In each of the following radioactive decay processes, supply the missing particle. a. \({ }^{73} \mathrm{Ga} \rightarrow{ }^{73} \mathrm{Ge}+\) ? b. \({ }^{192} \mathrm{Pt} \rightarrow{ }^{188} \mathrm{Os}+?\) c. \({ }^{205} \mathrm{Bi} \rightarrow{ }^{205} \mathrm{~Pb}+?\) d. \(^{241} \mathrm{Cm}+? \rightarrow{ }^{241} \mathrm{Am}\)

When a radioactive nucleus undergoes beta-minus decay, one of its neutrons is transformed into a proton. During this process, the nucleus emits two particles: an electron and an electron antineutrino (\(u_e\)). This type of decay results in an increase in the atomic number by one, while the mass number remains unchanged.

Imagine a crowded room where someone leaves (the neutron) and another person wearing a different shirt (the proton) enters immediately after; the room's population stays the same, but its composition changes. Similarly, in beta-minus decay, a neutron (\(n\)) becomes a proton (\(p\)), emitting an electron (\(e^-\))—referred to as a beta particle—and an antineutrino. This is what happens with \(^{73}_{31}Ga\) turning into \(^{73}_{32}Ge\).

Applications of beta-minus decay include carbon dating, where scientists can measure the age of archaeological finds by analyzing the decay of \(^{14}C\) into \(^{14}N\).

Alpha decay is another radioactive process whereby an unstable nucleus releases an alpha particle, which consists of two protons and two neutrons (\(^4_2He^{2+}\)), effectively decreasing the atom's mass by four units and the atomic number by two.

It's as if a family of four—two parents and two children—moves out of a building, leading to a notable drop in the building's occupancy. For example, when \(^{192}_{78}Pt\) undergoes alpha decay, it emits an alpha particle and transforms into \(^{188}_{76}Os\). This phenomenon is a common source of emitted radiation in heavy elements such as uranium and thorium, and it's utilized in smoke detectors that contain americium-241.

In contrast to beta-minus decay, beta-plus decay involves the transformation of a proton inside the nucleus into a neutron, accompanied by the release of a positron (\(e^+\)) and a neutrino (\(u_e\)). This ultimately leads to a decrease in the atomic number by one, leaving the mass number unaltered.

Using our previous analogy, this would be like someone putting on a different shirt, effectively changing their appearance. In beta-plus decay, a proton (\(p\)) changes into a neutron (\(n\)), releasing a positron—the positively charged counterpart to the electron—and a neutrino. For example, \(^{205}_{83}Bi\) decays into \(^{205}_{82}Pb\). Beta-plus decay is an essential aspect of the operation of Positron Emission Tomography (PET) scanners in medical imaging.

The process of electron capture is somewhat akin to beta-plus decay. However, rather than emitting a positron, the nucleus draws an inner orbital electron into its midst. This captured electron then combines with a proton to form a neutron and a neutrino. The atomic number decreases by one, while the mass number stays the same.

Imagine a game of tag where, instead of running away, the target willingly joins the seeker. During electron capture, a proton in the nucleus 'tags' an electron, usually from the closest orbit, leading to the emission of a neutrino and the formation of a new neutron. When \(^{241}_{96}Cm\) captures an electron, it results in \(^{241}_{95}Am\). Electron capture is an important decay mode for isotopes that are unable to emit a positron due to energy considerations, and it also plays a crucial role in the stellar nucleosynthesis of elements inside stars.