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Ch.21 - Nuclear Chemistry
Brown - Chemistry: The Central Science 15th Edition
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Tutti i libri di testoBrown 15th EditionCh.21 - Nuclear ChemistryProblema 37
Capitolo 21, Problema 37

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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1
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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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.
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Zero-Order Half-life

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.
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Rate of Radioactive Decay

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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Beta Decay
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