Question

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. \(^{241} \mathrm{Cm}+? \rightarrow{ }^{241} \mathrm{Am}\)

Short answer

a. β¯ (electron) + νₑ (electron antineutrino) b. α (alpha particle) c. β⁺ (positron) + νₑ (neutrino) d. e⁻ (electron) + νₑ (neutrino)

Step-by-step solution

a. Finding the missing particle in the decay of \({ }^{73} \mathrm{Ga} \rightarrow{ }^{73} \mathrm{Ge}+\)

In this decay process, the mass number is conserved (73), while the atomic number increases by 1 (from 31 for Ga to 32 for Ge). This change in atomic number is characteristic of beta-minus decay, in which a neutron is transformed into a proton and emits an electron (known as a beta particle) and an electron antineutrino. So, the missing particle is a beta-minus particle (electron) and an electron antineutrino. Missing particle: β¯ (electron) + νₑ (electron antineutrino)

b. Finding the missing particle in the decay of \({ }^{192} \mathrm{Pt} \rightarrow{ }^{188} \mathrm{Os}+?\)

In this decay process, the mass number decreases by 4 (from 192 for Pt to 188 for Os) and the atomic number remains the same (from 78 for Pt to 76 for Os). This change in mass number is characteristic of alpha decay, in which two protons and two neutrons are emitted from the nucleus in the form of an alpha particle. Missing particle: α (alpha particle)

c. Finding the missing particle in the decay of \({ }^{205} \mathrm{Bi} \rightarrow{ }^{205} \mathrm{~Pb}+?\)

In this decay process, the mass number is conserved (205), while the atomic number decreases by 1 (from 83 for Bi to 82 for Pb). This change in atomic number is characteristic of beta-plus decay or positron emission, in which a proton is transformed into a neutron and emits a positron and a neutrino. So, the missing particle is a beta-plus particle (positron) and a neutrino. Missing particle: β⁺ (positron) + νₑ (neutrino)

d. Finding the missing particle in the reaction \(^{241} \mathrm{Cm}+? \rightarrow{ }^{241} \mathrm{Am}\)

In this reaction process, the mass number is conserved (241), while the atomic number decreases by 1 (from 96 for Cm to 95 for Am). This change in atomic number is characteristic of electron capture, in which a proton in the nucleus captures an electron from the inner electron shell and transforms into a neutron, emitting a neutrino. So, the missing particle is an electron (captured from the inner shell) and a neutrino in the reactants. Missing particle: e⁻ (electron) + νₑ (neutrino)

Key concepts

Beta-minus Decay

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

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.

Beta-plus Decay

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.

Electron Capture

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.