Explain with example `S_(N^1)` mechanism.

Explanation

`SN^(1)` reactions (Substitution Nuclephilic Unimolecular reactions) are those reactions which take place in two consecutive steps. The rate of the reaction depends only on the concentration of alkyl halide and is independent of the concentration of nucleophile. For example consider the action of nucleophile on alkyl halide.
` R -X + bar(N)u to underset("Rate" alpha"[Alkyl halide]")(R -Nu + : bar(X))`
Thus `SN^(1)` reactions show first order-kinetic. Let us consider the hydrolysis of tețitary butyl bromide with hydroxide ion.
`underset("tert-Butyl alcohol")( (CH_3)_3 C - Br + : bar(O)H to underset("tert-Butyl alcohol")((CH_3)_3 C-OH + bar(B)r)`
Mechanism : It is a unimolecular reaction and occurs in two steps. Step 1. Formation of carbocation.

This step is slow and hence is the rate-determining step of the reaction. Step II. Attack of nucleophile on carbocation. The carbocation formed in step! readily combines with nucleophile `(:barO H)` to form tertiary butylalcohol.

This step is fast and hence does not affect the rate of the reaction. Kinetics : Since the selfionisation of tertiary butylbromide is the slowest step of the reaction i.e., the rate determining step, therefore, it shows first order kinetic. In other words, the rate of reaction depends upon the concentration of tert-butyl bromide only.
Rate `mu [(CH_3)_3 C-Br]`
Rate `=K[(CH_3)_3 C-Br]`
All tertiary aikyl halides undergo substitution by `SN^(1)` mechanism