Gallium- 67 is used in nuclear medicine (Table 11.6). After...

Gallium- 67 is used in nuclear medicine (Table 11.6). After treatment, a patient's blood shows an activity of 30,000 counts per minute (counts/min). How long will it be before the activity decreases to about 7500 counts $/ \mathrm{min} ?$

Key Concepts

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation in the form of particles or electromagnetic waves. This concept forms the basis for understanding how the activity of a radioactive substance decreases over time and is central to applications in both nuclear physics and nuclear medicine.

Half-life

The half-life of a radioactive isotope is the time required for half of the radioactive nuclei in a sample to decay. It is a characteristic property of each isotope and is crucial for determining how quickly the activity of a radioactive material, such as in nuclear medicine, decreases with time.

Exponential Decay Formula

The exponential decay formula, A = A?e^(–?t), mathematically describes how the activity (or amount) of a radioactive substance decreases over time. This equation relates the initial activity (A?), the decay constant (?), and the elapsed time (t), allowing for the calculation of remaining activity after a given period.

Decay Constant

The decay constant (?) is a parameter that quantifies the rate of radioactive decay. It is related to the half-life by the formula ? = ln(2)/T?/?. Understanding the decay constant is essential for converting between half-life and the exponential decay equation used to predict the time evolution of a radioactive sample.

Logarithms in Decay Calculations

Logarithms are used in decay calculations to solve for time or the decay constant within the exponential decay equation. By taking the natural logarithm of both sides of the decay equation, one can isolate and calculate the time required for the radioactive activity to decrease to a specified level.

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