Cobalt-60 is a strong gamma emitter that has a half-life of 5.26 yr. The cobalt-60 in a radiotherapy unit must be replaced when its radioactivity falls to 75% of the original sample. If an original sample was purchased in June 2021, when will it be necessary to replace the cobalt-60?

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Identify the half-life of cobalt-60, which is given as 5.26 years.

Use the formula for exponential decay: \( N = N_0 \times (0.5)^{t/T_{1/2}} \), where \( N \) is the remaining quantity, \( N_0 \) is the initial quantity, \( t \) is the time elapsed, and \( T_{1/2} \) is the half-life.

Set \( N/N_0 = 0.75 \) because the activity falls to 75% of the original.

Rearrange the formula to solve for \( t \): \( 0.75 = (0.5)^{t/5.26} \).

Solve for \( t \) using logarithms: \( t = 5.26 \times \frac{\log(0.75)}{\log(0.5)} \).

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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 the radioactive nuclei in a sample to decay. For Cobalt-60, which has a half-life of 5.26 years, this means that after this period, only 50% of the original amount remains. Understanding half-life is crucial for calculating how long it takes for a radioactive substance to reach a certain level of activity.

Radioactive decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This decay occurs at a predictable rate, characterized by the half-life, and leads to a decrease in the amount of the radioactive isotope over time. In the context of Cobalt-60, it is important to know how much of the substance remains after a certain period to determine when it needs to be replaced.

Exponential decay

Exponential decay describes the process where the quantity of a substance decreases at a rate proportional to its current value. In radioactive decay, this means that the amount of Cobalt-60 decreases exponentially over time, allowing us to calculate the remaining quantity after a specific number of half-lives. This concept is essential for determining when the radioactivity of the cobalt-60 falls to 75% of its original amount.

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