The compound that will undergo S_N1 reaction with the fastest rate is

The correct compound is (1-bromoethyl)benzene (the structure shown in the fourth image, $C_6{H}_5\mathrm{CH}\left(\mathrm{Br}\right){\mathrm{CH}}_3$).

Reasoning:
The rate of an S_N1 reaction depends directly on the stability of the carbocation intermediate formed in the rate-determining step (loss of the leaving group, ${\mathrm{Br}}^{-}$). Let us analyze the carbocations formed by each of the given compounds:

1. (Bromomethyl)cyclohexane (first image):
This is a primary alkyl halide. Upon loss of the bromide ion, it forms a primary carbocation:
$C_6{H}_{11}{\mathrm{CH}}_2-\mathrm{Br}\to C_6{H}_{11}{\mathrm{CH}}_2^{+}+{\mathrm{Br}}^{-}$
Primary carbocations are highly unstable, resulting in an extremely slow rate of S_N1 reaction.

2. Bromocyclohexane (second image):
This is a secondary alkyl halide. Upon loss of the bromide ion, it forms a secondary cyclohexyl carbocation:
$C_6{H}_{11}-\mathrm{Br}\to C_6{H}_{11}^{+}+{\mathrm{Br}}^{-}$
A secondary carbocation is moderately stable, but lacks resonance stabilization.

3. Bromobenzene (third image):
This is an aryl halide. The carbon-bromine bond has partial double-bond character due to resonance, making it very strong. Additionally, forming a phenyl carbocation is extremely unfavorable:
$C_6{H}_5-\mathrm{Br}\nrightarrow C_6{H}_5^{+}+{\mathrm{Br}}^{-}$
Therefore, aryl halides do not undergo S_N1 reactions under normal conditions.

4. (1-Bromoethyl)benzene (fourth image):
This is a secondary benzylic halide. Upon loss of the bromide ion, it forms a secondary benzylic carbocation:
$C_6{H}_5\mathrm{CH}\left(\mathrm{Br}\right){\mathrm{CH}}_3\to C_6{H}_5{\mathrm{CH}}^{+}{\mathrm{CH}}_3+{\mathrm{Br}}^{-}$
This carbocation is highly stabilized by resonance conjugation with the pi-electrons of the aromatic benzene ring, allowing the positive charge to be delocalized over multiple ring positions. As a result, this intermediate forms very rapidly, leading to the fastest S_N1 reaction rate among the choices.