A family member is about to have a brain scan using technetium\(99^{*},\) an excited isotope with 6.01 -hour half-life. The hospital makes Tc-99* from the decay of molybdenum-99 \(\left(t_{1 / 2}=2.7 \text { days }\right)\) then delivers it to the nuclear medicine department. You're told that the \(\mathrm{Tc}-99^{*}\) will arrive 90 minutes after production, and that there must be 10 mg of it. The technician says she will produce \(12 \mathrm{mg}\) of \(\mathrm{Tc}-99^{* *} .\) Is that sufficient?

The concept of half-life is an integral part of understanding nuclear decay. It describes the time required for half of the unstable or radioactive atoms in a sample to undergo decay. In the context of our exercise, technetium-99*, an excited isotope used for brain scans, has a half-life of 6.01 hours. This means that every 6.01 hours, the quantity of technetium-99* will reduce by 50%. Knowing the half-life is crucial for determining how much of a substance will remain after a certain period, which is essential for medical applications where precise dosages of radioactive substances are critical.

When solving problems involving half-life, it is important to consider the time elapsed relative to the half-life of the substance. For example, if the technetium-99* needs to be available 90 minutes after production, we need to assess how this duration impacts the amount of the substance, taking into account its half-life, for appropriate diagnostic use.

The decay constant, represented as 'k' in nuclear decay equations, is a parameter that defines the rate at which a radioactive substance decays. It is inversely proportional to the half-life of the substance, which gives us the relationship: \(k = \frac{0.693}{t_{1/2}}\), where \(t_{1/2}\) is the half-life. The decay constant is pivotal because it offers a way to calculate the fraction of an element remaining at any given point without having to compute the effects of multiple half-lives.

In the solution to our problem, by knowing the half-life of technetium-99*, we were able to calculate the decay constant. This constant is then used in conjunction with the amount of time that has passed to determine how much of the substance will remain, ensuring that the delivered quantity is adequate for its intended use. It is a fundamental value that bridges the measured half-life and real-time decay observations.

Exponential decay is a process described by a mathematical model where the quantity of a radioactive substance diminishes at a rate proportional to its current amount. This decay occurs in a manner such that over equal time intervals, equal proportions of the substance decompose. The formula used to express this behavior is \(N = N0 \cdot e^{-kt}\), where \(N\) is the amount remaining after time \(t\), \(N0\) is the initial amount, \(e\) is the base of the natural logarithm, \(k\) is the decay constant, and \(t\) is the elapsed time.

In this real-world scenario, to ensure the patient receives the correct dosage during the brain scan, we use the exponential decay formula. By substituting the known values into the formula, including the decay constant we've calculated and the elapsed time between production and delivery, we can predict how much technetium-99* will actually be available for the scan, ensuring both safety and the effectiveness of the diagnostic procedure.