Write balanced nuclear equations for the following transformations: \((\mathbf{a})\) bismuth-213 undergoes alpha decay; \((\mathbf{b})\) nitrogen-13 undergoes electron capture; \((\mathbf{c})\) technicium-98 undergoes electron capture; \((\mathbf{d})\) gold-188 decays by positron emission.

Alpha decay is a type of radioactive decay where an unstable nucleus emits an alpha particle, which consists of two protons and two neutrons — essentially a helium nucleus. This process transmutes the original element into a new element with an atomic number decreased by two and a mass number decreased by four. It's one of the most common forms of natural radioactive decay.

For example, with bismuth-213 undergoing alpha decay, we write the equation:
\[^{213}_{83}Bi \rightarrow ^{209}_{81}Tl + ^4_2He\].
This shows that bismuth-213 loses an alpha particle and becomes thallium-209 in the process. Such equations are crucial in understanding how radioactive materials change over time and can be used to calculate the half-lives of radioactive isotopes.

Electron capture is another fascinating process in nuclear chemistry where an inner orbital electron is captured by the nucleus of its own atom. During this interaction, a proton in the nucleus is transformed into a neutron while the electron essentially 'disappears', resulting in the emission of an electron neutrino.

For both nitrogen-13 and technetium-98 examples, the equations are as follows:
\[^{13}_7N + ^0_{-1}e \rightarrow ^{13}_6C\]
and
\[^{98}_{43}Tc + ^0_{-1}e \rightarrow ^{98}_{42}Mo\].
These reactions demonstrate the conversion of one element to another through the reduction of the atomic number by one without changing the mass number. Electron capture is particularly important in the field of nuclear medicine, where it's used in diagnostic imaging.

Positron emission, also known as beta plus decay (\(\beta^+\) decay), occurs when a proton in a nucleus is transformed into a neutron and a positron, which is the antiparticle of an electron. The positron is expelled from the atom, and its ejection helps to conserve energy, momentum, and angular momentum in the system.

The equation for gold-188 decaying by positron emission looks like this:
\[^{188}_{79}Au \rightarrow ^{188}_{78}Pt + ^0_{+1}e\].
This process decreases the atomic number by one while the mass number remains unchanged. Positron emission is used in positron emission tomography (PET), a powerful medical imaging technique that provides three-dimensional images of metabolic processes in the body.

Nuclear chemistry dives into the heart of atoms, where it examines and explains the reactions and processes that occur within the nucleus. It encompasses a broad range of topics, including radioactive decay, nuclear fission and fusion, and radiochemical techniques.

Understanding nuclear chemistry is vital not only for solving nuclear equations but also for harnessing nuclear energy, developing new medical treatments, and managing radioactive materials safely. It allows scientists to predict the behavior of radioactive elements, design nuclear reactors, and understand the effects of radiation on biological systems.