Question

I-131 has a half-life of \(8.04\) days. If a sample initially contains \(30.0\) g of I-131, approximately how much I131 will be left after 32 days?

Short answer

After 32 days, approximately 1.875 g of I-131 will be left.

Step-by-step solution

Understanding the Half-Life Concept

The half-life of a radioactive isotope is the time it takes for half of the radioactive atoms in a sample to decay. In this case, the half-life of I-131 is 8.04 days.

Calculate the Number of Half-Lives

To find the number of half-lives that will have passed after 32 days, divide the total time elapsed by the half-life period: Number of half-lives = Total time / Half-life = 32 days / 8.04 days.

Perform the Calculation

The result from Step 2 will be the number of half-lives that have passed. To find the remaining amount of I-131, we use the formula: Remaining amount = Initial amount * (1/2)^(Number of half-lives).

Finding the Remaining Amount of I-131

Substitute the initial amount of I-131 and the number of half-lives into the formula to calculate the remaining I-131: Remaining amount = 30.0 g * (1/2)^(Number of half-lives).

Key concepts

Radioactive Isotope Decay

Radioactive isotopes, also known as radioisotopes, are atoms with an unstable nucleus that change into a more stable form by emitting radiation in a process called radioactive decay. This decay occurs because the atomic configuration is not energetically favorable, and through the process, the isotope emits particles such as alpha particles, beta particles, or gamma rays.

The rate at which a radioactive isotope decays is measured by its half-life, which is the time it takes for half of the radioactive atoms present in a sample to decay into a different isotope or element. This property is intrinsic to each isotope, meaning that regardless of the amount of the substance, the half-life remains constant. For example, if you start with a certain mass of a radioactive isotope, after one half-life, only half of that mass will remain unchanged; the other half will have decayed into another material.

Half-Life Calculation

Calculating the half-life of a radioactive substance is a key step in understanding how radioactive decay behaves over time. It is both a useful tool in scientific contexts, and has practical implications in fields such as medicine, archaeology, and environmental science. To calculate the half-life, you use the equation \(N(t) = N_0 \times (1/2)^{t/T}\), where \(N(t)\) is the remaining quantity of the isotope at time \(t\), \(N_0\) is the initial quantity, and \(T\) represents the half-life period.

The calculation encompasses dividing the elapsed time by the half-life of the isotope to determine the number of half-lives that have passed. By raising one half to the power of the number of half-lives, we can find the fraction of the original sample that remains after a certain period. For example, after two half-lives, one fourth of the original substance would remain, as the substance halves once, and then halves again.

I-131 Decay

Iodine-131 (I-131) is a radioactive isotope of iodine commonly used in medicine, particularly in the diagnosis and treatment of thyroid conditions. The half-life of I-131 is approximately 8.04 days, indicating that after this period, half the atoms in a given sample will have decayed into another isotope, typically Xenon-131.

In medical applications, understanding the decay of I-131 is crucial for determining dosage and assessing therapeutic effectiveness. For instance, in the treatment of hyperthyroidism, the dose of I-131 must be carefully calculated to ensure the absorption of a sufficient amount to ablate, or destroy, the appropriate thyroid tissue, while minimizing radiation exposure to the rest of the body.

Exercise Application

According to the exercise, to determine the amount of I-131 left after 32 days, we first calculate the number of half-lives that have elapsed (4 half-lives, since 32 divided by 8.04 is roughly 4). By applying the half-life calculation formula, we find that after 32 days, the remaining mass of I-131 is \(30.0 \text{ g} \times (1/2)^4\), which is \(30.0 \text{ g} \times \frac{1}{16}\), or about 1.875 grams. This leaves us with a clear understanding of how I-131 decays over a specified period and helps us calculate the remaining radioactive material, which is crucial for safety and treatment efficacy in medical scenarios.