C—C bonds and C—H bonds are extremely common in organic molecules. If you remember my post showing how skeletal structures work, you should recall that these bonds are only noted with points and intersections for the former and are left to inference with the latter. Such structures are "skeletal" because they really do show the hydrocarbon skeleton of a molecule. All those C—C and C—H bonds are usually pretty stable. They can contribute to the chemistry of a molecule, but not nearly so prominently as functional groups.
I won't attempt to list every functional group here, because there are lots and lots of them and your puny brain would probably die or something. But I will list some of them. First, I want to introduce a new notation. Actually, I'm not sure if I already introduced it, but I'm too lazy to go back and check and you probably forgot about it anyway. The letter "R" is often used to denote the rest of a molecule apart from a functional group, especially if what remains is a plain old hydrocarbon. This can be convenient for situations when it's only the functional group we care about and drawing the rest of the molecule would be impractical or wouldn't even make sense. If we do this more than once though, and the groups being condensed are not identical, using "R" to denote both would be inappropriate, but "R" for one and "R'" (R prime) for another is fine.
So for an example, I've decided to use 2-butoxyethanol because I've used it to clean graffiti off the walls in the restroom at work.
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Aliphatic hydrocarbons
This term comes from the Greek aleiphas, meaning "fat." Actually, many of the properties fats have come from the long hydrocarbon chains they possess. If an aliphatic hydrocarbon has no π-bonds (that is, no double or triple bonds), it is an alkane. Branches and rings might be present and do affect the properties of alkanes, but they're still called alkanes, although a compound with a ring might be said to be a cycloalkane.
But if π-bonds are present, no matter how few there are or how big the rest of the molecule is, it's not an alkane anymore. A double bond means that the molecule is an alkene. So using what we learned earlier, a molecule that contains this functional group: R2C=CR2 (different groups are all "R" here because noting them all with superscripts would be ridiculously clunky) is an alkene. The double bond counts as a functional group. Specifically it is the alkenyl functional group.
Similarly, a triple bond is a functional group. Something with R—C≡C—R' functional group is an alkyne (no matter how many double bonds or single bonds it has). And this is an alkynyl functional group.
Aromatic hydrocarbons
The only hydrocarbons I know of that are not classed as aliphatic are ones containing aromatic rings. You might be anticipating this from the trend with what I've said regarding alkenes and alkynes, but the presence of even a single aromatic ring in an otherwise aliphatic molecule means that the compound is considered aromatic and not aliphatic. Aromaticity is a tricky concept though, and we're not covering it just yet. So for now, the only aromatic ring we'll concern ourselves with is the benzene ring.
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Another well-known aromatic hydrocarbon with its own common name is toluene. It's just like benzene, but where benzene has six hydrogens, one attached to each carbon, toluene replaced on hydrogen with a methyl group, a carbon bonded to three hydrogens. So it's like R—Ph with "R" being "CH3" except this is a special case and gets its own name. If toluene acts as a functional group itself, with one of the hydrogens on the carbon outside the ring being replaced by a bond to the rest of molecule, and I'll draw a picture just to be sure you're following me...
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Now for some more functional groups...
Alkyl halide (halo group)
R—X (where X is a halogen). It could be an alkyl fluoride, alkyl chloride, alkyl bromide, or alkyl iodide depending on which halogen is attached.
Alcohol (hydroxyl group)
R—OH
Ether (alkoxy group)
R—O—R
Amine (amino group)
R—NH2 (primary) or R2NH (secondary) or R3N (tertiary)
Thiol (mercapto group)
R—SH
Sulfide (alkylthio group)
R—S—R'
Aldehyde (carbonyl group)
R—CHO (for those of you who can't deal with condensed structures very well, the carbon is double-bonded to the oxygen, bonded to the hydrogen, and bonded to the R group, so it could also be R—CH=O)
Ketone (another carbonyl group)
R—CO—R' (or R—C=O—R'). The difference between an aldehyde and a ketone is that in an aldehyde, the carbobyl group is on the end of a chain, but in a ketone, it's in the middle of a chain.
Carboxylic acid (carboxyl group)
R—COOH (like an aldehyde with the hydrogen on the carbonyl carbon replaced by an hydroxyl group, alternatively it's like an alcohol with an oxygen double-bonded to the terminal carbon)
Ester (ester group)
R—COOR' (like a carboxylic acid, but with the hydrogen on the oxygen replaced by a hydrocarbon group).
Amide (carboxamide group)
R—CONH2 (primary) or R—CONHR (secondary) or R—CONR2 (tertiary). Not to be confused with amines, which don't have that oxygen double bonded to the carbon that nitrogen is attached to.
What's that? You want more? Fine. Next time, I'll post some more functional groups for you.
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