Write the balanced nuclear equation for each of the following nuclear reactions: (a) electron capture by iron-55, (b) beta emission by potassium-42, (c) positron emission by ruthenium-93, (d) alpha emission by californium- 251 .

Electron capture is a fascinating process that occurs inside an atom's nucleus. Atomic nuclei are made of protons and neutrons, with protons being positively charged and neutrons electrically neutral. Sometimes, an atom's nucleus captures one of its own electrons, typically from the innermost shell.

This captured electron combines with a proton to produce a neutron and a neutrino, a nearly massless particle. What's spectacular about electron capture is that it causes the atom to transform into a different element because the number of protons, which determines the element's identity, decreases by one. However, the mass number, which is the total number of protons and neutrons, doesn't change.

For example, in the electron capture by iron-55, one electron (\(e^-\)) is captured by the nucleus (\(^{55}_{26}Fe\)), resulting in a manganese-55 atom (\(^{55}_{25}Mn\)) and the emission of an electron neutrino (\(u_e\)).

Beta emission is another type of nuclear decay which plays a key role in the natural transmutation of elements. Unlike electron capture, beta emission increases the atomic number of an element by one, while keeping the mass number unchanged. This increase is due to the conversion of a neutron in the nucleus into a proton, which is accompanied by the release of a beta particle (which is an electron) and an antineutrino.

Consider potassium-42 (\(^{42}_{19}K\)). It emits a beta particle (\(e^-\)), along with an antineutrino (\(\bar{u}_e\)), resulting in calcium-42 (\(^{42}_{20}Ca\)). This process is a crucial part of many natural radioactive series and has applications in medical imaging and treatment, as well as carbon dating in archaeology.

Positron emission is quite the opposite of beta emission. During positron emission, a proton inside the nucleus is transformed into a neutron and releases a positron (\(e^+\)), which is the antimatter counterpart of the electron, along with a neutrino. This nuclear reaction, again, leaves the mass number constant but lowers the atomic number by one, leading to the transformation into a new element.

For instance, ruthenium-93 (\(^{93}_{44}Ru\)) undergoes positron emission to become technetium-93 (\(^{93}_{43}Tc\)). Positron emission is not just a theoretical concept but is employed in positron emission tomography (PET), a cutting-edge medical imaging technique that helps detect diseases in the body.

Alpha emission is another way that unstable nuclei can reach a more stable state. An alpha particle, which consists of two protons and two neutrons bound together (\(^{4}_{2}He\)), is ejected from the nucleus. This emission results in the reduction of both atomic and mass numbers—by two and four, respectively. Alpha particles are relatively heavy and carry a double positive charge.

Californium-251 (\(^{251}_{98}Cf\) demonstrates alpha emission by releasing an alpha particle to form curium-247 (\(^{247}_{96}Cm\)). This process is crucial for understanding the synthesis and decay of heavy elements and is a common type of radiation in radioactive decay series seen in elements such as uranium and radium.