a) Write `S_(N)1` mechanism for the hydrolysis of 2-Bromo-2-methyl propane. Why are `S_(N)1` reactions generally carried in polar protic solvents? 

Explanation

`SN^(1)` mechanism follows first order kinetics
`underset("2 - Bromo -2 methyl propane")(CH_(3)-overset(CH_(3))overset("|  ")underset(CH_(3))underset("|  ")"C "-Br+NaOH)rarr underset("2 - metylpropan -2- ol")(CH_(3)-overset(CH_(3))overset("|  ")underset(CH_(3))underset("|  ")"C "-OH+NaBr)`
2-Bromo-2 methyl propane on hydrolysis with aqueous `NaOH` gives 2-metylpropan 2- o1
Mechanism `SN^(1)` mechanism involves two steps
I Step (Slow step)
Tertiary butyl bromide ionises slowly to given `SP^(2)` hybridised planar 2-methylpropane. cation

II Step (Slow step)
The nucleophile `OH^(-)` from aqueous NaOH attacks planar carbocation on either side to give (tertiary butyl alcohol) 2-Metylpropane cation

`3^(@)` Butyl alcohol obtained is a racemic mixture.
Protic solvents solvate halide ion and provide energy so generally carried in polar protic solvents.