Place the following molecules or ions in order of decreasing bond length: (a) the \(\mathrm{CO}\) bond in \(\mathrm{CO}, \mathrm{CO}_{2}, \mathrm{CO}_{3}{ }^{2-} ;\) (b) the \(\mathrm{SO}\) bond in \(\mathrm{SO}_{2}, \mathrm{SO}_{3}, \mathrm{SO}_{3}^{2-}\); (c) the \(\mathrm{CN}\) bond in \(\mathrm{HCN}, \mathrm{CH}_{2} \mathrm{NH}\), \(\mathrm{CH}_{3} \mathrm{NH}_{2}\). Explain your reasoning.

Bond order is a critical concept in chemistry that helps explain the stability and length of chemical bonds between atoms. It's determined by the number of chemical bonds between a pair of atoms. For instance, a single bond has a bond order of one, a double bond has a bond order of two, and a triple bond has a bond order of three. Generally, the higher the bond order, the stronger and shorter the bond will be.
A good way to remember this is to think about the bond as a road connecting two cities: the more lanes (bonds) it has, the easier (stronger) it is to travel (hold together) between the cities (atoms). So, when you're comparing molecules to understand their bond lengths, check their bond orders first. A molecule with a higher bond order will have a shorter bond length compared to one with a lower bond order. This relationship is crucial for explaining why, for instance, the bond length of CO in carbon monoxide (with a triple bond) is shorter than the bond in CO32- (which has an average bond order less than double due to resonance structures).

Resonance structures are like different sketches of the same molecule showing alternative ways electrons can be arranged with bonds and lone pairs. These structures are a tool that chemists use to depict the delocalization of electrons within a molecule. In molecules where resonance is a factor, like CO32- or SO32-, the actual structure is a blend, or a resonance hybrid, of all the possible resonance forms.
Think of resonance structures as different facial expressions of a person. Each expression might show different emotions (arrangements of electrons), but it's still the same person (molecule). Just like a neutral face might be the average of all expressions, the true structure of a molecule with resonance is an average of its resonance forms rather than any individual one. Because of this averaging, the bond order in such molecules is not a whole number, and this 'partial' bond order affects the bond length, usually making it longer compared to structures without resonance.

Molecular geometry is the three-dimensional arrangement of atoms in a molecule. It's determined by various factors including the number of bonds, the presence of lone pairs, and the molecule's electron-pair geometry. The VSEPR theory (Valence Shell Electron Pair Repulsion theory) is a model used to predict the geometry of molecules. It states that electron pairs around a central atom will arrange themselves as far apart as possible to minimize repulsion.
The shape of a molecule has a direct impact on its physical and chemical properties, including the bond length. For example, in CO2, the carbon atom forms double bonds with each oxygen and the molecule has a linear shape, which influences the bond lengths. In contrast, molecules with bent or trigonal shapes can have different bond lengths due to the angles between the bonds and the relative positions of the atoms. Understanding the shape of the molecule provides insight into the bond lengths and their relationship within the molecule.