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. \(^{129} \operatorname{Sb}(\beta)\)

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.

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: \(n \rightarrow p^+ + \beta^-\)

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, \(^{129}\textrm{Te}\))