Problem 77 - Introductory Chemistry Atoms First

Question

Suppose you have \(100 \mathrm{~g}\) of \({ }_{53}^{123} \mathrm{I}\). How much of it will be left after \(26.2 \mathrm{~h}\) ? After \(39.3 \mathrm{~h}\) ? [The half-life of \({ }_{53}^{123} \mathrm{I}\) is \(13.1 \mathrm{~h}\).]

Step-by-Step Solution

Answer

After 26.2 hours, 25 grams of the radioactive isotope will be left, and after 39.3 hours, 12.5 grams of the radioactive isotope will be left.

Step 1: Write down the given information

We are given: - Initial amount of the radioactive isotope: \(100 \mathrm{~g}\) - Time period 1: \(26.2 \mathrm{~h}\) - Time period 2: \(39.3 \mathrm{~h}\) - Half-life: \(13.1 \mathrm{~h}\)

Step 2: Use the decay formula

The decay formula is given by: \(A(t) = A_0 \cdot (1/2)^{\frac{t}{T}}\) where: - \(A(t)\) is the remaining amount of the radioactive substance at time \(t\), - \(A_0\) is the initial amount of the radioactive substance, - \(T\) is the half-life of the radioactive substance, and - \(t\) is the time that has passed. For our problem, \(A_0 = 100\) and \(T = 13.1\).

Step 3: Find the remaining amount for time period 1

Plug in the values for time period 1: \(A(26.2) = 100 \cdot (1/2)^{\frac{26.2}{13.1}}\) \(A(26.2) = 100 \cdot (1/2)^{2}\) \(A(26.2) = 100 \cdot (1/4)\) \(A(26.2) = 25 \mathrm{~g}\) So, after 26.2 hours, 25 grams of the radioactive isotope will be left.

Step 4: Find the remaining amount for time period 2

Plug in the values for time period 2: \(A(39.3) = 100 \cdot (1/2)^{\frac{39.3}{13.1}}\) \(A(39.3) = 100 \cdot (1/2)^{3}\) \(A(39.3) = 100 \cdot (1/8)\) \(A(39.3) = 12.5 \mathrm{~g}\) So, after 39.3 hours, 12.5 grams of the radioactive isotope will be left. To summarize, after 26.2 hours, 25 grams of the radioactive isotope will be left, and after 39.3 hours, 12.5 grams of the radioactive isotope will be left.

Key Concepts

Understanding Half-Life

The concept of half-life is key to understanding radioactive decay. It represents the time it takes for half of a radioactive substance to transform into another form or element. This transformation process is constant, meaning that the half-life remains the same no matter how much of the substance is present.

In the given problem, the half-life of iodine isotope _{53}^{123}I is 13.1 hours. This means every 13.1 hours, half of the iodine decays.

If you start with 100 grams, you'll have 50 grams after 13.1 hours. Understanding half-life helps predict how long a substance will remain active.

Exploring the Decay Formula

The decay formula allows us to calculate the remaining amount of a radioactive substance over time. It is expressed as: \[ A(t) = A_0 \cdot \left(\frac{1}{2}\right)^{\frac{t}{T}} \] Here:

• \(A(t)\) is the amount left after time \(t\).
• \(A_0\) is the initial quantity.
• \(T\) is the half-life.

This formula works by multiplying the initial amount by the fraction \((1/2)^{t/T}\), reducing it by half-life intervals. It's a reliable tool for understanding how quickly a substance will diminish.

Radioactive Isotopes Explained

Radioactive isotopes are atoms with unstable nuclei that release radiation as they decay. They are present in various elements and used in medical treatments and industries.

The iodine isotope _{53}^{123}I is an example. It decays steadily, adhering to its half-life of 13.1 hours.

This predictable decay is crucial for applications where precise timing matters, such as medical imaging or treatment. Understanding isotopes helps grasp how different atoms behave and their potential uses.

Calculating Time Periods

Calculating the time period during which a radioactive substance decays involves assessing how many half-lives have passed.

For the provided problem, you need to determine how many half-lives fit into the given time periods (26.2 and 39.3 hours).

• In 26.2 hours, about 2 half-lives (\(26.2/13.1\)) occur.
• In 39.3 hours, about 3 half-lives (\(39.3/13.1\)) occur.

These calculations help predict the remaining amount of substance using the decay formula. This method reveals how fast a substance decays, vital for planning and safety measures.