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It seems that if a carbocation intermediate is formed through solvolysis, then the product(s) are going to always be a pretty even mix of SN1 and E1 products.
Are there any conditions that greatly favor one mechanism? I thought that a very bulky solvent - like t-$\ce{BuOH}$ - would favor E1. However, the carbocation is $\ce{sp^2}$ hybridized, so there still wouldn't be a lot of steric hindrance to prevent SN1, right?
In a practical way, you would have to consider each particular case, there are tons of reasons that could make a reaction to yield mostly E1 or SN1 product (it will be always a mixture, that can vary from 50:50 to 0.01:99,9, I hope I'm making myself clear, they are always competing reactions).
But there are several factors that contribute to make one of the two more favorable than the other, for example:
The kind of the reactant employed:
If you use hydrogen halides, ($\ce{HCl, HBr, HI}$) you will mostly obtain SN1 products.
If you use acids like $\ce{H2SO4, H3PO4~or~TsOH}$ you will mostly obtain E1.
Why is this? The reagents/acids of the first case, are very good nucleophiles. The reagents lisetd on the second case, are awful nucleophiles. A good nucleophile is key for a substitution. But for elimination you only need something that can abstract a product in the alpha position to the leaving group.
On the other hand, very strong and small bases, can favour an elimination through E2 mechanism, with almost no substitution taking place. E1 mechanisms is usually observed only under mild basic conditions.
Solvents, like you say, also can favour one type or another:
Solvents with low nucleophility also decreases the % of substitution reaction.
Both E1 and SN1 are favoured in protic solvents, in general.
To sum up, you will always have both products, but yes, there are factors that can be modified in order to maximize one of the two reactions. There are some general considerations, like the ones I posted, but in practice, each reaction is a world.
If you use hydrogen halides, (HCl, HBr, HI) you will mostly obtain Sn1 products. and A good nucleophile is key for a substitution. need some comment: Hydrogen halides, at least HBr and HI (HCl not so much) are good nucleophiles and good nucleophiles abet only $\mathrm{S}_{\mathrm{N}}2$ reactions. The rate of an $\mathrm{S}_{\mathrm{N}}1$ reaction remains mostly unaffected by choosing a good over a bad nucleophile because the intermediate carbocation is so reactive that it reacts even with very weak nucleophiles.
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