#58) Sulfur is an interesting element because it may exist in a wide variety of formal oxidation states.  Stated another way, it bonds to a lot of different elements, in a lot of different ways!

a) SO32-: this is the sulfite ion, the sum of the OS must equal the charge on the ion, in this case -2.  Since the OS of each oxygen is -2, the sulfur must be in the +4 oxidation state.

OS(S) + 3 x OS(O) = -2
OS(S) + 3 x (-2) = -2
OS(S) - 6 = -2
OS(S) = +4

b) S2O32-: this is the thiosulfate ion...  the avg OS for each S is +2.

2 x OS(S) + 3 x OS(O) = -2
2 x OS(S) + 3 x (-2) = -2
2 x OS(S) - 6 = -2
2 x OS(S) = +4
OS(S) = +2

The reality is one sulfur is in the "0" oxidation state, while the other is in the +4 oxidation state.  There is one central sulfur atom bound to all the remaining atoms.  The other sulfur is bound to sulfur alone, and nothing else!

c) S2O82-: sulfur is in the +7 oxidation state, on average (as far as you can tell at this point in your chemical careers!)

2 x OS(S) + 8 x OS(O) = -2
2 x OS(S) + 8 x (-2) = -2
2 x OS(S) - 16 = -2
2 x OS(S) = +14
OS(S) = +7

In reality, this is the peroxodisulfate ion and has the structure shown below:

The central O atoms are bound to one another, and must have a higher oxidation state than -2...  they are -1 each.  Each sulfur exhibits a +6 state.

d) HSO4-: this is the bisulfate, or hydrogensulfate, ion.  The sulfur is in the +6 oxidation state.

OS(H) + OS(S) + 4 x OS(O) = -1
1 + OS(S) + 4 x (-2) = -1
1 + OS(S) - 8 = -1
OS(S) = +6

e) S4O62-: The sum of the OS on each atom = the charge on the ion!

4 x OS(S) + 6 x OS(O) = -2
4 x OS(S) + 6 x (-2) = -2
4 x OS(S) - 12 = -2
4 x OS(S) = +10
OS(S) = +2.5

Each S is in the +2.5 oxidation state, on average.  Note that, since there is no such thing as a "half" electron, this cannot actually be the case.  This ion is termed a "polythionite" ion and resembles the peroxodisulfate ion depicted above, but with two sulfur atoms bridging the ends instead of two oxygens.  These central sulfurs are in the "0" oxidation state, and hence the outer S atoms must be in the +5 oxidation state.