[banzaiben]

Hey everybody,
I'm doing practice synthesis problems for an upcoming midterm (2nd quarter O-Chem, undergraduate) and I wanted to see if I was stretching the boundaries of certain reaction mechanisms for a particular problem (see below). 

The reaction mechanism I came up with goes as follows:
1) Add NaOH as a nucleophile to replace Bromine in an S_N2 Reaction
2) Add H₂SO₄ to induce an S_N1 mechanism and create a carbocation in the place of the hydroxyl group
- Allow the cyclobutane to bond with the carbocation to create a bicyclo[4.3.0]nonane (the shortest bridge being a double bond, but I don't know how to name that)
4) Add Borane and hydrogen peroxide to add an OH and H in syn conformation to the double bond to achieve the product

So my main question is: is it likely that the cyclobutane would open to bond with cyclohexane's carbocation, or is it more likely that there would just be a hydrogen shift?

I'd also like to add that this is my first post, so if I'm doing anything wrong or could make it easier for people to answer my question, please let me know.
Thanks

[orgopete]

What do you think would happen if you just solvolyzed the bromide in ethanol/water?

[discodermolide]

You do have a very strained spiro ring system here.

[banzaiben]

What do you think would happen if you just solvolyzed the bromide in ethanol/water?

I suppose that either the cyclohexane/cyclopentane complex would form an alkene at the connection (as stated above) or it could possibly add an OH at the bond.  My problem with solvolysis was that it's very distinct in pointing out that the H and OH are in s-cis formation in the final product, and inducing an S_N1 reaction would just give a racemic mixture, so I wanted to do a syn addition to an alkene.  Is there something I'm missing?
Thanks for the prompt responses.

[orgopete]

I too thought the carbocation would rearrange. Although elimination could occur, you should check the stoichiometry of the solvolysis reaction.

R₂CHCR₂Br  + H₂O R₂C=CR₂ or R₂CHCR₂OH + ??

What effect might that missing chemical have on the product(s)?

[banzaiben]

What effect might that missing chemical have on the product(s)?

Oh I see, so HBr created during the reaction would most likely just remove the hydroxyl group.  In that case, I should probably just seek out a different mechanism that involves Sn2 instead of Sn1 because this is starting to seem like a bit of a stretch. 

Thanks for all the help, asking questions instead of giving answers definitely did more for my understanding of the problem.

[orgopete]

You are getting closer. Why do you think HBr will add? If it did, wasn't the starting material a secondary bromide? If you formed a tertiary bromide, wouldn't it solvolyze faster? The solvolysis is being done with water present. Won't any HBr present actually simply be present as Br^-? Won't water pick up the proton? Can't hydronium ion shift the equilibrium away from or convert any alkene toward an alcohol?