For each of the following molecules or ions of sulfur and oxygen, write a single Lewis structure that obeys the octet rule, and calculate the oxidation numbers and formal charges on all the atoms: (a) SO \(_{2},(\mathbf{b}) \mathrm{SO}_{3},(\mathrm{c}) \mathrm{SO}_{3}^{2-}\) (d) Arrange these molecules/ions in order of increasing \(S-O\) bond length.

We will start by drawing Lewis structures of the molecules/ions. (a) SO2: O=S=O (b) SO3: O=S(=O)=O (c) SO3^2-: O=S(-O^−2)=O Next, we will calculate the oxidation numbers for each atom in the molecules/ions.

Oxidation numbers are hypothetical charges assigned to atoms, based on their electronegativity. (a) SO2: Oxygen: -2 (each oxygen atom) Sulfur: +4 (sum of oxygen charges = -4, sulfur has to cancel it out) (b) SO3: Oxygen: -2 (each oxygen atom) Sulfur: +6 (sum of oxygen charges = -6, sulfur has to cancel it out) (c) SO3^2-: Oxygen: -2 (each oxygen atom except the one with extra lone pair) and -1 (oxygen with extra lone pair) Sulfur: +6 (sum of oxygen charges = -5, sulfur has to cancel it out, and there's a -1 charge on the ion) Now, we move to calculating the formal charges for all atoms.

Formal charges are the charge on an atom if all shared electrons were equally shared between atoms. (a) SO2: Oxygen: 0 (6 - 6) Sulfur: 0 (6 - 6) (b) SO3: Oxygen: 0 (6 - 6) Sulfur: 0 (6 - 6) (c) SO3^2-: Oxygen: -1 (6 - 7) for the oxygen with extra lone pair; 0 (6 - 6) for the other two oxygen atoms Sulfur: 0 (6 - 6) Lastly, we will arrange the molecules/ions in order of increasing S-O bond length.

To determine the S-O bond length order, we will consider the multiple bonds and the resonance structures. (a) SO2: There is one double bond between S and each O atom, the bond lengths will be shorter due to double bonds. (b) SO3: There are two resonance structures for SO3. Each S-O bond is a double bond in the resonance structure. The bond lengths will be shorter as compared to single bonds and will be similar to SO2 bond lengths. (c) SO3^2-: There are three resonance structures for SO3^2-. In each resonance structure, sulfur has one double bond with one oxygen atom and single bonds with other two oxygen atoms. This makes the average bond order between S-O higher than a single bond but less than a double bond. So, the bond lengths will be longer compared to the S-O bond length in SO2 and SO3. Based on this information, the order of increasing S-O bond lengths is as follows: SO2 < SO3 < SO3^2-