A small region of a cancer patient's brain is exposed for 24.0 min to 475 Bq of radioactivity from \({ }^{60}\) Co for treatment of a tumor. If the brain mass exposed is \(1.858 \mathrm{~g}\) and each \(\beta^{-}\) particle emitted has an energy of \(5.05 \times 10^{-14} \mathrm{~J},\) what is the dose in rads?

Radioactivity is the process by which unstable atomic nuclei lose energy by emitting radiation. This decay happens over time, transforming the original nucleus into a more stable state. There are different types of radioactive decay, including alpha, beta, and gamma decay. The rate at which these decays happen is measured in units called Becquerels (Bq), where one Bq equals one decay per second. Understanding radioactivity is crucial in fields such as medicine, where it is used in radiation therapy to treat cancers.

Beta particles are high-energy, high-speed electrons or positrons emitted during radioactive decay processes like beta decay. In the specific case of our exercise involving \textsuperscript{60}Co, beta particles are emitted which have significant energy. Each \textbeta^{-} particle emitted in our exercise has an energy of \( 5.05 \times 10^{-14}\ \mathrm{J} \). Calculating the number of such emissions helps determine the total energy released, which is crucial for calculating the radiation dose.

The radiation dose measures the amount of energy deposited by ionizing radiation in a given mass of tissue. There are different units to measure this dose. The gray (Gy) is a standard unit that represents one joule of radiation energy deposited per kilogram of tissue. However, in the exercise, it converts to rads, where 1 Gy equals 100 rads. Understanding this conversion is essential to comprehend the magnitude of radiation exposure.

Radiation therapy is a treatment method that uses high doses of radiation to kill or damage cancer cells. It works by damaging the DNA inside these cells, causing them to die or stop dividing. The effectiveness of radiation therapy depends on delivering the precise amount of radiation to the targeted area. In the exercise, the patient's brain mass exposed to radioactivity is treated with specific calculations to determine the dose required for effective treatment, ensuring minimal damage to surrounding healthy tissue.

Energy conversion in the context of the exercise involves determining the total energy released during radioactive decay and then converting that energy into a form that can be used to measure the radiation dose. The total energy released by beta particles must be calculated first, followed by converting this energy into grays and finally into rads. This process involves several steps: starting from calculating the activity, determining the number of decays, finding the total energy, and converting it to the relevant radiation dose units. These conversions ensure that the calculated radiation dose accurately reflects the energy absorbed by the tissue, which is vital for effective cancer treatment.