Draw Lewis structures and determine the formal charge on each atom in (a) \(\mathrm{NO}^{+}\); (b) \(\mathrm{NH}_{2}^{-}\); (c) \(\mathrm{CH}_{3}^{-}\).

Lewis structures are diagrammatic representations of the bonding between atoms within a molecule and the lone pairs of electrons that may exist in the molecule. They play a vital role in predicting the geometry of a molecule, the reactivity, and the physical properties it may have. To draw a Lewis structure, count the total number of valence electrons that each atom contributes to the molecule or ion. Include any extra electrons resulting from negative charges or subtract electrons equal to the positive charge.

For instance, in the exercise provided, \(\mathrm{NO}^{+}\) loses an electron due to its positive charge, affecting the electron arrangement. Once the electrons are counted, they are arranged around atoms, pairing them to form chemical bonds. Remaining unpaired electrons are depicted as dots representing lone pairs. Electrons are placed to satisfy the requirements of the octet rule wherever possible, with elements trying to complete an octet of electrons in their valence shell.

Valence electrons are the outermost electrons of an atom and are crucial to forming chemical bonds. They are the electrons involved when atoms bond with each other. The number of valence electrons determines an atom's chemical properties and its ability to bond with other atoms. Main group elements have valence electrons in the same number as their group number. For example, nitrogen is in group 15, so it has 5 valence electrons.

The first step in solving our exercise is tallying up the valence electrons. For the \(\mathrm{NH}_{2}^{-}\) ion, nitrogen brings 5 valence electrons, and each hydrogen brings 1. The negative charge indicates an additional electron, making it 5 + 1 + 1 + 1 = 8 valence electrons in total. With this count, we start drawing the Lewis structure.

The octet rule is a chemical rule of thumb that reflects observation that atoms of main-group elements tend to bond in such a way that each atom has eight electrons in its valence shell. This gives the atom the electron configuration of a noble gas, making it energetically stable. There are exceptions, such as hydrogen and helium, which are stable with a duet, or two electrons. In the case of \(\mathrm{CH}_{3}^{-}\), carbon aims to achieve an octet by forming four bonds, one with each hydrogen, and it accommodates the extra electron from the negative charge as a lone pair.

When calculating formal charges, the octet rule helps in predicting the most likely arrangement of electrons, guiding where to place lone pairs and how many bonds each atom probably has.

Electron arrangement refers to how electrons are distributed in an atom’s orbital and how they are shared or paired between atoms in a molecule. Lewis structures help visualize this distribution by using dots to represent electrons. In the three molecules from our exercise, drawing the Lewis structures requires an understanding of how many bonds each atom commonly forms. Nitrogen typically forms three bonds and has one lone pair, oxygen forms two bonds and has two lone pairs, and carbon prefers four bonds with no lone pairs unless there is a charge or other compensatory factors.

After bonds are formed using the shared electron pairs, the remaining valence electrons are assigned as lone pairs to the respective atoms, following the octet rule where applicable. \(\mathrm{NO}^{+}\), for instance, will have a differing electron arrangement compared to a neutral molecule due to its positive charge and less electron density.