A 68-mg sample of a radioactive nuclide is administered to a patient to obtain an image of her thyroid. If the nuclide has a half-life of 12 hours, how much of the nuclide remains in the patient after 4.0 days?

The half-life of a nuclide is a critical concept in the context of radioactive decay. It refers to the time required for half of the atoms in a radioactive sample to decay. This is a fixed property for each radioactive isotope and is crucial in applications across various fields such as medicine, archaeology, and environmental science.

In our exercise scenario, when a 68-mg sample has a half-life of 12 hours, it tells us that every 12 hours, the amount of the nuclide will reduce to half of its previous quantity. This pattern continues at regular intervals, leading to an exponential decrease in the remaining amount of the substance over time. Understanding half-life is essential for safely managing the use of radioactive materials within the body, such as the diagnostic imaging example provided in the exercise.

By grasping the concept of half-life, students can predict and calculate how much of a radioactive substance will remain after a given period – a principle used to determine correct dosages in medical treatments, date archaeological findings, or gauge the safety of radioactive exposure over time.

Radioactive decay follows an exponential pattern which can be mathematically modeled using the exponential decay formula. The general formula is expressed as:\[ P = P_0 \times \left(\frac{1}{2}\right)^n \]where:\

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• \(P\) is the final amount remaining after decay,\
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• \(P_0\) is the initial amount of the nuclide,\
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• \(n\) represents the number of half-lives that have passed.\
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To visualize how this works, imagine a graph where the y-axis represents the quantity of the nuclide and the x-axis represents time. The curve starts at the initial quantity \(P_0\) and decreases sharply, then gradually flattens as the quantity approaches zero. This demonstrates how the quantity of the substance decays more quickly at the start and slows down as there is less substance left to decay.

The exponential decay model is powerful because it can predict the amount of a radioactive substance at any point in time, provided we know its half-life and the initial quantity. This concept emphasizes that the rate of decay is proportional to the quantity of nuclide remaining, a key understanding for the accurate computation of decay in various practical contexts.

Conversion of time units is often necessary in radioactive decay problems, as half-lives can be given in various time units (seconds, minutes, hours, days, years, etc.), and the measurement of time elapsed may not match these units. Consistency in units is essential to perform accurate calculations.

In the exercise, the nuclide has a half-life of 12 hours, but the period over which we want to determine the remaining nuclide is given in days. To solve the problem, converting days to hours is a necessary step, which is achieved by multiplying the number of days by 24 hours per day:\[ 4.0 \text{ days} \times 24 \frac{\text{hours}}{\text{day}} = 96 \text{ hours} \]Once the conversion is complete, the calculation uses a consistent unit (hours in this case), making it possible to determine the number of half-lives that have occurred and apply the exponential decay formula. This example illuminates how time conversion is a fundamental skill in scientific calculations, ensuring that the math aligns accurately with the real-world phenomena being analyzed.