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The half-life of a substance is the time it takes for half of the atoms in a radioactive material to decay. Essentially, after one half-life, only half of the original radioactive atoms remain. This concept helps us understand how quickly a substance undergoes radioactive decay.
The half-life of a radionuclide is a constant value that does not change with different conditions, such as temperature or pressure. This means that each radioactive material has its own unique half-life.
For example, in this problem, the radionuclide Cu-64 has a half-life of 12.7 hours. Therefore, every 12.7 hours, half of the Cu-64 will have decayed. This is a crucial feature used in calculating how much of a substance remains after a given period of time.

The decay constant, often denoted by the symbol \(\lambda\), is a probability parameter in radioactive decay equations that measures the likelihood of decay per unit time. In simpler terms, it's a factor that indicates how fast the radioactive substance is decaying.
The formula to calculate the decay constant is \(\lambda = \frac{\ln(2)}{t_{1/2}}\), where \(t_{1/2}\) represents the half-life of the radionuclide.
For Cu-64, given its half-life is 12.7 hours, the decay constant can be computed as follows:

• \(\lambda = \frac{\ln(2)}{12.7} \approx 0.0546 \, \text{h}^{-1}.\)

A higher value of \(\lambda\) means a faster decay, while a lower \(\lambda\) implies a slower decay process.

The exponential decay formula is a mathematical expression used to model the process of radioactive decay. It allows us to determine the amount of a substance remaining after a certain period.
The formula is written as: \(N(t) = N_0 \, e^{-\lambda t}\), where:

• \(N(t)\) is the remaining quantity at time \(t\).
• \(N_0\) is the initial quantity.
• \(\lambda\) is the decay constant.
• \(e\) is the base of the natural logarithm.

Using this formula, we can calculate the remaining mass of Cu-64 at different times:

• At 14 hours: \(N(14) = 5.50 \, \text{g} \times e^{-0.0546 \times 14} \approx 2.629 \, \text{g}.\)
• At 16 hours: \(N(16) = 5.50 \, \text{g} \times e^{-0.0546 \times 16} \approx 2.4915 \, \text{g}.\)

This formula clearly shows how the mass of the radioactive material decreases over time.

Cu-64, or Copper-64, is a radionuclide known for its radioactive properties. It is utilized in various fields, including medicine, where it's used in diagnostic imaging and cancer treatment.
Cu-64 decays primarily by positron emission, making it valuable in positron emission tomography (PET) scans, a type of imaging test.
A sample initially containing a specific mass of Cu-64 can be expected to lose mass over time as it decays.
In the exercise, we started with a 5.50g sample of Cu-64 at time zero. After a period of decay, from 14 hours to 16 hours:

• The remaining mass was calculated twice using the exponential decay formula. First, at 14 hours, and then at 16 hours.
• The difference between the two masses indicated the amount of Cu-64 that decayed during this time, showing the practical application of understanding radionuclides in real-world situations.

Understanding the decay behavior of Cu-64 is essential for accurately employing it in both scientific research and clinical applications.