PET studies require fluorine-18, which is produced in a cyclotron and decays with a half-life of 1.83 hours. Assuming that the \(\mathrm{F}-18\) can be transported at \(60.0 \mathrm{miles} /\) hour, how close must the hospital be to the cyclotron if \(65 \%\) of the \(\mathrm{F}-18\) produced makes it to the hospital?

The half-life of an isotope is the time it takes for half of the atoms in a sample to decay. This is a critical concept in understanding radioactive substances, as it helps us predict how quickly a radioactive isotope will lose its radioactivity. Fluorine-18, for example, has a half-life of 1.83 hours, meaning that after this period of time, only 50% of its initial quantity remains.

Understanding half-life is crucial in numerous applications such as medicine, where radioactive isotopes are used in diagnostics and treatment. For instance, in Positron Emission Tomography (PET) scans, knowing the half-life of fluorine-18 allows healthcare professionals to calculate how much of the substance will reach the patient's body in a useful state. This calculation directly influences how much of the isotope needs to be produced and how quickly it must be delivered to where it is needed.

The decay constant, often symbolized as 'k', is a probability factor that defines the rate at which a sample of radioactive substance will decay. It's an intrinsic property of each isotope and can be calculated from the half-life using the formula: \[ k = \frac{\ln(2)}{t_{1/2}} \], where \( t_{1/2} \) is the half-life. In our exercise, we determine the decay constant of fluorine-18 to use in further calculations.

The decay constant gives us an indication of the stability of an isotope; the higher the decay constant, the more quickly the substance will decay. For PET studies, knowing this constant is essential since it helps in determining the time frame in which the isotope remains viable for medical imaging. A precise calculation ensures that the isotopes will be used effectively, minimizing any waste due to decay.

The exponential decay equation describes how the quantity of a radioactive substance decreases over time. It's represented by the equation \( N(t) = N_0 e^{-kt} \), where \( N_0 \) is the original amount of substance, \( k \) is the decay constant, and \( t \) represents time. In the context of our PET study example, we use this equation to find the time at which 65% of the initial fluorine-18 remains.

The beauty of this equation lies in its ability to provide a snapshot of the radioactive substance at any given moment. With exponential decay, the rate of decay is proportional to the remaining quantity, meaning that as time passes, the substance decays more slowly. This is essential in fields requiring precise measurement of radioactive materials, and especially in medical applications, where it's vital to calculate the exact time span during which the radioactive material is effective for diagnostics or treatment.