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My organic textbook states the pathway for propyl tert-butyl ether in dilute sulfuric acid will follow E1 with a tertiary carbocation intermediate. It mentions that Sn1 will only occur with good nucleophiles.
It does not explain why however. Why is this any different than a tertiary haloalkane in water? I expect that the alcohol is a good leaving group, similar to water.
Vollhardt, P.; Schore, N. Molecules: Structure and Function, 7th edition; W. H. Freeman: New York, 2014, p350
It isn't necessarily the same situation as the haloalkane in water. In the sulfuric acid case, there is a large amount of acid present, and as you rightly say the alcohol and water are similar leaving groups, so the SN1 product is easily decomposed back to the tertiary carbocation. As a result, the mechanism tends towards the thermodynamic product, which apparently is the alkene (probably for entropic reasons.
Compare this with the alkyl halide in water. Here, the halide is a much better leaving group than water. Additionally, there is a low concentration of acid in solution (at most 1 equivalent, from the generation of HX). This means that loss of water from the SN1 product is much less favourable.
That said, I'm sure you'd get a mixture of E1 and SN1 products from the haloalkane reaction you describe, I merely wanted to point out that the ether and haloalkane cases aren't as similar as you may have presumed.
