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 that contains the radioactive material?

Short answer

The rock is 20,000 years old.

Step-by-step solution

Understanding the Half-Life

A half-life is the time required for half of the radioactive nuclei in a sample to decay to form a stable daughter product. In this case, the half-life of the isotope is 10,000 years.

Determine the Ratio in Present Terms

If the current ratio of the parent isotope to the daughter product is \(1:3\), it means that for every 1 part of the parent isotope, there are 3 parts of the daughter product.

Convert the Ratio to Decay Stages

A ratio of \(1:3\) means there are 4 parts in total: 1 part parent and 3 parts daughter. This indicates that the sample went through enough half-lives to result in \(\frac{1}{4}\) of the original parent parent isotope remaining.

Calculate the Half-Lives Elapsed

The fraction \(\frac{1}{4}\) is equal to \(\left(\frac{1}{2}\right)^2\). This indicates that 2 half-lives have passed.

Calculate the Age of the Rock

Each half-life is 10,000 years, so if 2 half-lives have passed, the age of the rock is \(2 \times 10,000 = 20,000\) years.

Key concepts

Half-Life Calculation

Understanding the concept of half-life is essential for radioactive decay problems. A half-life is the period during which half of a sample of a radioactive substance decays into a stable daughter product. This decay process continues until most of the radioactive material has transformed into a stable form. In simple terms, if you start with a certain amount of a radioactive isotope, half of that amount will decay over one half-life period.

Let's look at an example: If a radioactive isotope has a half-life of 10,000 years, after one half-life (10,000 years), only half of the original substance remains radioactive. After two half-lives (20,000 years), only one-quarter remains, and after three half-lives (30,000 years), only one-eighth remains. You can model this decay process using formulas like:

• The fraction remaining after n half-lives: \(\left(\frac{1}{2}\right)^n\)
• Total decayed material: \(1 - \left(\frac{1}{2}\right)^n\)

By understanding half-life, you can determine how long a sample has been decaying and subsequently deduce the age of the material.

Isotope Dating

Isotope dating, often known as radiometric dating, is a powerful tool used by scientists to determine the age of rocks and fossils using radioactive isotopes. This method relies on comparing the abundance of a parent isotope to its stable daughter product. As radioactive isotopes decay at a known rate (half-life), scientists measure the ratio of parent to daughter isotopes to estimate the time that has passed since the rock or material formed.

The key to isotope dating lies in establishing the initial ratio of parent to daughter isotopes in the sample. Once scientists have this information, they can apply the principles of half-life to assess how much time has passed. For instance, if a sample begins with 100% parent isotopes and zero daughter isotopes, and the current ratio reveals considerable decay, the age of the sample can be calculated by determining the number of half-lives that have elapsed.
Thus, the equation:

• Age of the sample = Number of half-lives elapsed × Half-life of the isotope

This approach allows for precise dating of geological events and formations.

Geochronology

Geochronology is the science of determining the age of rocks, minerals, and sediments. Utilized in both archaeology and geology, it provides a timeline for Earth’s history. Geochronology combines principles of radioactive decay, stratigraphy, and other techniques to date formations over vast timescales.

Key methods used in geochronology include:

• Radiometric dating, which determines ages based on radioactive decay (as discussed in isotope dating).
• Dendrochronology, dating tree rings to the exact year they were formed.
• Stratigraphy, examining rock layers (strata) for relative dating.

Geochronologists use these methods to construct chronological sequences and interpret Earth's history. By understanding geochronology, scientists can piece together geological and climatic changes, evolutionary shifts, and the timing of key events that shaped life on our planet. This field helps us appreciate the vastness of geological time scales and the processes that have influenced the Earth's evolution.