As a result of being exposed to the radiation released during the Chernobyl nuclear accident, the dose of iodine- 131 in a person's body is \(7.4 \mathrm{mC}(1 \mathrm{mC}=1 \times\) \(10^{-3} \mathrm{Ci}\) ). Use the relationship rate \(=\lambda N\) to calculate the number of atoms of iodine- 131 to which this radioactivity corresponds. (The half-life of \({ }^{131} \mathrm{I}\) is 8.1 days.)

Understanding the half-life of isotopes is essential when studying radioactivity. The half-life is the amount of time required for half of the radioactive isotope present to decay into a different element or isotope. This concept is crucial because it allows us to predict how long a radioactive material will remain hazardous and how long it can be useful, for example in medical applications.

For instance, the half-life of Iodine-131, an isotope used in medical diagnostics and treatment, is 8.1 days. This relatively short half-life implies that after 8.1 days, only half of the original amount of Iodine-131 remains, the rest having decayed. After another 8.1 days, only a quarter would be left, and so on, exponentially decreasing the quantity over time.

The ability to calculate the remaining quantity of an isotope after a certain period involves a simple mathematical formula: \( N_t = N_0 (1/2)^{t/T} \), where \( N_t \) is the remaining quantity, \( N_0 \) is the initial quantity, \( t \) is the elapsed time, and \( T \) is the half-life of the isotope.

The decay constant, represented by the Greek letter λ (lambda), is a probability rate at which a single atom of a radioactive isotope will decay per unit of time. The decay constant is inversely proportional to the half-life, which means that isotopes with a shorter half-life have a higher decay constant.

The formula that relates decay constant and half-life is given as: \( \text{Decay constant }(\text{λ}) = \frac{ \text{ln}(2)}{\text{Half-Life (T)}} \). By knowing the decay constant, we can calculate the activity of a sample, which is the average number of decays per second and is given by the relationship: Activity (Rate) = λN, where N is the number of undecayed atoms.

In the context of the Iodine-131 example, a precise calculation of the decay constant allows us to determine the activity and, subsequently, the number of atoms undergoing decay, facilitating the application of radiation safety measures and correct dosage in therapeutic uses.

A Becquerel (Bq) is the SI unit for radioactivity and represents one decay per second. Conversion between different units of radioactivity is crucial for clear communication and calculation in nuclear medicine, health physics, and radiological science.

To convert from the traditional unit Curie (Ci) to Becquerel (Bq), you can use the conversion factor \( 1 \text{Ci} = 3.7 \times 10^{10} \text{Bq} \). It is important to realize that when dealing with radioactivity, large numbers are typical due to the atomic scale at which these processes occur. For example, in the textbook solution, an activity of 7.4 mC (millicuries) equates to an activity of over 200 million Bq. This conversion is pivotal in international settings, as Becquerel is the standard system, while the Curie is still used in some countries.

In handling problems that involve calculating activity and decay, students should make sure to convert to Becquerel if not already in SI units. This simplifies the process of finding the number of atoms and any subsequent calculations that use the decay constant or the half-life.