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 the Concept of Half-Life

The half-life of a substance is the time it takes for half of the radioactive parent isotopes to decay into stable daughter isotopes. Given in this problem, the half-life is 10,000 years.

Set up the Parent to Daughter Formula

The ratio of radioactive parent atoms to stable daughter atoms is given as 1:3. This implies there is 1 part parent isotope and 3 parts daughter isotope. The total parts are 1 + 3 = 4 parts.

Calculate the Fraction of Parent Isotopes Remaining

Since the ratio is 1:3, the remaining parent isotopes are 1 part out of a total of 4 parts. Thus, the fraction remaining is \( \frac{1}{4} \).

Use the Exponential Decay Formula

The decay of radioactive isotopes is expressed as \( N_t = N_0 (0.5)^{t/T} \), where \( N_t \) is the fraction remaining, \( N_0 \) is the initial amount, \( t \) is time, and \( T \) is the half-life. We solve for \( t \) such that \( \frac{1}{4} = (0.5)^{t/10000} \).

Solve the Equation

Set \( (0.5)^{t/10000} = \frac{1}{4} = (0.5)^2 \). Therefore, \( \frac{t}{10000} = 2 \). Solve for \( t \), we get \( t = 2 \times 10000 = 20000 \) years.

Key concepts

Half-Life

The half-life of a radioactive isotope is the time it takes for half of the original radioactive atoms, known as the parent isotopes, to decay into another form, often called the daughter isotopes. In the context of radioactive decay, understanding the half-life gives a reliable estimation of how quickly a particular substance undergoes this transformation. For example, if an isotope has a half-life of 10,000 years, it will take 10,000 years for half of the parent isotopes to transform into daughter isotopes.

This concept is crucial in calculating the age of an object containing a radioactive material. With every passing half-life, the amount of parent isotopes decreases by half, which allows scientists to unravel the mystery of ancient samples, like rocks, by observing the ratio of parent to daughter isotopes.

Parent Isotope

A parent isotope is the original radioactive isotope that undergoes decay. These isotopes are unstable and over time, they will transform into a more stable element known as the daughter isotope. In any radioactive decay process, the parent isotope is where the chain reaction begins and evolves.

In practical applications, such as determining the age of rocks, scientists look at the amount of remaining parent isotope compared to the amount that has decayed. For example, in the exercise provided, the ratio of parent to daughter isotopes is crucial to finding the age of the rock. With a ratio of 1:3, it implies a certain number of half-lives have elapsed since the rock was formed, which can be calculated using the exponential decay formula.

Daughter Isotope

The daughter isotope is the product of the radioactive decay of a parent isotope. These isotopes are typically more stable than their parent isotopes. As the parent isotopes decay over time, they eventually turn into daughter isotopes, which accumulates and provides clues about the age of the sample.

By examining the ratio of daughter to parent isotopes, scientists can make inferences about the history of a rock or any other object containing the radioactive material. In the exercise, if the daughter to parent ratio is 3:1, it indicates that a significant portion of the parent atoms has already decayed into daughter atoms, helping to determine that the age of the rock is 20,000 years.

Exponential Decay Formula

The exponential decay formula is a mathematical equation used to predict the amount of a radioactive substance that remains after a given period. It is written as \( N_t = N_0 (0.5)^{t/T} \), where:

• \( N_t \) is the remaining amount of the parent isotope.
• \( N_0 \) is the initial amount of the parent isotope.
• \( t \) is the time that has elapsed.
• \( T \) is the half-life of the substance.

This equation models the process of exponential decay, which simply means that the quantity decreases by a consistent percentage over regular intervals, specifically by half in this context. To find the age of a rock in our exercise, this formula solves for \( t \) using the known ratio of remaining parent isotopes, enabling the calculation of elapsed time since the rock formed. This allows scientists to date rocks and understand the history of our planet.