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

Understand Half-Life Concept

In this problem, the half-life given is 10,000 years. This means after every 10,000 years, half of the radioactive isotope remains, while the other half decays into the stable daughter product.

Initial Understanding of Ratio

The problem states that the ratio of radioactive parent isotope to stable daughter product is 1:3. This indicates that out of a total of 4 parts, 1 part remains as the parent isotope and 3 parts have turned into the daughter isotope.

Set Up the Initial Amounts

Assume initially there were 4 parts of the radioactive material. After some time, 1 part remains as the radioactive parent, implying 3 parts have decayed.

Calculate Number of Half-Lives

Initially, there were 4 parts of the parent isotope. After each half-life, the amount of radioactive material is halved. Thus: - After 1 half-life: 2 parts would remain. - After 2 half-lives: 1 part would remain. Therefore, 2 half-lives are required to reduce from 4 parts to 1 part.

Calculate Age of the Rock

Since one half-life is 10,000 years, and it took 2 half-lives for the ratio to become 1:3, the rock is 2 half-lives old.Therefore, the age of the rock is:\(2 \times 10,000 = 20,000\) years.

Key concepts

Understanding Half-Life

Half-life is the time required for half of a sample of a radioactive substance to decay into its stable form. This transformation occurs as atoms of the radioactive isotope release energy, transforming into the stable daughter product. Understanding half-life involves grasping that with each passing half-life, the quantity of the radioactive isotope left in the material decreases by half. For example, if a sample initially contains 100 atoms of a radioactive isotope, after one half-life only 50 atoms would remain. After two half-lives, only 25 atoms would be left. Each isotope has a unique half-life, and it remains consistent despite changes in chemical conditions. Knowing the half-life helps scientists calculate the age of rocks or fossils and provides insight into the duration of geological processes.

Role of Radioactive Isotopes

Radioactive isotopes are unstable nuclei that decay into a more stable form over time. This decay process releases energy in the form of radiation. Each radioactive isotope decays at its own fixed and predictable rate, which is measured in terms of its half-life. In geology and archaeology, radioactive isotopes play a crucial role in dating ancient materials. By studying the ratio of remaining radioactive isotope to the amount of stable daughter product, scientists can estimate the age of the material. This is a key aspect of techniques like radiometric dating. Some commonly used radioactive isotopes in dating include Carbon-14, Uranium-238, and Potassium-40. Each of these isotopes is effective for dating different types and ages of materials due to their varying half-lives.

Formation of Stable Daughter Products

When a radioactive isotope decays, it transforms into a stable daughter product. This occurs as the unstable nucleus changes its structure, often by emitting particles and energy. The stable daughter product no longer undergoes radioactive decay, thus marking the endpoint of the decay process for that particular isotope. In radioactive dating, the formation of stable daughter products helps in calculating the elapsed time since the formation of the rock. By measuring the ratio of the remaining radioactive isotope to the stable product, scientists can gauge how many half-lives have passed, thereby estimating the age of the rock.

Determining the Age of Rocks

To find out the age of a rock containing radioactive material, scientists use the principles of radioactive decay. Using the known half-life of the radioactive isotope, they calculate how many half-lives have elapsed. First, the ratio of the radioactive isotope to the stable daughter product in the rock sample is determined. For example, a ratio of 1:3 indicates that one-quarter of the original radioactive isotope remains. Next, scientists calculate the number of half-lives needed to reach the current ratio. Once this number is found, it is multiplied by the half-life duration to date the rock's formation. Using this method accurately helps unravel the history of the Earth's geology, allowing for a timeline of events that shaped the planet to be established.