Problem 47 - Chemistry A Molecular Approach

Question

A radioactive sample contains \(1.55 \mathrm{~g}\) of an isotope with a halflife of 3.8 days. What mass of the isotope remains after 5.5 days?

Step-by-Step Solution

Answer

After 5.5 days, approximately 0.671 g of the isotope remains.

Step 1: Understanding the Half-life Concept

The half-life of a radioactive substance is the time it takes for half of the radioactive atoms to decay. The amount of substance remaining after a certain amount of time can be calculated using the formula: \(N(t) = N_0 (1/2)^{(t/T)}\) where \(N(t)\) is the remaining quantity after time \(t\), \(N_0\) is the initial quantity, and \(T\) is the half-life period.

Step 2: Identify Known Values

From the problem statement, identify the known values: initial amount of isotope \(N_0 = 1.55\) g, half-life \(T = 3.8\) days, and the elapsed time \(t = 5.5\) days.

Step 3: Calculate the Number of Half-lives

Calculate the number of half-lives that have elapsed by dividing the time passed by the half-life of the isotope: \(number\_of\_halflives = t/T = 5.5\text{ days} / 3.8\text{ days}\).

Step 4: Apply the Half-life Formula

Use the formula with the known values to calculate the remaining mass: \(N(t) = 1.55 (1/2)^{(5.5/3.8)}\).

Step 5: Calculate the Remaining Mass

Compute the remaining mass using a calculator, and make sure to observe correct significant figures in your answer.

Key Concepts

Understanding Half-Life

The half-life of a radioactive substance is a crucial concept in understanding radioactive decay. It's defined as the time needed for half of the radioactive atoms in a sample to decay. This means after one half-life, the amount of the original substance is reduced by half. In real-world applications, half-life is used to determine the longevity and decay of radioactive materials, which can range from seconds to millions of years.

For example, if a substance has a half-life of 3.8 days, after 3.8 days only half of the original substance will remain. After another 3.8 days, half of what remained will decay, leaving us with a quarter of the initial quantity. This process continues exponentially, which brings us to another key concept in the study of radioactive decay - exponential decay.

Radioactive Isotopes and Decay

Radioactive isotopes, or radionuclides, are variants of chemical elements that have unstable nuclei. This instability causes them to lose energy by emitting radiation in the form of particles or electromagnetic waves. This process, known as radioactive decay, transforms the radioactive isotope into a different element or a different isotope of the same element.

Radiation emitted during decay can be in the form of alpha particles, beta particles, or gamma rays, with each type of radiation having different properties and levels of penetration. The decay of radioactive isotopes has various applications, such as dating archaeological findings via carbon-14, treating cancer with targeted radiation therapy, and powering deep space probes with isotopes like plutonium-238.

Exponential Decay

Exponential decay is the process by which a quantity decreases at a rate proportional to its current value. In the context of radioactive decay, this concept is important because it describes how the amount of a radioactive isotope diminishes over time. The formula for exponential decay is expressed as: \(N(t) = N_0 (1/2)^{(t/T)}\), where \(N(t)\) represents the remaining quantity of the isotope after time \(t\), \(N_0\) is the initial quantity, and \(T\) is the half-life.

When we plot this decay on a graph, the curve smoothly drops off, getting closer to zero as time extends but never actually reaching zero. This mathematical model is essential for predicting how long it will take for a substance to reach a certain level of decay, which has implications from medical dosing to the safe handling of nuclear waste.

Significant Figures

In science and engineering, significant figures are used to indicate the precision of a measured or calculated quantity. They include all the digits that are known reliably, plus one last digit that is somewhat uncertain. When performing calculations, especially in chemistry and physics, it's critical to consider significant figures to ensure results are reported with the appropriate level of precision.

To adhere to the principles of significant figures during calculations, rules for rounding and handling of operations like addition, subtraction, multiplication, and division are followed. In the case of our radioactive decay problem, utilizing significant figures correctly is paramount to conveying the scientific accuracy of the remaining mass of isotope after a certain period. This attention to detail impacts everything from scientific research to the information provided on medication labels.