Propose a mechanism for each reaction, showing explicitly how the observed mixtures of products are formed.

a) $3-\mathrm{methyl}-\mathrm{but}-2-\mathrm{en}-1-\mathrm{ol}+\mathrm{HBr}\to 1-\mathrm{bromo}-2-\mathrm{ene}+3-\mathrm{bromo}-3-\mathrm{methyl}-1-\mathrm{ene}$

b)$2-\mathrm{methyl}-\mathrm{but}-3-\mathrm{en}-2\mathrm{a}-\mathrm{ol}+\mathrm{HBr}\to 1-\mathrm{bromo}-2-\mathrm{ene}+3-\mathrm{bromo}-3-\mathrm{methyl}-1-\mathrm{ene}$c)$\mathrm{cyclopenta}-1,3-\mathrm{diene}+{\mathrm{Br}}_2\to 3,4-\mathrm{dibromocyclopent}-1-\mathrm{ene}+3,5-\mathrm{dibromocyclopent}-1-\mathrm{ene}$

d)$1-\mathrm{cholorobut}-2-\mathrm{ene}+{\mathrm{AgNO}}_3+{\mathrm{H}}_2\mathrm{O}\to \mathrm{but}-2-\mathrm{en}-1-\mathrm{ol}+\mathrm{but}-3-\mathrm{en}-2-\mathrm{ol}$

e)$3-\mathrm{cholorobut}-1-\mathrm{ene}+{\mathrm{AgNO}}_3+{\mathrm{H}}_2\mathrm{O}\to \mathrm{but}-2-\mathrm{en}-1-\mathrm{ol}+\mathrm{but}-3-\mathrm{en}-2-\mathrm{ol}$

1. When is$3-\mathrm{methyl}-\mathrm{but}-2-\mathrm{en}-1-\mathrm{ol}$ reacted with HBr, two allylic cations are formed, and the nucleophile can attack either one. The mechanism of the reaction is shown as follows:

1. When $2-\mathrm{methyl}-\mathrm{but}-3-\mathrm{en}-2\mathrm{a}-\mathrm{ol}$is reacted with HBr, two allylic cations are formed, and the nucleophile can attack either one. The mechanism of the reaction is shown as follows

When$\mathrm{cyclopenta}-1,3-\mathrm{diene}$ isreacted with and${\mathrm{Br}}_2$, two allylic carbocations are formed, and the nucleophile can attack either one. The mechanism of the reaction is shown as follows:

When is $1-\mathrm{cholorobut}-2-\mathrm{ene}$reacted with${\mathrm{AgNO}}_3$and${\mathrm{H}}_2\mathrm{O}$, the same allylic carbocations as in are formed. The nucleophile can attack either one. The mechanism of the reaction is shown as follows:

When $3-\mathrm{cholorobut}-2-\mathrm{ene}$is reacted with${\mathrm{AgNO}}_3$and${\mathrm{H}}_2\mathrm{O}$, the same allylic carbocations as in are formed. The nucleophile can attack either one. The mechanism of the reaction is shown as follows: