Question

Write an equation describing the radioactive decay of each of the following nuclides. (The particle produced is shown in parentheses, except for electron capture, where an electron is a reactant.) a. \(^{68}\) Ga (electron capture) b. \(^{62} \mathrm{Cu}\) (positron) c. \(^{212} \operatorname{Fr}(\alpha)\) d. \(^{129} \operatorname{Sb}(\beta)\)

Short answer

The radioactive decay equations for the given nuclides are: a. Electron Capture for \(^{68}\textrm{Ga}\): \(^{68}\textrm{Ga} + e^- \rightarrow ^{68}\textrm{Zn}\) b. Positron Emission for \(^{62}\textrm{Cu}\): \(^{62}\textrm{Cu} \rightarrow ^{62}\textrm{Ni} + e^+\) c. Alpha Decay for \(^{212}\textrm{Fr}\): \(^{212}\textrm{Fr} \rightarrow ^{208}\textrm{At} + \alpha\) d. Beta Decay for \(^{129}\textrm{Sb}\): \(^{129}\textrm{Sb} \rightarrow ^{129}\textrm{Te} + \beta^-\)

Step-by-step solution

Identify Nuclide and Decay Type for each case

For each case, we have a nuclide that undergoes a specific type of decay (electron capture, positron emission, alpha decay, or beta decay). Let's first write down the given information for each case. a. \(^{68}\textrm{Ga}\) undergoes electron capture. b. \(^{62}\textrm{Cu}\) undergoes positron emission. c. \(^{212}\textrm{Fr}\) undergoes alpha decay. d. \(^{129}\textrm{Sb}\) undergoes beta decay.

Understand each decay type and its corresponding notation

Next, we need to understand each decay type and its corresponding notation for the decay process. 1. Electron Capture: In electron capture, the nucleus captures an inner-shell electron. This means that a proton in the nucleus converts into a neutron by absorbing an electron. The notation for electron capture is: \(p^+ + e^- \rightarrow n\) 2. Positron Emission: In positron emission, the nucleus emits a positron (\(e^+\)), which is an elementary particle with the same mass as an electron but with a positive charge. In this process, a neutron transforms into a proton, releasing a positron. The notation for positron emission is: \(n \rightarrow p^+ + e^+\) 3. Alpha Decay: In alpha decay, the nucleus emits an alpha particle, which is a helium nucleus with two protons and two neutrons (\(^4\textrm{He}\) or \(\alpha\)). The notation for alpha decay is: \(X \rightarrow Y + \alpha\) 4. Beta Decay: In beta decay, the nucleus emits a beta particle (\(\beta^-\)), which consists of an electron. In this process, a neutron converts into a proton, releasing an electron. The notation for beta decay is: \(n \rightarrow p^+ + \beta^-\)

Write the decay equations for each case

Now, we can write the decay equations for the given nuclides using the notations we have just discussed. a. Electron Capture for \(^{68}\textrm{Ga}\): \(^{68}\textrm{Ga} + e^- \rightarrow ^{68}\textrm{Zn}\) (One proton in \(^{68}\textrm{Ga}\) nucleus captures an electron and converts into a neutron, resulting in a new element, \(^{68}\textrm{Zn}\)) b. Positron Emission for \(^{62}\textrm{Cu}\): \(^{62}\textrm{Cu} \rightarrow ^{62}\textrm{Ni} + e^+\) (A neutron in \(^{62}\textrm{Cu}\) nucleus converts into a proton, emitting a positron, resulting in a new element, \(^{62}\textrm{Ni}\)) c. Alpha Decay for \(^{212}\textrm{Fr}\): \(^{212}\textrm{Fr} \rightarrow ^{208}\textrm{At} + \alpha\) (\(^{212}\textrm{Fr}\) nucleus emits an alpha particle, resulting in a new element, \(^{208}\textrm{At}\)) d. Beta Decay for \(^{129}\textrm{Sb}\): \(^{129}\textrm{Sb} \rightarrow ^{129}\textrm{Te} + \beta^-\) (A neutron in \(^{129}\textrm{Sb}\) nucleus converts into a proton, emitting a beta particle, resulting in a new element, \(^{129}\textrm{Te}\))

Key concepts

Electron Capture

Electron capture is a process in which an unstable atom's nucleus absorbs an inner-shell electron, leading to a transformation of a proton into a neutron. Contrary to many other forms of radioactive decay, no particles are ejected from the nucleus; instead, an orbiting electron is consumed.

