Question

The half-life of a radioactive isotope is 500 million years. Scientists testing a rock sample discover that the sample contains three times as many daughter atoms as parent isotopes. What is the age of the rock? a) 500 million years c) 1,000 million years b) 1,500 million years d) 2,500 million years

Short answer

The age of the rock is 1,000 million years.

Step-by-step solution

Understanding Half-Life

The half-life of a radioactive isotope is the time required for half of the parent isotopes in a sample to decay into daughter isotopes. In this case, the half-life is 500 million years.

Setting Up the Ratio

The problem states that there are three times as many daughter isotopes as parent isotopes. This means for every 1 remaining parent isotope, there are 3 daughter isotopes, making the total number of original parent isotopes 4. Therefore, the ratio of parent to daughter isotopes is 1:3.

Calculating the Number of Half-Lives

At the start (0 half-lives), there are only parent isotopes. After 1 half-life, there are twice as many parent isotopes as daughter isotopes (1:1 ratio). After 2 half-lives, the parent isotopes are equal to a quarter of the total (1:3 ratio), which matches the current situation. So, the rock sample has undergone 2 half-lives.

Calculating the Age of the Rock

Since each half-life is 500 million years and the rock sample has undergone 2 half-lives, the age of the rock is \[ 2 \times 500 \text{ million years} = 1000 \text{ million years}. \]

Choosing the Correct Answer

Based on our calculation, the age of the rock is 1,000 million years, which corresponds to option c).

Key concepts

Understanding Radioactive Decay

Radioactive decay is a natural process where unstable atomic nuclei lose energy by emitting radiation. This process transforms the parent isotope into a stable daughter isotope. The time it takes for half of the parent isotopes to convert into daughter isotopes is known as the half-life. Each radioactive isotope has a characteristic half-life, which is a constant value. It is crucial for understanding how long decaying processes take and is fundamental in calculating ages of rocks and other geological samples. As parent isotopes decay, the quantity of daughter isotopes increases, which allows scientists to estimate the age of materials by comparing these amounts.

Significance of Isotope Ratios

Isotope ratios are critical in understanding radioactive decay. They represent the proportions between parent and daughter isotopes in a given sample. For geologists, these ratios serve as a clock, helping to determine the age of rocks or fossils. In the original exercise, a ratio of 1:3 indicates that for every one parent isotope, there are three daughter isotopes. This suggests multiple half-lives have passed. Initially, there are no daughter isotopes, and over time, as more parent isotopes decay, the number of daughter isotopes increases relative to the remaining parent isotopes. Calculating these ratios helps scientists figure out how many half-lives have elapsed since the rock first formed.

Geochronology and Dating Methods

Geochronology is the science of determining the age of rocks, sediments, and fossils using various dating methods. One of the most common methods is radiometric dating, which uses the principle of radioactive decay and isotope ratios to estimate ages. In this method, scientists analyze the presence of parent and daughter isotopes within a sample. By understanding the half-life and current isotope ratio, they can back-calculate to determine how long the rock has existed. For example, knowing the isotope ratio in a rock and the half-life of the isotopes, as demonstrated in the exercise, provides an age calculation of 1,000 million years for the rock sample. This technique offers a precise dating mechanism, essential for constructing the geological history of Earth.