Question

A radioactive isotope has a half-life of 8 days. If today \(125 \mathrm{mg}\) is left over, what was its original weight 32 days earlier? (a) \(2 \mathrm{~g}\) (b) \(4 \mathrm{~g}\) (c) \(5 \mathrm{~g}\) (d) \(6 \mathrm{~g}\)

Short answer

The original weight was 2 grams.

Step-by-step solution

Understand the half-life concept

The half-life of a substance is the time it takes for half of it to decay. In this problem, the half-life of the radioactive isotope is 8 days.

Calculate the number of half-lives

Determine how many half-life periods have passed in the given time frame of 32 days. The formula for the number of half-lives is: \( n = \frac{t}{T_{1/2}} \), where \( t \) is time elapsed (32 days) and \( T_{1/2} \) is the half-life (8 days). So, \( n = \frac{32}{8} = 4 \).

Use exponential decay formula

The formula for radioactive decay is \( N = N_0 \times \left(\frac{1}{2}\right)^n \), where \( N \) is the remaining amount, \( N_0 \) is the original amount, and \( n \) is the number of half-lives. You know \( N = 125 \) mg and \( n = 4 \).

Solve for original weight

Rearrange the formula to solve for \( N_0 \): \( N_0 = \frac{N}{\left(\frac{1}{2}\right)^n} = \frac{125}{\left(\frac{1}{2}\right)^4} = 125 \times 2^4 = 125 \times 16 = 2000 \) mg. Convert \( 2000 \) mg to grams, which equals \( 2 \) grams.

Key concepts

Half-Life

The half-life of a substance is a fundamental concept in radioactive decay. It refers to the time required for half of a given amount of a radioactive substance to transform into a different form or decay product. For example, if you start with 100 grams of a radioactive substance, after one half-life, only 50 grams would remain.
This process is vital to understanding how substances change over time. In our exercise, the half-life of the isotope in question is 8 days. This means every 8 days, the amount of the radioactive material halves. After 16 days, you would have a quarter of the original amount left, and so on.
This systematic process allows scientists to predict the behavior of radioactive materials over time. It also helps in solving practical problems, like finding how much of a substance was present at an earlier time, given its decay over a certain period.

Exponential Decay

Exponential decay describes a process where the quantity of a substance decreases at a rate proportional to its current amount. This concept is elegantly captured in the radioactive decay formula: \[ N = N_0 \times \left(\frac{1}{2}\right)^n \]where:

• \( N \) is the remaining quantity of the substance.
• \( N_0 \) is the initial amount of the substance.
• \( n \) is the number of half-life periods that have passed.

In the example problem, you know the current amount remaining is 125 mg and that 4 half-life periods (as calculated by dividing 32 days by the half-life of 8 days) have passed. Using the formula, the exponential decay description allows us to work backward and find the initial amount. Therefore, you multiply the remaining amount by \( 2^4 \) to return to the original amount, effectively undoing the effect of the four half-lives.

Isotopes

Isotopes refer to atoms of the same element that have the same number of protons but different numbers of neutrons. This means they have different atomic masses. In the context of radioactive decay, isotopes play a crucial role, as not all isotopes of an element are stable.
Radioactive isotopes, also known as radioisotopes, are those that change over time by emitting radiation. This process alters the isotope into either a stable form or another radioactive isotope. In our example, the isotope we are dealing with is specifically engineered in a problem to illustrate decay via a predictable half-life period.
Isotopes have numerous applications, from medical treatments and diagnostic tests to dating archaeological artifacts. Understanding isotopes is essential as it forms a basis for studying how elements transform and how radioactive materials might behave in different environments.