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What is the difference between an electrophilic addition and a nucleophilic addition? Is there a rule or easy to understand pattern that can be used to differentiate between the two types?

Answer
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Hint: Electrophilic and nucleophilic additions are both addition reactions. They differ in the addition of the species. In electrophilic reactions an electrophile is generated, which is a species that accepts electrons, while in nucleophilic reactions the nucleophile is added, that is the species which donates electrons.

Complete answer:
The additional reactions can be classified into nucleophilic and electrophilic. As we know, nucleophiles are the species that donates the electrons, while electrophiles are the species that accept electrons. This suggests that nucleophiles are nucleus loving, while electrophiles are electron loving.
The difference between nucleophilic and electrophilic addition reactions is that:
A nucleophilic addition reaction has a nucleophile being added up. This nucleophile provides or donates electrons on the place of its addition.
While an electrophilic addition reaction has an electrophile, which is an electron deficient species that accepts electrons. It is added on the place as to provide electrons to this electrophile.
A simple way to look if it’s the electrophilic or a nucleophilic reaction is to look at the reactions and the species being added.
For nucleophilic addition, in reduction of aldehydes, hydrogen as a nucleophile is added. This hydrogen donates electrons to the carbonyl group of aldehyde. The reaction is:
$RHC=O+{{H}^{-}}\to RC{{H}_{2}}-{{O}^{-}}\xrightarrow{{{H}_{2}}O}RC{{H}_{2}}OH$
For electrophilic addition reaction in addition to bromine to the alkenes to form bromoalkanes. The bromine atom acts as electrophile. The cleavage of double bond provides electrons that are taken up by the bromine atom. The reaction is:
${{R}_{2}}C=C{{R}_{2}}+B{{r}_{2}}\to {{R}_{2}}BrC-{{C}^{+}}{{R}_{2}}+B{{r}^{-}}\to {{R}_{2}}BrC-CBr{{R}_{2}}$
Hence, nucleophilic and electrophilic addition reactions are differentiated.

Note:
One more way is to tell that the reaction is nucleophilic or electrophilic is to tell that the added group is electron rich or deficient. This can be identified by the Lewis structures of electrophile and nucleophiles. A nucleophile always consists of a lone pair that it can donate, while electrophile does not contain any lone pair of electrons.