Write a balanced equation for each of the following nuclear reactions: (a) bismuth-212 decays into polonium-212 (b) beryllium- 8 and a positron are produced by the decay of an unstable nucleus (c) neptunium- 239 forms from the reaction of uranium- 238 with a neutron and then spontaneously converts into plutonium-239 (d) strontium-90 decays into yttrium-90

Alpha decay is a type of radioactive decay wherein an unstable atomic nucleus emits an alpha particle, which consists of two protons and two neutrons, the same as a helium nucleus (\text{^{\textrm{4}}_{\textrm{2}}He}). This process reduces the mass number of the original nucleus by four and the atomic number by two. For instance, when bismuth-212 undergoes alpha decay, it transforms into lead-208 (\text{^{\textrm{208}}_{\textrm{82}}Pb}), expelling an alpha particle in the process.
The equation representing this decay should balance both mass and atomic numbers on each side, as shown below:
\[ \text{^{\textrm{212}}_{\textrm{83}}Bi} \rightarrow \text{^{\textrm{208}}_{\textrm{82}}Pb} + \text{^{\textrm{4}}_{\textrm{2}}He} \]
Alpha decay typically occurs in the heaviest elements found in nature and is one of the common modes of decay for radionuclides.

Beta plus decay is a nuclear process where a proton in the nucleus is transformed into a neutron, releasing a positron (\text{e^+}) and a neutrino. This type of decay results in the decrease of the atomic number by one, while the mass number remains unchanged, as the conversion involves particles within the nucleus without affecting its overall mass. For instance, a boron-8 nucleus can decay into a beryllium-8 nucleus, emitting a positron.
The balanced nuclear equation for this reaction is written as:
\[ \text{^{\textrm{8}}_{\textrm{5}}B} \rightarrow \text{^{\textrm{8}}_{\textrm{4}}Be} + \text{^{\textrm{0}}_{+1}e} \]
Beta plus decay is common in proton-rich nuclei and is an essential process for the production of many isotopes used in medical imaging and treatments.

Neutron capture is a nuclear reaction in which a neutron is absorbed by a nucleus, resulting in a compound nucleus. This process increases the atomic mass of the resulting isotope by one, but does not change the atomic number, as neutrons are neutrally charged. Moreover, neutron capture often occurs in reactors and stars, leading to the creation of heavier elements.
The formation of neptunium-239 from uranium-238 is an example of neutron capture, followed by beta decay, which is represented by the equations:
\[ \text{^{\textrm{238}}_{\textrm{92}}U} + \text{^{\textrm{1}}_{\textrm{0}}n} \rightarrow \text{^{\textrm{239}}_{\textrm{93}}Np} \]
and then:
\[ \text{^{\textrm{239}}_{\textrm{93}}Np} \rightarrow \text{^{\textrm{239}}_{\textrm{94}}Pu} + \text{^{\textrm{0}}_{-\textrm{1}}e} \]
This two-step process signifies the transformation of one element into another through neutron capture and subsequent decay.

Beta decay is a common decay process whereby a radioactive nucleus emits a beta particle, converting a neutron into a proton, or vice-versa, resulting in a change in the atomic number. Beta particles can be either electrons (\text{e^-}) in beta minus decay or positrons (\text{e^+}) in beta plus decay.
An example of beta decay is the transformation of strontium-90 into yttrium-90. In this case, a neutron in the strontium-90 nucleus changes to a proton, and an electron is emitted, as noted in the equation:
\[ \text{^{\textrm{90}}_{\textrm{38}}Sr} \rightarrow \text{^{\textrm{90}}_{\textrm{39}}Y} + \text{^{\textrm{0}}_{-\textrm{1}}e} \]
This decay results in an increase in the atomic number but the mass number remains unaffected. Beta decay plays a crucial role in the radioactive processes that heat the Earth's core and also has applications in medical therapies and archaeological dating.