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

Understand the Problem

We have a radioactive isotope with a half-life of 10,000 years. The current ratio of the radioactive parent isotope to the stable daughter product is 1:3. We need to find out how old the rock containing this isotope is.

Set Up the Ratio Rules

The original parent isotope becomes daughter isotope over time. A 1:3 ratio means there is 1 part parent and 3 parts daughter. Thus, the total parts (1 parent + 3 daughter) is 4 parts.

Determine Original Parent Amount

Initially, all 4 parts would have been parent isotopes. Over time, 3 of these parts have decayed to become the daughter isotope. This leaves 1 part remaining as the parent, matching the given ratio (1:3).

Calculate Decay Using Half-life

The 1 remaining part is \(\frac{1}{4}\) of the original amount. Since one half-life reduces the amount to \(\frac{1}{2}\), we consider that it requires two half-lives to reduce from \(1\) (original) to \(\frac{1}{4}\).

Calculate the Age of the Rock

Each half-life is 10,000 years. Since it took two half-lives to reach \(\frac{1}{4}\) of the original, the age of the rock is \(2 \, \times 10,000 = 20,000\) years.

Key concepts

Half-Life

The concept of half-life is central to understanding radiometric dating. Half-life is the amount of time it takes for half of a radioactive sample to decay into a more stable form. Imagine you have a pile of dominoes, and every hour half of them fall over.
After one hour, only half are left standing. After another hour, half of those remaining fall, and so on.
Understanding half-life helps us interpret how long a specific isotope has been decaying.

• Half-life is constant for a given isotope.
• It's independent of the initial amount of material.
• For our example, the isotope has a half-life of 10,000 years.

This means that every 10,000 years, the amount of parent isotope reduces to half, crucial for calculating the age of materials like rocks.

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. During this process, the parent isotope transforms into a daughter isotope. Think of it like a sand timer, where the sand (parent isotope) flows from the top bulb to the bottom bulb (daughter isotope).
The rate of this flow is what we describe as radioactive decay. Each type of isotope has a unique half-life, which dictates the speed of this transformation.

• Radioactive decay is a random, yet predictable process.
• It results in the formation of one or more different elements (daughter products).
• Decay rates allow the estimation of time elapsed since the formation of the parent isotope.

By knowing how much parent isotope remains, scientists can estimate the time passed since the rock was formed.

Parent to Daughter Ratio

The parent to daughter ratio in a radioactive material reveals significant information about its age. When a rock forms, it starts with a certain amount of radioactive parent isotopes. Over time, these isotopes decay into daughter isotopes.
By assessing the ratio of remaining parent isotopes to the produced daughter isotopes, we can estimate the time elapsed. In our exercise, the given ratio of 1:3 indicates that one part of the original parent isotope remains, while three parts have decayed into the daughter isotope.

• Initial ratio is always parent to zero daughter isotopes.
• The ratio grows over time as decay proceeds.
• A 1:3 ratio implies \(\frac{1}{4}\) of the original parent isotope remains.

Understanding this ratio is crucial as it directly aids in calculating the age of the material using the known half-life.