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

Identify the Concept

The problem is based on the concept of radioactive decay where a radioactive parent isotope decays into a stable daughter product. The half-life is the time required for half of a quantity of a radioactive isotope to decay.

Understand the Given Information

The half-life of the isotope is 10,000 years. The ratio of radioactive parent to stable daughter is 1:3.

Determine the Fraction of Parent Remaining

Since the ratio of the parent isotope to the daughter product is 1:3, the fraction of the parent isotope remaining is $\frac{1}{1+3}=\frac{1}{4}$.

Relate Remaining Fraction to Half-Life

The remaining fraction $\frac{1}{4}$ means that ${\left(\frac{1}{2}\right)}^n=\frac{1}{4}$, where $n$ is the number of half-lives that have elapsed. Therefore, $n=2$ since ${\left(\frac{1}{2}\right)}^2=\frac{1}{4}$.

Calculate the Age of the Rock

Each half-life is 10,000 years. Thus, if 2 half-lives have elapsed, the age of the rock is $2\times 10,000$ years.

Key concepts

Half-Life

The concept of half-life is an essential part of radioactive decay. It refers to the amount of time it takes for half of a sample of a radioactive isotope to decay into a stable state. For example, in the given exercise, the radioactive isotope has a half-life of 10,000 years. This means that at the end of 10,000 years, half of the original amount of the isotope would have decayed into a stable form.

Understanding half-life allows scientists to determine the age of rocks and fossils by observing how much of the radioactive isotope remains in a sample. In the exercise, we used the knowledge of half-life to find out that 2 half-lives have passed, equating to 20,000 years, by noticing that only a quarter of the radioactive parent isotope remains.

Some key points about half-life include:

• It is a fixed value unique to each radioactive isotope.
• Independent of the initial amount of the material.
• Used to estimate the decay over time and the age of materials.

By recognizing the decay pattern, you can estimate the age of geological samples, unlocking the history stored within them.

Radioactive Isotope

A radioactive isotope, also known as a radioisotope, is an atom that has excess nuclear energy, making it unstable. This instability prompts it to undergo radioactive decay, emitting radiation in the form of particles or electromagnetic waves, until it transforms into a stable isotope.

In the exercise, the hypothetical radioactive isotope undergoes such a decay process, resulting in the formation of a stable daughter product. This change is core to understanding how radioactive isotopes can be used as natural clocks to date ancient materials.

Characteristics of radioactive isotopes include:

• Their nuclei are unstable and will decay over time.
• They emit specific types of radiation such as alpha, beta, or gamma rays.
• Each has a distinct half-life which helps identify it.

Radioactive isotopes are invaluable for various applications, such as dating archeological finds, tracing chemical and biological processes, and even in medical treatments where precise control in decay is necessary.

Stable Daughter Product

During radioactive decay, a radioactive parent isotope transforms into a stable daughter product over time. This transformation process is critical in radioactive dating techniques, which utilize the known ratio of parent to daughter isotopes to determine ages.

In the exercise's context, the ratio of radioactive isotope to stable daughter product is 1 to 3. This indicates that a significant portion of the original material has decayed, leaving behind a copious amount of stable daughter product. This ratio is essential as it reveals the extent of decay and, subsequently, the age of the rock.

Key considerations about stable daughter products are:

• They mark the end of the decay process, being non-radioactive and stable.
• The ratio to parent isotopes provides information on the elapsed time since formation.
• The identification of product isotopes can help trace the decay pathway of the parent isotope.

Understanding stable daughter products allows us to "close the loop" in radioactive decay, using the measurement of these products to interpret the timeline of geological events accurately.