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 2016, 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: \( N(t) = N_0 \times (0.5)^{t/T_{1/2}} \), where \( N(t) \) is the remaining quantity, \( N_0 \) is the initial quantity, \( t \) is the time elapsed, and \( T_{1/2} \) is the half-life.

Set \( N(t) = 0.75 \times N_0 \) because the cobalt-60 needs to be replaced when its activity is 75% of the original.

Solve the equation \( 0.75 = (0.5)^{t/5.26} \) for \( t \) to find the time elapsed since the purchase.

Add the calculated time \( t \) to June 2016 to determine the replacement date.

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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 results in the transformation of the original isotope into a different element or isotope. In the context of Cobalt-60, it is important to know how much of the substance remains over time to assess its usability in applications like radiotherapy.

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 certain number of half-lives. This concept is essential for determining when the radioactivity of Cobalt-60 falls to 75% of its original level.

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Beta Decay

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