Determine the decay constant of radium- 226 , which has a half-life of \(1600 \mathrm{yr}\).

The concept of half-life is crucial in the study of radioactive substances like radium-226. The half-life is the time required for half of the radioactive atoms in a sample to decay. For radium-226, this period is 1600 years. This implies that after 1600 years, only half of the original quantity of radium-226 would remain in a sample, the rest having decayed into other elements.

Understanding the half-life of a radioactive isotope is important not only in the field of nuclear physics but also in geological dating, medical treatments, and safety considerations for radioactive materials. For students struggling with grasping the concept, it helps to visualize the half-life as a sort of 'radioactive clock' that ticks at a rate defined by the substance's decay constant.

Radioactive decay is a random and spontaneous process by which an unstable atomic nucleus loses energy by emitting radiation. This can occur in the form of alpha particles, beta particles, gamma rays, or other particles. For radium-226, it decays by emitting an alpha particle to become radon-222.

To provide a more concrete understanding, consider that each radioactive atom is like a ticking bomb with an uncertain timer. At some unpredictable point, it will 'explode,' or decay, into a more stable form. The rate of radioactive decay for a large number of atoms is defined by the decay constant, symbolized by \(\lambda\), which is mathematically related to the half-life. The decay constant represents the probability of decay of a single atom per unit time.

Exponential decay is a pattern of decrease in which a quantity diminishes at a rate proportional to its current value. This is described by the formula \(N(t) = N_0 \ : e^{-\lambda t}\function, which links the initial quantity\(N_0\), the number of remaining undecayed atoms at time \N\overn exponentially to time \t\and the decay constant\). In the context of radioactivity, this model reflects how the number of undecayed atoms in a substance decreases over time.

To make the concept more approachable, think of a car depreciating in value year over year, or the way the brightness of a cellphone screen fades with time, as both are practical examples of exponential decay. In the case of radium-226, the decay follows an exponential pattern, indicating that it takes the same amount of time for the material to decay from 100% to 50% as it does from 50% to 25%, and so on.