Question

A drug containing \({ }_{43}^{99} \mathrm{Tc}\left(t_{1 / 2}=6.05 \mathrm{~h}\right)\) with an activity of \(1.50 \mu \mathrm{Ci}\) is to be injected into a patient at \(9.30 \mathrm{a} . \mathrm{m} .\) You are to prepare the sample \(2.50 \mathrm{~h}\) before the injection (at 7: 00 a.m.). What activity should the drug have at the preparation time (7:00 a.m.)?

Short answer

Question: Determine the activity of the drug containing ${{}_{43}^{99}} \mathrm{Tc}$ at the preparation time (7:00 AM) given its half-life is 6.05 hours and its activity at 9:30 AM is 1.50 µCi. Answer: The activity of the drug containing ${{}_{43}^{99}} \mathrm{Tc}$ at the preparation time (7:00 AM) should have an activity of 2.11 µCi.

Step-by-step solution

Understand the half-life concept

Half-life is the time required for a quantity (in this case, the activity of the drug) to reduce to half its initial value. We are given that the half-life of \({ }_{43}^{99} \mathrm{Tc}\) is 6.05 hours.

Identify the known values

We know the following values: - Half-life of \({ }_{43}^{99} \mathrm{Tc}\) (\(t_{1/2}\)) is 6.05 hours, - Activity at 9:30 AM is 1.50 µCi, - We need to find the activity at 7:00 AM (2.5 hours before 9:30 AM).

Calculate the decay constant

First, we need to calculate the decay constant (k) using the half-life formula: \(k = \frac{ln(2)}{t_{1/2}}\) Plug in the given half-life and solve for k: \(k = \frac{ln(2)}{6.05} = 0.1147 \mathrm{~h}^{-1}\)

Calculate the activity at 7:00 AM

Now, we will use the following equation to find the activity of the drug at 7:00 AM (A_t): \(A_t = A_0 e^{-kt}\) Where: - \(A_t\) is the activity at time t, - \(A_0\) is the initial activity, - \(k\) is the decay constant, - \(t\) is the time difference between the initial time and desired time. Since we are given the activity at 9:30 AM (A_0) and we want to find the activity at 7:00 AM (A_t), we can rewrite the equation as: \(A_{7\:00\:\text{AM}} = A_{9\:30\:\text{AM}} e^{kt}\) Now, plug in the known values and solve for the activity at 7:00 AM: \(A_{7\:00\:\text{AM}} = 1.50 \mu\text{Ci} \cdot e^{0.1147 \cdot 2.5} = 2.11 \mu\text{Ci}\) Therefore, the activity of the drug containing \({ }_{43}^{99} \mathrm{Tc}\) at the preparation time (7:00 AM) should have an activity of 2.11 µCi.

Key concepts

Half-Life

The concept of half-life is fundamental in understanding radioactive decay. It is defined as the time required for a radioactive substance to decrease to half its original amount. This is a crucial factor in nuclear physics, medicine, and radiocarbon dating because it informs us about the rate at which a substance undergoes radioactive decay. For instance, if a drug has a half-life of 6 hours, this means that every 6 hours, the amount of the drug remaining will be half of what it was at the start of the period. In medical terms, when planning treatments or diagnostic tests with radioactive tracers, knowing the half-life of the substance allows practitioners to calculate the optimal time for administration and predict the radiation exposure.

When solving problems involving half-life, it helps to visualize the process as a decay curve, showing how the amount of a substance decreases over time. The half-life remains constant regardless of the initial quantity, making it an intrinsic property of the radioactive isotope.

Decay Constant

The decay constant, often symbolized by the letter 'k', is a probability factor that provides the rate at which a radioactive isotope decays. It is inversely related to the half-life and allows us to express radioactive decay in mathematical terms. You can calculate the decay constant using the formula:
\[k = \frac{\ln(2)}{t_{1/2}}\]
Where \(\ln(2)\) is the natural logarithm of 2 (approximately 0.693) and \(t_{1/2}\) is the half-life of the substance. A high decay constant implies that the radioactive decay happens quickly, meaning the substance will have a short half-life and vice versa. This constant is a critical part of the exponential decay formula, which is used to predict the remaining amount of a radioactive substance at any given time.

Exponential Decay

Exponential decay describes a process where the decrease in the amount of a radioactive substance is proportional to the amount present at any time. This form of decay is characterized by a rapid decline that gradually becomes less steep over time. Mathematically, it is represented by the formula:
\[A_t = A_0 e^{-kt}\]
Here, \(A_t\) is the amount of the substance at time \(t\), \(A_0\) is the initial amount, \(k\) is the decay constant, and \(e\) is the base of the natural logarithms. In the context of radioactive drugs, exponential decay allows us to calculate, as in our exercise, how much of the substance will be active at a certain point, ensuring that the right amount is administered for medical imaging or treatment purposes.

Understanding the principles of exponential decay is not only important in the context of radioactive substances but also in many other fields like economics, biology, and electrical engineering where similar decay patterns can be observed.

Activity of a Radioactive Substance

The activity of a radioactive substance refers to the number of decay events that occur in a given unit of time. It is a direct measure of the radiation a radioactive substance emits and is measured in units like curies (Ci) or becquerels (Bq). The activity decreases over time as the substance decays, following an exponential trend. This rate of decay is influenced by the decay constant and can be calculated for any time using the aforementioned exponential decay formula.

For example, in the medical field, the activity must be known to ensure that the correct dosage is administered for both the efficacy of a treatment and the safety of the patient. Misjudging the activity can lead to under-treatment or expose the patient to unnecessary radiation risks. Calculating the activity at a specific time ahead of a medical procedure ensures the right amount of radioactive substance is used.