Cobalt-60, which undergoes beta decay, has a half-life of 5.26 yr. (a) How many beta particles are emitted in 600 s by a 3.75 -mg sample of \(^{60} \mathrm{Co} ?(\mathbf{b})\) What is the activity of the sample in \(\mathrm{Bq}\) ?

The concept of half-life is pivotal in the field of nuclear physics and plays a crucial role in understanding radioactive decay. A half-life is the duration required for half the atoms in a radioactive sample to decay. This time period is a fixed characteristic for each radioactive isotope, such as Cobalt-60's 5.26 years. To calculate the half-life in practical situations, one must often convert years to seconds to match standard scientific units.

Determining the half-life is not just academic; it has real-world applications, such as dating archaeological artifacts, medical diagnoses through radiopharmaceuticals, and controlling the dosage in radiation therapy. By knowing the half-life, scientists and engineers can predict how long a radioactive material will remain active or hazardous, plan the safe storage of nuclear waste, and track environmental contamination.

The decay constant, denoted by the symbol \(λ\), is the probability per unit time that an atom will decay. It is intimately linked to the concept of half-life, as the two can be used to calculate each other. The formula to find the decay constant using the half-life is expressed as \(λ = \frac{\ln(2)}{t_{1/2}}\), where \(\ln(2)\) represents the natural logarithm of 2.

The importance of the decay constant lies in its use in calculating both the activity of a radioactive sample and predicting future behavior of a sample. For example, in the case of Cobalt-60, once \(λ\) is known, number crunching through various formulas lets us calculate how intense the radioactive source is and how many atoms decay over a specific time, providing critical information for safety and scientific measurements.

The activity of a radioactive sample provides a measure of the rate at which the atoms within the sample are decaying. It's conventionally expressed in becquerels (Bq), where one Bq corresponds to one decay per second. Calculating the activity involves using the decay constant and the current number of undecayed atoms in the sample. The fundamental equation for activity is \(A = λ × N_t\), with \(N_t\) being the number of atoms at time \(t\).

The practical applications for understanding the activity of a sample are numerous. For instance, in medicine, it allows medical professionals to determine the correct dosage for patients undergoing treatment. In environmental science, it helps in monitoring and managing exposure to radiation. Also, in nuclear power generation, it ensures the reaction rates are kept within safe operating limits.