In terms of notations, electron capture can be represented as follows:
\[ p^+ + e^- \rightarrow n \]
Here, \( p^+ \) denotes a proton, \( e^- \) an electron, and \( n \) a neutron. As a result of this process, the atomic number decreases by one, changing the element to a different one on the periodic table, while the mass number remains unchanged. For example, when gallium-68 (\(^{68}\textrm{Ga}\)) undergoes electron capture, it transforms into zinc-68 (\(^{68}\textrm{Zn}\)) as depicted:
\[ ^{68}\textrm{Ga} + e^- \rightarrow ^{68}\textrm{Zn} \]
Understanding electron capture is significant in various fields like nuclear physics and medical diagnostics.

Positron Emission

Positron emission, also known as beta-plus decay (\(\beta^+\)-decay), is a type of radioactive decay where an unstable nucleus emits a positron. The positron is a particle with the same mass as an electron, but with a positive charge.

The end result of positron emission is that one of the atom's neutrons is transformed into a proton. The equation representing this process is:
\[ n \rightarrow p^+ + e^+ \]
Through this transformation, the atom increases its atomic number by one but does not change its mass number. Taking copper-62 (\(^{62}\textrm{Cu}\)) as an example, when it undergoes positron emission, it decays into nickel-62 (\(^{62}\textrm{Ni}\)):
\[ ^{62}\textrm{Cu} \rightarrow ^{62}\textrm{Ni} + e^+ \]
Positron emission has implications for medical imaging techniques such as positron emission tomography (PET), which utilizes this decay to diagnose and monitor diseases.

Alpha Decay

Alpha decay is a common mode of radioactive decay in which an unstable nucleus ejects an alpha particle, consisting of two protons and two neutrons (essentially, a helium-4 nucleus).

This type of decay results in a new nucleus with a mass number reduced by four and an atomic number lowered by two. The general equation for alpha decay is expressed as:
\[ X \rightarrow Y + \alpha \]
An example of this decay process is with francium-212 (\(^{212}\textrm{Fr}\)), which decays into astatine-208 (\(^{208}\textrm{At}\)):
\[ ^{212}\textrm{Fr} \rightarrow ^{208}\textrm{At} + \alpha \]
Alpha particles are relatively large and carry a double positive charge. Although they can be harmful if inhaled or ingested, their low penetrative power renders them generally harmless to human skin. Understanding alpha decay is crucial in areas such as radiometric dating and radiation therapy.

Beta Decay

Beta decay occurs when a neutron in an unstable atomic nucleus transforms into a proton, emitting an electron (beta particle) and an antineutrino in the process. It's a way for the nucleus to achieve a more stable neutron-to-proton ratio.

The general equation for this transformation is:
\[ n \rightarrow p^+ + \beta^- \]
In this expression, \( p^+ \) represents the resulting proton and \( \beta^- \) stands for the emitted electron. Beta decay increases the atomic number by one while the atomic mass remains unchanged. As seen in the decay of antimony-129 (\(^{129}\textrm{Sb}\)), it transforms into tellurium-129 (\(^{129}\textrm{Te}\)):
\[ ^{129}\textrm{Sb} \rightarrow ^{129}\textrm{Te} + \beta^- \]
The study of beta decay is significant for nuclear power generation, nuclear medicine, and understanding fundamental particles in physics.