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In order to identify the particles produced, we need to first identify the decay process occurring in each case. \((\mathbf{a})\) Sodium-24 decays to magnesium-24: This is a beta decay process, as the atomic number (number of protons) increases by 1 while the atomic mass (number of protons and neutrons) remains the same. \((\mathbf{b})\) Mercury-188 decays to gold-188: This is also a beta decay process, as the atomic number increases by 1, and the atomic mass remains the same. \((\mathbf{c})\) Iodine-122 decays to xenon-122: In this decay process, the atomic number decreases by 1 while the atomic mass remains the same. This is a positron emission process (beta-plus decay). \((\mathbf{d})\) Plutonium-242 decays to uranium-238: This decay process involves the atomic number decreasing by 2, and the atomic mass decreasing by 4. This indicates an alpha decay process. Now that the decay processes are identified, we can determine the particles produced during each decay.

Based on the type of decay identified in Step 1, we can now identify the particle produced in each decay process. \((\mathbf{a})\) Sodium-24 decays to magnesium-24: Beta decay produces an electron \(\beta^{-}\) (a beta-minus particle). \((\mathbf{b})\) Mercury-188 decays to gold-188: Beta decay produces an electron \(\beta^{-}\) (a beta-minus particle). \((\mathbf{c})\) Iodine-122 decays to xenon-122: Positron emission produces a positron \(\beta^{+}\) (a beta-plus particle). \((\mathbf{d})\) Plutonium-242 decays to uranium-238: Alpha decay produces an alpha particle, which consists of 2 protons and 2 neutrons, represented by \(_2^4\text{He}\). And there we have it! We've identified the particle produced during each decay process: an electron (\(\beta^{-}\)) for \((\mathbf{a})\) and \((\mathbf{b})\), a positron (\(\beta^{+}\)) for \((\mathbf{c})\), and an alpha particle for \((\mathbf{d})\).