Describe the bonding in the \(\mathrm{O}_{3}\) molecule and the \(\mathrm{NO}_{2}^{-}\) ion using the localized electron model. How would the molecular orbital model describe the \(\pi\) bonding in these two species?

First, we need to draw the Lewis structures for O₃ and NO₂⁻ molecules. For O₃, the Lewis structure is: \[ \underset{ : }{ O } - \overset{ + }{ \underset{ : }{ O } } - \overset{ - }{ \underset{ : }{ O } } \] For NO₂⁻ ion, the Lewis structure is: \[ \underset{ : }{ N } - \overset{ + }{ \underset{ : }{ O } } - \overset{ - }{ \underset{ : }{ O } } \]

Using the Valence Shell Electron Pair Repulsion (VSEPR) theory, we can see that both molecules have one central atom with two other atoms attached and one lone pair of electrons. Therefore, both molecules have a bent VSEPR structure. The hybridization of both central atoms in O₃ and NO₂⁻ is sp².

In the localized electron model, we consider electron pairs to be localized between the bonded atoms. In both O₃ and NO₂⁻ molecules, the central atoms form two sigma (σ) bonds using their sp² hybrid orbitals, and one pi (π) bond using their remaining p orbital. The π bond in both molecules is delocalized over the three atoms in a resonance structure, resulting in a partial double bond character between all three atoms.

In the molecular orbital model, we consider the molecular orbitals formed by the combination of atomic orbitals. In O₃ and NO₂⁻, the π bonding results from the overlap of the p orbitals on the three atoms involved. This creates a delocalized π bond which extends over all three atoms, effectively lowering the energy of the molecule and stabilizing it. In summary, the localized electron model describes the bonding in O₃ and NO₂⁻ using sigma bonds formed by sp² hybrid orbitals and a delocalized π bond using p orbitals, which are in resonance over the three atoms. The molecular orbital model describes the same π bonding as a delocalized bond that results from the overlap of the p orbitals of the involved atoms, contributing to molecular stability.