The isotopes \({ }^{17} \mathrm{~F}^{20} \mathrm{~F}\), and \({ }^{21} \mathrm{~F}\) are all radioactive, decaying either by beta emission or by positron emission. Name the decay process for each isotope.

In the intricate world of nuclear chemistry, beta emission stands out as a fundamental radioactive decay process. When a nucleus undergoes beta emission, it emits a beta particle, which is a high-speed electron or positron. This process happens when there is an imbalance between the number of protons and neutrons within an atomic nucleus. To correct this imbalance, a neutron in the nucleus transforms into a proton and in the process, emits an electron (also known as a beta-minus particle).

The emit of an electron increases the atomic number by one, however, the mass number remains unchanged because the loss of a neutron is balanced by the creation of a proton. Mathematically, this can be depicted by the nuclear reaction:\[ _{Z}^{A}X \rightarrow _{Z+1}^{A}Y + _{-1}^{0}e \]
In this equation, ‘X’ represents the parent atom, ‘Y’ depicts the new atom formed, '_{-1}^{0}e' is the emitted beta particle, 'Z' is the atomic number, and 'A' is the mass number. Beta emission is a critical concept as it fundamentally alters the chemical element of the atom, leading to transmutation. This decay process also aids in the quest for stable configurations by nuclei with an excess of neutrons.

Positron emission is the twin brother of beta emission with an optimistic twist. It involves the decay of a nucleus by the emission of a positron, the antiparticle or antimatter counterpart of the electron. This occurs when a proton within the atomic nucleus is transformed into a neutron and a positron is emitted to conserve charge. Unlike beta emission which increases the atomic number, positron emission decreases it by one. Therefore, the daughter nucleus forms an element with an atomic number less by one. The general equation for positron emission is:\[ _{Z}^{A}X \rightarrow _{Z-1}^{A}Y + _{+1}^{0}e \]
Here 'X' is the original atom, 'Y' is the new atom created, and '_{+1}^{0}e' signifies the positron. The mass number 'A' does not change. Positron emission is prevalent in isotopes with low neutron-to-proton ratios, and it's an attempt by nuclei to achieve a more stable state by reducing protons.

Fluorine, with its symbol 'F' and atomic number 9, is not content with just one form; it flaunts a variety of isotopes. Isotopes are atoms of the same element that have different numbers of neutrons, thus differing in mass numbers. Fluorine's most common isotope is fluorine-19, which is stable. However, when you encounter isotopes like fluorine-17, fluorine-20, and fluorine-21, you stumble upon their radioactive nature. These unstable versions of fluorine undergo radioactive decay to achieve stability.

Fluorine-17 and fluorine-21 experience beta emission, resulting in an increase in their atomic number, leading to the formation of neon isotopes. Meanwhile, fluorine-20 takes the route of positron emission, bringing its atomic number down and creating an isotope of oxygen. These decay processes are critical for understanding the principles of nuclear transformations and provide invaluable insights into the behavior of radioactive isotopes.