Prove mathematically that it is more energetically favorable for a fluorine atom to take an electron from a sodium atom than for a fluorine atom to take an electron from another fluo- rine atom.

Electron transfer between atoms occurs during chemical reactions, which are the foundation of chemistry. The energetics of electron transfer are essential for understanding why certain chemical processes happen and others do not. This depends on the balance between the ionization energy, the energy required to remove an electron from an atom, and the electron affinity, the energy released when an atom gains an electron.

When dealing with the transfer of an electron from one atom to another, two main energy parameters need to be considered: the ionization energy of the donor atom and the electron affinity of the acceptor atom. The overall energy change for the transfer can be determined by subtracting the electron affinity of the acceptor from the ionization energy of the donor:\[\begin{equation} \Delta E = IE_{\text{donor}} - EA_{\text{acceptor}} \end{equation}\]Where \(\Delta E\) is the energy change, \(IE_{\text{donor}}\) is the ionization energy of the donor atom, and \(EA_{\text{acceptor}}\) is the electron affinity of the acceptor atom. If the result is a positive value, energy is absorbed for the transfer; if negative, energy is released.

Energetically favorable reactions are typically those that release energy, indicating that the formation of products is more stable than the reactants. Understanding these energy changes is crucial in predicting the likelihood of an electron transfer reaction, which has practical applications in the field of chemistry.

Fluorine is a highly electronegative element, meaning it has a strong tendency to gain electrons. The ionization of fluorine refers to the process of removing an electron to form a positively charged ion. However, due to fluorine's high electronegativity, its ionization energy is significantly high, meaning it requires a substantial amount of energy to strip an electron away from a fluorine atom.

The process of a fluorine atom losing an electron - its ionization - is not a naturally favored process. In chemical terms, this is because the removal of an electron from a fluorine atom disrupts the stable electron configuration of the atom which is already close to the noble gas configuration. Essentially, you'd be taking an electron from an electron-hungry atom, effectively forcing it into a less stable, higher-energy state which it resists.

Understanding this concept is important when examining electron transfer reactions involving fluorine. Due to the unfavorable nature of fluorine ionization, a fluorine atom is much more likely to be on the receiving end of electron transfer, thereby acting as an electron acceptor and achieving a more stable, lower-energy state as a fluoride ion (F^{-}). This characteristic underlies many of the chemical behaviors of fluorine and its compounds.

Sodium, on the other hand, is an element with a single electron in its outermost shell, making it quite prone to losing that electron and becoming a positively charged ion, or cation. The ionization energy of sodium represents the energy required to remove that outermost electron. Relative to other elements, sodium's ionization energy is low, because removing the single outer electron achieves a stable electron configuration, similar to that of the noble gas neon.

This low ionization energy of sodium explains why sodium readily undergoes ionization to form Na^+ ions. The energetics in this case favor the loss of an electron as it results in a more stable electron configuration and lowers the overall potential energy of the atom. Thus, in an electron transfer scenario, as was shown in the exercise, sodium atoms act as electron donors.

When considering the interaction between sodium and fluorine, sodium's low ionization energy coupled with fluorine's high electron affinity makes the transfer of an electron from sodium to fluorine energetically very favorable. The result is the creation of sodium ions (Na^+) and fluoride ions (F^-), a process that is key to the formation of compounds like sodium fluoride (NaF), which is widely used in toothpaste and water fluoridation.