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Addition Reactions - involve the conversion of a π bond into 2 new σ bonds

General form: A + B → C

Eg. CH3-CH=CH-CH3 + HCl → CH3-CH2-CHCl-CH3

Substitution Reactions - involve the no change in bonding - one σ bond replaces another

General form: A + B → C + D

Eg. CH3-CHBr-CH2-CH3 + KOH(aq) → CH3-CH(OH)-CH2-CH3 + KBr

Elimination Reactions - reverse of addition, in that two σ bonds are lost, replaced by a new π bond

General form: A → B + C

Eg. CH3-CH(OH)-CH2-CH3 -- conc. H2SO4 --> CH3-CH=CH-CH3 + H2O

Rearrangement / Isomerisation - process in which a single substance changes structure, A → B. Such a reaction may involve changes in bond / type, though this is not necessary. These reactions are comparatively rare.

Eg. CH3-CH2-CH2-C(OH)=CH2 → CH3-CH2-CH2-C(=O)-CH3

These are the four "prototypical" reactions, though several others which can be categorised as one of these are generally referred to by other names.

Eg. CH3-CH(OH)-CH3 -- H2SO4 / K2Cr2O7 --> CH3-C(=O)-CH3

could be described as an elimination reaction, but would usually be called an oxidation

Eg. CH3-C(=O)-CH3 -- 1. LiAlH4 2. H^+ / H2O --> CH3-CH(OH)-CH3

could be described as a (nucleophilic) addition reaction, but would usually be called a reduction

Eg. CH3-C(=O)-OH + CH3-OH -- H2SO4 / Δ / reflux --> CH3-C(=O)-O-CH3 + H2O

could be described as a substitution reaction, but would usually be called a condensation

Another important category of organic reactions are straight-forward Lowry-Bronsted acid-base reactions:

Eg. (CH3-CH2)3N + HCl → (CH3-CH2)3NH^+ + Cl^-

Note that there are also some reactions that are difficult to characterise in a simple way, like the following reactions requiring catalysis:

stilbene + ethylene → styrene

C6H5-CH=CH-C6H5 + CH2=CH2 → 2 C6H5-CH=CH2

but-1-yne + water → butanone

CH3-CH2-C≡CH + H2O → CH3-CH2-C(=O)-CH3

(this is actually an addition reaction followed by an isomerisation)

CH3-CH2-C(=O)-CH3 + NH2-OH → CH3-CH2-C(=N-OH)-CH3 + H2O

the pinacol to pinacolone rearrangement

CH3-C(CH3)(OH)-C(CH3)(OH)-CH3 → CH3-C(CH3)2-C(=O)-CH3

which is an elimination reaction that involves an isomerisation

... I add these last few just to illustrate that the general types are a useful tool / guide for understanding organic chemistry, but they are not the be-all and end-all.

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