Problem 7 - Biology Today and Tomorrow with Physiology

Question

If the half-life of a radioisotope is 20,000 years, then a sample in which three-quarters of that radioisotope has decayed is _______ years old. a. 15,000 b. 26,667 \(c .30,000\) d. 40,000

Step-by-Step Solution

Answer

d. 40,000 years

Step 1: Understand the definition of half-life

A half-life is the time required for a quantity to reduce to half its initial value. For a radioisotope, it is the time taken for half of the radioactive atoms to decay.

Step 2: Calculate the number of half-lives for three-quarters decay

If three-quarters of the radioisotope has decayed, it means one-quarter of the original amount remains. This is equivalent to saying that the original quantity has halved once (leaving one-half), and then halved again (from one-half to one-quarter). Thus, the number of half-lives that have passed is 2.

Step 3: Calculate the sample age

Each half-life for this radioisotope is 20,000 years. Since 2 half-lives have passed, the age of the sample is \(2 \times 20,000 = 40,000\) years.

Key Concepts

Radioisotope Decay

Radioisotopes are atoms with unstable nuclei that lose their excess energy through a process known as decay. Over time, these atoms transform into different elements or isotopes that are stable. This behavior is central to understanding radioisotope behavior and its applications in fields like medical diagnostics and environmental dating, such as carbon dating.
Radioactive decay is a random process at the level of single atoms, meaning it's impossible to predict when a particular atom will decay. However, on a statistical level we can predict the decay of a large number of atoms, which is expressed in terms of half-life. The half-life of a radioisotope is the time it takes for half of the radioactive atoms in a sample to decay. This concept helps us to gauge the age or growth of materials over large periods of time.

• Radioisotopes transform into more stable forms over time.
• The process is predictable statistically, based on the concept of half-life.
• This principle is useful in various scientific and industrial applications.

Radioactive Decay Calculation

Calculating radioactive decay involves using the half-life information to determine how long a process has been occurring. For instance, if you know a material's half-life, you can determine how long it has been decaying based on how much remains.
In the given exercise, three-quarters of a radioisotope has decayed, which implies one-quarter remains intact. To determine the age, we must understand how many half-lives it takes to reach one-quarter of the original amount.

• Each half-life decreases the remaining quantity by half.
• From full to one-half is one half-life, and from one-half to one-quarter is another.
• Thus, two half-lives mean the sample has been around for twice the half-life duration.

To find the age of the sample when two half-lives have passed, we multiply the half-life (20,000 years in this case) by 2: \[\text{Age} = 2 \times 20,000 = 40,000 \text{ years}\].

Nuclear Physics Concepts

Nuclear physics helps to explain the behavior of radioactive materials. It encompasses principles such as nuclear decay, half-life, and atomic transformations. At its core, nuclear physics explores how unstable atoms release energy to become stable. By studying this, we can better understand phenomena within various scientific disciplines.
The nucleus of an atom holds protons and neutrons together with significant forces. However, when it comes to radioisotopes, these nuclei can become unstable due to an imbalance of these internal components. To reach stability, energy is released in the form of radiation, a process known as decay.

• Nuclear decay: Process of releasing energy to stabilize an atom.
• Half-life: Time needed for half of a radioactive sample to decay.
• Applications: Used widely in medicine, archaeology, and more.

Understanding these concepts can illuminate how elements transform over time and offer insights into both ancient and modern processes. Nuclear physics not only allows precise age determination, but its underlying principles also enable advancements in both research and practical applications.