Part 6: MECHANISMS
Note that no other reagents are needed in order to complete any of these sequences, you should only be using what is there.

A
The reaction is that of a terminal alkyne with a strong base to give a carbanion that is a good nucleophile. When this is reacted with an alkyl bromide, the nucleophilic C reacts with the electrophilic C to form a new C-C bond in an SN2 type reaction and hence a new alkyne. This is an alkylation of a terminal alkyne.

Alkylation of a terminal alkyne
Notes : 
1.  NaNH2 is essentially ionic in that it can be thought of as Na+ and -NH2 .  If you consider it as ionic with formal full charges, don't show a covalent bond.
2.  Since the alkyl halide here is primary, it will NOT form a carbocation so the reaction is not an SN1.
3.  Remember, electron rich to electron poor.

B
The reaction is a radical addition of HBr to an alkene which is a radical chain process.  Since it is a radical process, there are no carbocations formed nor are there any bromide ions formed. Review formal charges ? The regioselectivity is accounted for by the addition of Br first and the preferential formation of a more stable radical as a result of that. 

Radical addition of HBr
Notes : 
1.  UV light is the radical initiator here. No peroxides are shown or hence needed (answer the question asked !)
2.  So the uv light causes homolytic cleavage of the H-Br bond (the weakest bond in the system)
3.  The Br radical adds to the alkene first giving the radical that is more highly alkyl substituted because that is the more stable radical. The accounts for the observed anti-Markownikov regioselectivity.
4.  This alkyl radical then reacts with another molecule of HBr to give the product. This keeps the process going as it regenerates another Br radical.
5.  Why doesn't it react with the H radical ? The concentration of the H radical is very low and the likelihood of the alkyl radical and H radical colliding is very very low.
6.  The alkene pi bond does not dissociate homolytically to give a diradical (that would be a very unstable species)
C
The bridged nature of the product and the diene / dienophile portions of the starting material, plus the heat and the formation of a cyclohexene type system should suggest an intramolecular variation of a Diels-Alder reaction with furan as the diene component :

Diels-Alder reaction




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