Nickel-65 beta decays by electron emission with decay constant \(\lambda=0.275 \mathrm{h}^{-1} .\) (a) Identify the daughter nucleus. (b) In a sample of initially pure \(\mathrm{Ni}-65,\) find the time when there are twice as many daughter nuclei as parents.

Understanding the radioactive decay equation is crucial in grasping how unstable atoms transform over time. In the case of beta decay, as we've seen with Nickel-65, the equation that describes the process is \( N(t) = N(0) e^{-\lambda t} \), where \( N(t) \) is the number of parent nuclei at time \( t \), and \( N(0) \) is the number of parent nuclei at the beginning of the observation. The constant \( e \) is the base of the natural logarithm, which arises naturally in processes of growth and decay.

The decay constant, \( \lambda \), is specific to each radioactive isotope, representing how quickly the substance decays. It essentially sets the clock for the half-life of the material — the time it takes for half of the parent nuclei to transform into daughter nuclei. By rearranging the equation and applying the characteristics of logarithms, we can solve for \( t \), the time it takes to reach a certain condition, like in our exercise where we determined when there were twice as many daughter nuclei as the remaining parent nuclei.

The decay constant \( \lambda \) plays a pivotal role in nuclear physics, characterizing the rate at which a radioactive substance undergoes decay. It is an intrinsic property of the radioactive isotope in question, and it's inversely proportional to the half-life, which is the time required for half of the radioactive atoms to decay.

To put it simply, a higher decay constant implies that the substance will decay more rapidly; conversely, a lower decay constant indicates a slower rate of decay. In the exercise featuring Nickel-65, the decay constant is given as \( \lambda = 0.275 \mathrm{h}^{-1} \), signifying that every hour, a proportion of the Nickel-65 nuclei will decay to the daughter product following the exponential decay law. This constant enables us to quantitatively express the speed at which parent nuclei transform into daughter nuclei over time.

The daughter nucleus is the product of a radioactive decay process. In beta decay, this occurs when a neutron in the parent atom changes into a proton and an electron, causing a shift in identity for the element. The electron is ejected—this is the beta particle—while the proton remains, slightly increasing the atomic number of the element, thereby forming a new element.

For instance, Nickel-65, which beta decays, sees one of its neutrons turned into a proton, thus transforming into Copper-65. This Copper-65 is the daughter nucleus. It's important to note that although the atomic number changes, the mass number, or the sum of protons and neutrons, remains the same in the case of beta decay. Recognizing the daughter nucleus is essential for understanding the products and implications of radioactive transformations.