[elerock21227]

Oxonium part:

Say there is a t-butyl alcohol reacting with  a non-nucleophilic acid like H₂SO₄.

The alcohol is protinated, forming a good leaving group and leaving. A tertiary carbocation is formed and H₂O grabs the Hydrogen on the β-carbon and elimintion occurs . The products are an alkene and H₃O+

I'm wondering why elimination product is the major yield of this rxn? What is stopping the carbocation on the substrate from going SN₁ and reverting back to a t-butyl alcohol ?

Active ester part:

If there is a primary alcohol reacting with Phosphorous Oxychloride & pyridine the reaction yields E₂ products. However, if that primary alcohol reacts with Thionyl chloride or Phosphorous trihalide, SN₂ products are yielded.

Why there are 2 different mechanisms possible?

[Doc Oc]

For the first question, there is a lot of other factors that come into play, the most significant being the strength of the base and the polarity of the solvent.  Check out this table, I think it gives a decent summary on considerations of S_N1 vs E1

http://www.wfu.edu/~wrightmw/chm122/handouts/sn1sn2e1e2%20summary.pdf

The 2nd question is somewhat related, there's a strong base in the 1st reaction condition, but in the 2nd there's an absence of strong base and you're left with something that is a better nucleophile than base.