Write a nuclear equation for the indicated decay of each nuclide. $$ \begin{array}{l}{\text { a. } U-234 \text { (alpha) }} \\ {\text { b. } T h-230 \text { (alpha) }} \\ {\text { c. } P b-214 \text { (beta) }} \\\ {\text { d. } N-13 \text { (positron emission) }} \\ {\text { e. } C r-51 \text { (electron capture) }}\end{array} $$

Nuclear stability is a balancing act, and when a nuclide lies on the heavier side, it often undergoes alpha decay to shed its excess weight and reach a more stable state. Alpha decay involves the emission of an alpha particle, effectively a helium-4 nucleus, composed of 2 protons and 2 neutrons.For example, when Uranium-234 (U-234) goes through alpha decay, it transforms into Thorium-230 (Th-230) by ejecting an alpha particle. In doing so, the Uranium's mass number drops by 4 (to account for the 2 protons and 2 neutrons lost) and its atomic number decreases by 2 (reflecting the loss of the 2 protons). It's like a family waving goodbye to their two children as they head off to college; the family dynamic changes. Likewise, when Thorium-230 (Th-230) decays, it emits an alpha particle as well, birthing Radium-226 (Ra-226). Parent nuclides 'shedding their weight' to become more stable is at the core of alpha decay.

In contrast to the heavyweights performing alpha decay, certain nuclides suffer from a neutron-proton imbalance that tilts towards neutrons. Beta decay serves as the corrective mechanism by which a neutron is transformed into a proton with the emission of an electron, known as a beta particle. It's akin to a crowded dance floor where one person decides to switch dance styles to better fit with the group.Take Lead-214 (Pb-214) for example. It can't keep its neutron-proton tango balanced, so it undergoes beta decay. This neutron-to-proton transformation nudges the atomic number up by one, bringing Pb-214 to Bismuth-214 (Bi-214) while the mass remains constant at 214. This 'dance switch' allows the nuclide to achieve a more stable state with a more even mix of protons to neutrons.

Conversely, some nuclides may find themselves with an excess of protons, which disrupts the stable dance of nuclear forces. Positron emission is one way to redress this imbalance. Similar to beta decay, but with a twist, a proton in the nucleus is converted into a neutron, releasing a positively charged positron—the antimatter twin of the electron.Nitrogen-13 (N-13) represents this scenario where a proton in its nucleus taps out, becoming a neutron and releasing a positron. The atomic number, therefore, decreases by one, as one proton leaves the atomic roster. In this case, N-13 (with 7 protons) turns into Carbon-13 (C-13), ending up with 6 protons. Such a rebalancing act helps the nucleus hit its proverbial sweet spot of stability.

Electron capture could be seen as the introverted cousin of positron emission. Instead of releasing a positron, an unstable nucleus with a proton surplus 'grabs' an electron from its own electron cloud, usually from the innermost shell. This electron is then merged with a proton to form a neutron, effectively decreasing the atomic number by one.A palpable example is Chromium-51 (Cr-51). It's like a shopper who realizes they have one too many items in their cart and opts to put something back, resulting in Chromium capturing an electron which combines with a proton. This yields Vanadium-51 (V-51), with one less proton thus one less atomic number while retaining the original mass number. It's a subtle yet profound change that leads the unstable nuclide on a path towards a more serene nuclear existence.