# What does 4n+2 mean?

The other night a student came to me with a question about aromaticity.

“There’s one thing I don’t get”, she said. “They say a molecule has to have 4n+2 electrons. How do you find n?”

Teachable moment! “n” is not a characteristic of the molecule! Let me explain.

In order for a molecule to be aromatic, it has to have the following characteristics:

• It must be cyclic
• It must be conjugated (i.e. all atoms around the ring must be able to participate in π-bonding through resonance)
• It must be flat
And, it must have a certain number of π-electrons. This is known as Huckel’s rule. The number of π electrons must equal one of the numbers in this series:
2, 6, 10, 14, 18….and so on. For example, we can find aromatic molecules with 2 pi electrons, 6 pi electrons, 10 pi electrons, 14 pi electrons, 18 pi electrons, and so on. But we have never found aromatic molecules with 0, 1, 3, 4, 5, 7, 9, 11, 12, 13, 15, 16, 17 (and so on) pi electrons. Those numbers are not in the series.
To reprise: the number of pi electrons in an aromatic molecule will always be found in the series  [2, 6, 10, 14, 18 …and so on]
However, there has to be a better way of expressing it than [2, 6, 10, 14, 18… and so on”]
There is! This is where we use algebra. this is where n is going to come in – we are going to use math (algebra) to replace “2, 6, 10, 14, 18… and so on” with a condensed formula.
2, 6, 10, 14, 18… is an example of an algebraic series.
Algebraically, you can express this with the formula (4n +2), where n is a natural number (0, 1, 2, 3…)
Let’s plug in different values of n (we will put n in bold)
For the formula 4n + 2
For n = 0, we get (4 × 0 + 2) = 2
For n = 1, we get (4 × 1 + 2) = 6
For n = 2, we get (4 × 2 + 2) = 10
For n = 3, we get (4 × 3 + 2) = 14
For n = 4, we get (4 × 4 + 2) = 18
We can keep going, but do you get the idea?
By using the formula [4n +2], we are expressing the same idea as  [2, 6, 10, 14, 18… and so on] but it is a lot more precise.

So “n” comes from algebra, NOT from chemistry.

Molecules that have the 3 characteristics listed above (cyclic, conjugated, flat) and have this number of π electrons [4n +2] will be aromatic. The letter “n” is not a characteristic of the molecule! [Thanks to commenters Shawn and Egon Willenhagen for  suggested edits and to Jamey for a correction]