Problem 6 - Biology Concepts and Applications

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

The sample is 40,000 years old. (Option d)

Step 1: Understand the Concept of Half-Life

The half-life of a substance is the time it takes for half of a sample to decay. In this problem, the half-life is given as 20,000 years.

Step 2: Analyze the Problem Context

Since three-quarters of the sample has decayed, only one-quarter of the sample remains. This implies the sample has gone through multiple half-life periods.

Step 3: Calculate the Number of Half-Lives

If one-quarter of the original sample remains, the sample has gone through two half-life periods. After one half-life, half remains; after two half-lives, one-quarter remains.

Step 4: Compute the Sample's Age

Since each half-life is 20,000 years and the sample has gone through two half-lives to reach one-quarter of the original amount, multiply: \(2 \times 20,000 = 40,000\) years.

Key Concepts

Half-Life

When discussing radioactive decay, the concept of half-life is central. It represents the time it takes for half of a given amount of a radioisotope to decay.
Imagine you start with 100 grams of a substance with a half-life of 20,000 years. In 20,000 years, only 50 grams would remain.

• After another 20,000 years (a total of 40,000 years), the amount would reduce to 25 grams.
• This progressive reduction continues as time passes.

Understanding half-life allows you to predict how long a substance will take to decrease to a certain amount.
It is crucial in fields like archaeology and geology, where determining the age of artifacts or geological samples depends on this concept.

Radioisotope

A radioisotope is a version of an element that has an unstable nucleus.
This instability leads to radioactive decay, where the nucleus loses energy by emitting radiation.

• Each radioisotope has a unique half-life that determines how quickly it decays.
• Some have half-lives of seconds, while others can span millions of years.

Radioisotopes are used in many practical applications, including medical treatments, power generation, and scientific research.
In dating techniques, radioisotopes like carbon-14 are vital in determining the age of ancient organic materials.
Understanding the behavior of radioisotopes helps scientists accurately estimate time frames in which historical or geological events occurred.

Age Calculation

Calculating the age of a sample using radioactive decay involves understanding how much of the radioisotope has decayed over time.
If a known fraction of the sample has decayed, you can compute its age by determining how many half-lives have passed.

• For example, if three-quarters of a sample has decayed, it means two half-lives have elapsed.
• This is because after one half-life, only half remains, and after the second half-life, that remaining half decays by half again, leaving one-quarter.

Using this information, simply multiply the number of half-lives by the length of each half-life.
In the case of the exercise above, since two half-lives of 20,000 years have passed because three-quarters of the sample has decayed, the sample is 40,000 years old. Age calculation through these means provides insight into the history of ancient objects and timelines of natural events.