At 8:00 a.m., a patient receives a 1.5-mg dose of I-131 to treat thyroid cancer. If the nuclide has a half-life of eight days, what mass of the nuclide remains in the patient at 5:00 p.m. the next day? (Assume no excretion of the nuclide from the body.)

Verified step by step guidance

Determine the total time elapsed from 8:00 a.m. on the first day to 5:00 p.m. the next day. This is 1 day and 9 hours, or 33 hours.

Convert the elapsed time from hours to days, since the half-life is given in days. There are 24 hours in a day, so divide 33 hours by 24 to get the time in days.

Use the formula for radioactive decay: \( N = N_0 \times (\frac{1}{2})^{\frac{t}{t_{1/2}}} \), where \( N_0 \) is the initial mass, \( t \) is the time elapsed, and \( t_{1/2} \) is the half-life.

Substitute the known values into the formula: \( N_0 = 1.5 \text{ mg} \), \( t = \text{time in days} \), and \( t_{1/2} = 8 \text{ days} \).

Calculate the remaining mass \( N \) using the formula, which will give you the mass of I-131 remaining in the patient after the specified time.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Half-life

Half-life is the time required for half of a sample of a radioactive substance to decay. In this context, the half-life of I-131 is eight days, meaning that after eight days, only half of the original amount will remain. This concept is crucial for calculating the remaining mass of the nuclide after a specific period.

Exponential Decay

Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value. For radioactive substances, this means that the amount of the substance decreases exponentially over time, following the formula N(t) = N0 * (1/2)^(t/T), where N0 is the initial amount, t is the elapsed time, and T is the half-life.

Time Calculation

Accurate time calculation is essential for determining how many half-lives have passed. In this scenario, the time from 8:00 a.m. to 5:00 p.m. the next day is 33 hours, or approximately 1.375 days. Understanding how to convert time into half-lives allows for the correct application of the half-life concept to find the remaining mass of I-131.

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