The half-life of cobalt--60 is 10.47 min. How many milligrams of \(\mathrm{Co}-60\) remain after 104.7 min if you start with 10.0 \(\mathrm{mg}\) of \(\mathrm{Co}-60\) ?

Radioactive decay is the spontaneous process by which an unstable atomic nucleus loses energy by emitting radiation in the form of particles or electromagnetic waves. This process is random and unpredictable for any single atom; however, for a large number of atoms, statistically, it follows a predictable pattern described by the half-life of the isotope.

Radioactive decay is fundamental in nuclear physics and chemistry, and it is the underlying mechanism for various natural phenomena as well as applications ranging from medical treatments to energy generation in nuclear reactors. When studying a particular isotope, such as cobalt-60, understanding its decay helps us to predict how long it will remain active or hazardous and guides its usage in industry or medicine.

In educational contexts, to aid comprehension, we may visualize this process by comparing it to a 'timer', where each tick represents a half-life, and with each tick, the total amount of the substance is halved. This helps in grasping the concept that the material doesn't disappear all at once but rather decreases exponentially over time.

Exponential decay is a decrease in an amount at a rate proportional to the current value, leading to a decline that can be described by a mathematical function in the form of a continuously decreasing curve. In the context of half-life problems, this form of decay is significant because it quantifies how substances like radioactive isotopes diminish over time.

The mathematical representation is often expressed with the formula: \( N = N_0 \cdot (1/2)^n \) where \( N \) is the remaining quantity after \( n \) half-lives, and \( N_0 \) is the initial quantity. This function shows that after each half-life period, the substance's quantity is halved from its previous amount. For learners, visualizing the process on a graph with a steep decline initially that levels off can help convey the concept of exponential decline.

To further illustrate, if 10.0 mg of a substance had a half-life of 1 hour, after an hour there would be 5.0 mg left, after two hours 2.5 mg, and so on. In practice, this mathematical model is crucial for understanding how quickly a radioactive material becomes safe or for planning treatments in radiation therapy.

Cobalt-60 (Co-60) is a synthetic radioactive isotope of cobalt, produced by bombarding cobalt-59 with neutrons in a reactor. With a half-life of 5.27 years, Co-60 emits beta particles and gamma rays as it decays. It's a powerful source of gamma rays and is used in various applications including cancer treatment, industrial radiography, and food irradiation.

When solving problems involving Co-60, it's crucial to understand both its physical properties and the math that governs its decay. Knowing its half-life is key to determining how long it remains effective or dangerous. For instance, in the given exercise, we can calculate the remaining Co-60 after a certain period by understanding the exponential nature of its decay. This makes Co-60 an excellent example when teaching about nuclear chemistry and the practical applications of radioactive decay.

Educators often use real-world isotopes like Co-60 in exercises to provide students with an insight into how nuclear physics applies outside of the textbook, solidifying their understanding through tangible examples. It also underscores the importance of rigorous safety protocols when handling or storing radioactive materials, due to their long-term decay characteristics.