Predict the product(s) and provide the mechanism for each reaction below. What do the mechanisms have in common?

a.

b.

c.

d.

Reaction of the epoxide occur with the acid and opening of the epoxide occur and reaction followed ${\mathbf{SN}}_{\mathbf{1}}$ mechanism that attack is occur from the tertiary side of the ether oxygen to form tertiary carbocation and hydroxy group is formed on the another side of the ether oxygen.

Reaction proceeds by ${\mathrm{SN}}_1$pathway and we know that the attack of the nucleophile is occurred from the less hindered side but here the nucleophile addition occur from the more hindered side and hydroxy form on the less hindered side due to the formation of stable carbocation as shown below:

Ring opening of epoxide

As per the epoxide give one side of the ether oxygen has methyl group and other side has oxygen only so reaction has to occur from less hindered side but as per the carbocation formation the stable carbocation occur from the more hindered side and hydroxy group form on less hindered side and the nucleophile that is halide ion is an attack on more hindered side as shown:

Ring opening of epoxide

The reaction of epoxide in presence of alkyl halide leads to the opening of the epoxide from the more hindered side that is like the left of the ether oxygen has methyl substituent so that halide is added on that side and the hydroxyl group is formed on the cyclohexane ring as given:

Epoxide ring opening

In the acid-catalyzed reaction the halide has to attack from the less hindered side but due to the presence of the tertiary group it follow ${\mathrm{SN}}_1$mechanism in which formation of carbocation occur and a tertiary carbocation is more stable than other so that the halide attack occurs from the more hindered side as shown below:

Epoxide ring opening