Question

A radioactive isotope having a half-life of 3 days was received after 12 days. It was found that there were \(3 \mathrm{~g}\) of the isotope in the container. The initial mass of the isotopes when packed was (a) \(12 \mathrm{~g}\) (b) \(24 \mathrm{~g}\) (c) \(36 \mathrm{~g}\) (d) \(48 \mathrm{~g}\)

Short answer

The initial mass of the isotopes when packed was 48 g.

Step-by-step solution

Determine the number of half-lives

Calculate the number of half-lives that have passed by dividing the total time elapsed by the half-life of the isotope. Formula: Number of half-lives = Total time elapsed / Half-life duration.

Apply the half-life formula

Use the formula that relates the final amount of substance to the initial amount and the number of half-lives: Final amount = Initial amount * (1/2)^Number of half-lives.

Solve for the initial amount

Rearrange the formula to solve for the initial amount by dividing the final amount of the isotope by (1/2) raised to the power of the number of half-lives.

Key concepts

Half-Life Calculation

Understanding the half-life of a radioactive isotope is fundamental in the fields of nuclear physics and chemistry. Half-life is defined as the time required for half of the radioactive nuclei in a sample to undergo decay. This concept allows scientists and engineers to predict how long an isotope will remain active or how quickly it will become safe to handle.

In the given problem, the students are tasked with determining the initial mass of a radioactive isotope based on its half-life and the remaining mass after a certain period. To calculate this, we use the formula: \[\text{Final amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^{\text{Number of half-lives}}\] This formula encapsulates the essence of half-life calculation as it shows how the amount of a substance diminishes over each half-life period.

In a practical sense, if you know the half-life of an isotope and the amount remaining after a period of time, you can work backwards to find out how much of the substance you began with. This calculation is critical in various applications, including dating archaeological findings, medical dosing of radioactive isotopes, and managing nuclear waste.

Radioactive Isotope Decay

Radioactive isotopes, also known as radioisotopes, are unstable atoms that degrade over time through a process called radioactive decay. This decay process is spontaneous and results in the emission of radiation as well as the transformation of the original isotope into a different element or a different isotope of the same element.

Each radioisotope has a characteristic decay rate, which is expressed by its half-life. The decay process follows an exponential pattern—a fact that underlies the mathematical models used to describe and quantify it.

Visualizing Decay

Imagine a large number of identical, unstable atomic nuclei in a sample. At the start, all of these nuclei are undecayed. As time passes, a fraction of them will decay, transforming into another element or a different isotope. The rate of decay is highest at the start and exponentially decreases as fewer undecayed nuclei remain. This visualization helps to understand why, after one half-life, exactly half of the original nuclei would remain.

Exponential Decay

The principle of exponential decay is not only central to radioactive decay but to a vast array of natural processes, from the cooling of a hot object to the discharge of a capacitor in electronics. Exponential decay describes a process of reduction at a rate proportional to the current value. Mathematically, this is often represented in the form of a constant base (in the case of half-life calculations, the base is \(\frac{1}{2}\)) raised to the power of the number of decay periods.

The equation for exponential decay allows us to predict the future value of a quantity based on its current value and the rate of decay. In the context of the exercise provided, the equation can be used to calculate the initial mass of a radioactive isotope given its half-life and remaining mass after a set period. It's a powerful tool for understanding how quantities diminish over time, and it's beautifully simple in its construction, representing complex natural procedures in a straightforward and calculable form.