half-lives (\( n \)) that have passed, set up the equation: \( \frac{1}{4} = \left(\frac{1}{2}\right)^n \). Solve using logarithms: \( n = \log_{0.5}(0.25) \). Thus, \( n = 2 \). This means 2 half-lives have passed.

Calculating the Age of the Rock

The age of the rock can be found by multiplying the number of half-lives (2) by the half-life of the isotope (10,000 years): \( \text{Age} = 2 \times 10,000 = 20,000 \ ext{years} \).

Key concepts

Radioactive Isotopes

Radioactive isotopes, often referred to as radioisotopes, are versions of chemical elements with unstable nuclei. They undergo radioactive decay to form stable isotopes. During this process, they emit radiation and slowly transform into a different nuclear configuration. This decay continues until a stable form is reached.
A natural abundance exists of isotopes within elements, and some isotopes are naturally radioactive. These isotopes are fundamental in areas like medicine, archaeology, and geology, particularly in radiometric dating, which helps in determining the age of rocks and fossils.
Understanding the behavior of radioactive isotopes provides insights into geological timelines and helps scientists estimate the age of materials that contain these isotopes. When a radioactive isotope decays, it transforms into a 'daughter' product, often another element. For example, the exercise gives a scenario where a radioactive parent isotope decays into three parts of a stable daughter product, which is crucial for calculating the age of geological samples.

Half-life

The concept of half-life is central to understanding radioactive decay. Half-life refers to the time it takes for half of a given amount of a radioactive isotope to decay into its daughter product. It's a consistent and measurable rate, unique for each isotope, and is a critical component in radiometric dating.
In the example given, the half-life of the radioactive isotope is 10,000 years. This means that every 10,000 years, half of the initial amount of the radioactive isotope would have decayed. If you start with a certain quantity, after one half-life, you'll have about 50% of the original isotope left.
This concept helps in understanding how ages of rocks are calculated. As the exercise illustrates, calculating the number of half-lives that have passed allows you to determine how old the rock is. Here, it was determined that two half-lives, or 20,000 years, have passed for the sample in question.

Decay Formula

The decay formula is mathematically expressed to calculate the remaining quantity of a radioactive isotope over time. It is written as \( \frac{N}{N_0} = \left( \frac{1}{2} \right)^n \), where:

• \( N \) represents the amount of radioactive isotope remaining at time \( t \).
• \( N_0 \) is the initial amount of the radioactive isotope.
• \( n \) stands for the number of half-lives that have elapsed.

In the exercise, it was observed that the ratio of remaining parent isotope to the total was \( \frac{1}{4} \). By setting this in the formula, \( \frac{1}{4} = \left( \frac{1}{2} \right)^n \), you can solve for \( n \) using logarithms.
This equation reveals the exponential nature of radioactive decay, showing how with each half-life, the isotope's quantity decreases by half. It's a powerful tool in dating geological formations, as demonstrated by calculating the age of the rock to be 20,000 years in this context.