Problem 5 - Biology Today and Tomorrow Without 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

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

Step 1: Understanding Half-Life

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

Step 2: Fraction Decayed

Since three-quarters of the radioisotope has decayed, it means that only one-quarter of the original amount remains. In terms of power of halves, this translates to \((\frac{1}{2})^2\), meaning 2 half-lives have passed.

Step 3: Calculate Age of the Sample

Given that 2 half-lives have occurred, and each half-life is 20,000 years, the age of the sample can be calculated by multiplying the number of half-lives by the half-life duration: \(2 \times 20,000 = 40,000\) years.

Key Concepts

half-life

Let's begin with understanding the concept of half-life. The half-life of a radioactive substance is the time required for half of it to decay into a new, more stable form. This process is consistent and predictable in many radioactive materials. For example, if the half-life of a radioisotope is 20,000 years, half of any given amount will decay within that time frame, no matter the quantity.
This decay process is exponential and not linear, meaning the time for each halving is the same, even as the quantity reduces further. This predictable timeframe allows scientists to use half-lives in various fields, from archaeology to medicine, to accurately date objects and calculate the remaining radioactive material.

radioisotope

Radioisotopes are simply isotopes that are radioactive. Isotopes are variants of elements with the same number of protons but different numbers of neutrons. What makes a radioisotope unique is its instability; it wants to reach a more stable state.
Because of this instability, radioisotopes emit particles and energy in the form of radiation, a process called decay. Use of radioisotopes is widespread in science and industry including:

• Carbon dating in archaeology.
• Tracers in biochemical research.
• Radiotherapy in medicine for cancer treatment.

Each radioisotope has its own distinct half-life, characteristic of its decay process, which helps scientists predict how quickly a sample will lose its radioactivity.

decay calculation

Calculating decay involves understanding the half-life concept and applying it to find out how old a radioactive sample might be. When you know how many half-lives have passed, you can determine the age of the sample.
For example, if three-quarters of a radioisotope has decayed, it means only one-quarter of the original amount remains. In terms of powers of two, that's \(\left(\frac{1}{2}\right)^{2}\), indicating two half-lives have passed.
Now, simply multiply the number of half-lives by the half-life duration. As in our problem: with a half-life of 20,000 years and two half-lives passed, the object is 40,000 years old. This calculation is crucial in fields like geology and biology where estimating the age of items or samples is important.