Write Lewis structures for the following ions: (a) \(\mathrm{NO}_{2}^{+},\) (b) \(\mathrm{S}_{2}^{2-},\) (c) \(\mathrm{BrF}_{2}^{+}\), (d) \(\mathrm{SCN}^{-}\). Show formal charges.

Understanding valence electrons is fundamental when we delve into the world of chemistry, especially when it comes to writing Lewis structures for molecules and ions. Valence electrons are the outermost electrons of an atom that are involved in chemical bonding. An atom’s chemical properties are essentially defined by its valence electrons, because they are the electrons that have the potential to engage in chemical reactions.

Take, for example, the molecule of \( \mathrm{NO}_{2}^{+} \) from the exercise. Nitrogen has 5 valence electrons in its outer shell, while oxygen has 6. As we create the Lewis structure, these valence electrons are what we use to form bonds or assign as lone pairs around the atoms. The more valence electrons there are, the more bonds an atom can form, generally speaking. In the case of ions, the charge on the ion can either remove or add valence electrons. For instance, the positive charge on \( \mathrm{NO}_{2}^{+} \) means that one electron is removed from the total count of valence electrons.

A crucial skill in evaluating Lewis structures is keeping track of these valence electrons, as they will determine the bonding patterns and the number and types of bonds that atoms can form, as seen in each solution step of the given exercise.

The concept of formal charge comes into play when we want to understand the distribution of electrons in molecules and ions beyond what the valence electrons show us. The formal charge tells us whether an atom is electron-rich or electron-deficient in a molecule when compared with its pure, isolated form. This idea is vital to our analysis because molecules and ions strive for the lowest possible energy state, and this usually means the lowest formal charges possible.

To calculate the formal charge, we use the formula: \( \text{Formal charge} = \text{Valence electrons} - \frac{1}{2} \times \text{Number of bonding electrons} - \text{Number of non-bonding electrons} \). As demonstrated in the solutions for \( \mathrm{S}_{2}^{2-} \) and \( \mathrm{SCN}^{-} \) ions, assigning formal charges can help check the validity of the Lewis structure. Atoms in stable molecules usually have a formal charge of zero, but in complex ions, some atoms may carry a charge. Ensuring that the sum of the formal charges equals the charge of the molecule or ion confirms that we've accounted for all the valence electrons correctly. The formal charges in the exercise solutions reflect this careful balancing act.

The octet rule is a chemical rule of thumb that reflects an atom’s tendency to prefer to have eight electrons in its valence shell, which gives it the same electronic configuration as a noble gas. This configuration is what most atoms aim for since it's associated with a stable, low-energy state.

In the case of \( \mathrm{NO}_{2}^{+} \), we see that despite following the octet rule for oxygen by forming two bonds with nitrogen, the nitrogen atom doesn't achieve a complete octet, showing that there are exceptions to this rule – especially in molecules with an odd number of electrons or where the central atom can have more than eight electrons (expanded octet). Similarly, in the \( \mathrm{BrF}_{2}^{+} \) ion, bromine is an element that can have more than eight electrons around it because it is in the third period, where d-orbitals are available for bonding.

It is important to remember that the octet rule doesn’t always strictly apply, but it serves as a guide in many Lewis structures. In the case of ions like \( \mathrm{S}_{2}^{2-} \) and \( \mathrm{SCN}^{-} \), we diligently distribute the electrons to satisfy the octet for each atom where possible, while also considering the overall charge of the ion.