Give one example each of (a) \(\alpha\) -emission, (b) \(\beta^{+}\) -emission, and (c) K-capture. Write the equation for these nuclear changes.

Imagine an unstable nucleus in need of shedding some weight to become stable. This is where alpha emission comes into play. Consider it as a nucleus throwing out a helium nucleus, which is made up of two protons and two neutrons. We call this a helium nucleus an alpha particle (\text{He}).

Let's make this idea concrete with an example. Uranium-238 (U), a heavy guy sitting with too many protons and neutrons, manages to slim down by emitting an alpha particle and transforms into Thorium-234 (Th). This can be visualized through the nuclear equation:\[ _{92}^{238}\text{U} \rightarrow _{90}^{234}\text{Th} + _{2}^{4}\text{He} \].

This type of emission is key in understanding the puzzle of nuclear changes and is vital to topics such as radioactivity and nuclear chemistry.

Now let's switch our attention to positron emission, also known as beta-plus decay. In this scenario, you have a proton-rich nucleus that wants to restore balance by converting one of its excess protons into a neutron. In the process, it releases a positron - the antimatter twin of an electron - and a neutrino, a very light, neutral particle that rarely interacts with matter. This process can be captured in a simple formula:\[ _{Z}^{A}\text{X} \rightarrow _{Z-1}^{A}\text{Y} + _{+1}^{0}\text{e} + u \].

If this seems a bit abstract, here's an example: Carbon-11, with too many protons, can undergo this decay to become Boron-11:\[ _{6}^{11}\text{C} \rightarrow _{5}^{11}\text{B} + _{+1}^{0}\text{e} + u \].
Understanding positron emission is not only a cornerstone of nuclear chemistry but also has practical applications in medical imaging, such as positron emission tomography (PET scans).

Picture a nucleus with one too many protons, which makes it a bit wobbly. It can stabilize itself simply by grabbing one of its own electrons, usually from the closest orbit, the K-shell, and pulling it into the nucleus. This process is known as K-capture or K-electron capture.

The nucleus does a bit of alchemy by turning a proton into a neutron and releasing energy, often in the form of an X-ray. The notation is:\[ _{Z}^{A}\text{X} + _{-1}^{0}\text{e} \rightarrow _{Z-1}^{A}\text{Y} + u \].

For example, Beryllium-7 (\text{Be}) can snatch an electron to become stable Lithium-7 (\text{Li}):\[ _{4}^{7}\text{Be} + _{-1}^{0}\text{e} \rightarrow _{3}^{7}\text{Li} + u \].
K-capture is another fascinating aspect of nuclear changes and radioactivity that shows how the interplay between electrons and protons can lead to the transmutation of elements.

All the above-mentioned decay processes are examples of nuclear changes. These changes are at the heart of nuclear chemistry, allowing us to peek into the atomic realm and see how unstable atoms pursue stability. From alpha particles jetting off like tiny space probes to protons transforming into neutrons, each nuclear change rearranges the very identity of atoms.

One fascinating fact about these changes is that they're responsible for the elements we see around us today. The elements formed in stars through fusion and other nuclear reactions later change through decay processes, providing us with a universe rich in chemical diversity.

Radioactivity is the spontaneous emission of radiation from an unstable atomic nucleus. It's a natural phenomenon that occurs because atoms are striving for a more stable state. Whether through alpha particles, positrons, or capturing electrons, every radioactive decay process is a step towards stability.

While radioactivity can be dangerous due to the energy released during decay, it's also a powerful tool for scientists. We use it to date ancient artifacts with radiocarbon dating, treat cancer through radiation therapy, and produce energy in nuclear reactors. Understanding radioactivity is not just about knowing the hazards; it's about harnessing the power of atoms for the benefit of society.

Finally, let’s bring all these concepts together under the umbrella of nuclear chemistry, a field that dives into the reactions and changes within the nucleus of an atom. It stretches beyond the mere shuffling of electrons in chemical reactions to the transformations that dictate the identities of elements themselves.

Through this lens, we see how elements come into being, alter, and sometimes vanish, giving rise to entirely new substances. This is the domain where we can understand not just the chemical behavior of elements but their very origins and fate in the vast timeline of the cosmos.

Nuclear chemistry bridges the gap between the physics of atomic particles and the practical application of chemistry in the real world, making it a key subject in advanced scientific study and industry applications alike.