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In which of the reactions below is the first compound in the equation NOT oxidized? enter image description here

The correct choice is (D), which can be seen through process of elimination, but I was wondering how to do this through the carbon oxidation number?

I feel that to know the oxidation number of nitrogen, I need the oxidation number of carbon, but to know the oxidation number of carbon, I need to know the oxidation number of nitrogen.

Or is the correct way to think about this problem that: (D) didn't gain any oxygen atoms, therefore, wasn't oxidized and leave it at that?

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There are several ways to determine oxidation number/state. The most useful method for organic (covalent) compounds uses the following three rules, and works for most elements engaged in covalent bonding (just replace carbon with some other element):

  1. Every bond between a carbon atom and a more electronegative atom increases the oxidation number of that carbon atom by 1.
  2. Every bond between a carbon atom and a less electronegative atom (hydrogen is most common) decreases the oxidation number of that carbon atom by 1.
  3. Every bond between a carbon atom and another carbon atom has no effect on the oxidation number of the carbon atom (although we could get more complicated since carbon atoms with different "hybridizations" have different electronegativities).

This approach is equivalent to assigning all of the electrons in a covalent bond to the more electronegative atom for purposes of determining oxidation number. This page on Master Organic Chemistry has a number of examples.

Here's an example from the Wikipedia article on the matter, using acetic acid:

enter image description here

Atom            Bonds to   Contribution to Ox# 
H (methyl)       C         +1 (2.5 > 2.1)
          TOTAL Ox#        +1 
C (methyl)       H         -1 (2.1 < 2.5)
                 H         -1
                 H         -1
                 C         +0 (same atom)
          TOTAL Ox#        -3
C (carboxyl)     O         +1 (3.5 > 2.5)
                 O         +1 (double bond counts as 2 bonds)
                 O         +1
                 C         +0
          TOTAL Ox#        +3
O (carbonyl)     C         -1 (2.5 < 3.5)
                 C         -1
          TOTAL Ox#        -2
O (hydroxyl)     C         -1 
                 H         -1 (2.1 < 3.5)
          TOTAL Ox#        -2
H (hydroxyl)     O         +1 (3.5 > 2.1)
          TOTAL Ox#        +1
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Or is the correct way to think about this problem that: (D) didn't gain any oxygen atoms, therefore, wasn't oxidized and leave it at that?

This is the simplest way of thinking of the problem.

Many introductory organic chemistry texts define oxidation as carbon forming additional bonds to oxygen, and reduction as carbon forming additional bonds to hydrogen. So if we have a reaction converting ethanol into acetaldehyde, this would be an oxidation; the carbon previously connected to the $\ce{-OH}$ group is now connected to oxygen with two bonds instead of one.

We can generalize these statements to say that oxidation involves the formation of bonds to a more electronegative element, and that reduction involves the formation of bonds to a less electronegative element.

With this in mind ...

In your problem, specifically, choice (D), carbon previously has two bonds to oxygen and ends up with two bonds to nitrogen. This is not considered an oxidation because it started with two bonds to a more electronegative element and ended up with two bonds to a more electronegative element.

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