Iodine- 131 is used in the treatment of tumors in the thyroid gland. Its half- life is \(8.1\) days. Suppose that, due to a shipment delay, the I-131 in a hospital's pharmacy is \(2.0\) days old. (a) What percentage of the I-131 has disintegrated? (b) A patient is scheduled to receive \(15.0 \mathrm{mg}\) of \(\mathrm{I}-131 .\) What dosage (in milligrams) should the hospital pharmacist recommend for this patient if the 2 -day-old bottle of \(\mathrm{I}-131\) is used?

Half-life is a crucial concept in nuclear physics and chemistry, particularly when dealing with radioactive decay. It is defined as the amount of time required for half of a quantity of a radioactive isotope to decay. This characteristic decay time can tell scientists and medical professionals a lot about how a radioactive substance will behave.

For instance, in the given exercise, the half-life of Iodine-131 (I-131) is mentioned as 8.1 days. This means, after 8.1 days, only half of the original amount of I-131 would remain; the other half would have decayed into other elements or isotopes. Understanding the half-life helps determine the dosage and timing for medical treatments and the safe handling of radioactive materials.

The exponential decay formula is at the heart of calculating how much of a radioactive substance remains at any given time. It's based on the equation:
\[N(t) = N_0 e^{-\text{λ} t}\]
where \(N(t)\) is the remaining quantity at time \(t\), \(N_0\) is the original quantity, \(e\) is the base of the natural logarithm, and \(λ\) is the decay constant. This decay constant is directly related to the half-life of the substance and can be calculated using the formula:\[\text{λ} = \frac{\text{ln}(2)}{T_{1/2}}\]
The concept is crucial for all kinds of applications, from dating archaeological finds to calculating safe dosages of radioactive medicine. In the exercise, we used this formula to establish not only the percentage of I-131 that disintegrated over two days but also to adjust the patient's dosage accordingly.

Isotopes have a wide range of applications in medicine, where they often serve as tracers or treatment agents due to their radioactive properties. For diagnostics, isotopes can be used in imaging to pinpoint areas within the body where cells may be behaving unusually. In treatments, such as with Iodine-131, isotopes can target specific cells, like cancerous cells in the thyroid gland, to deliver localized radiation therapy.

In the context of our exercise, I-131's application in treating thyroid tumors is a classic example of therapeutic use. The adjusted dosage calculation considers the decay of the isotope, ensuring the patient receives an effective amount that correlates with the decay that has occurred over time. This precision is essential for the isotope to function as intended without causing unnecessary exposure to radiation.