Arrange the compounds of each set in order of reactivity towards $S_N2$ displacement: $1-$Bromobutane, $1-$Bromo$-2,2-$dimethylpropane, $1-$Bromo$-2-$methylbutane, $1-$Bromo$-3-$methylbutane.
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$\text{CH}_3-\text{CH}_2-\text{CH}_2-\text{CH}_2-\text{Br}\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1-\text{Bromobutane}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \text{CH}_3\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ |\\ \text{CH}_3-\text{CH}-\text{CH}_2-\text{CH}_2-\text{Br}\\ 1-\text{Bromo}-3-\text{methylbutane}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{CH}_3\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ |\\ \text{CH}_3-\text{CH}_2-\text{CH}-\text{CH}_2-\text{Br}\\ 1-\text{Bromo}-2-\text{methylbutane}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \text{CH}_3\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ |\\ \text{CH}_3-\text{C}-\text{CH}_2-\text{Br}\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ | \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{CH}_3\\1-\text{Bromo}-2,2-\text{methylpropane}$
The steric hindrance to the nucleophile in the $S_N2$ mechanism increases with a decrease in the distance of the substituents from the atom containing the leaving group. Further, the steric hindrance increases with an increase in the number of substituents. Therefore, the increasing order of steric hindrances in the given compounds is as below:
$1-$Bromobutane $< 1-$Bromo$-3-$methylbutane $< 1-$Bromo$-2-$methylbutane $< 1-$Bromo$-2, 2-$dimethylpropane
Hence, the increasing order of reactivity of the given compounds towards $S_N2$ displacement is:
$1-$Bromo$-2, 2-$dimethylpropane $< 1-$Bromo$-2-$methylbutane $< 1-$Bromo$-3-$ methylbutane $< 1-$ Bromobutane.
$\ \ \ \ \ \ \ \ \ \ \ \ \ \text{CH}_3\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ |\\ \text{CH}_3-\text{CH}-\text{CH}_2-\text{CH}_2-\text{Br}\\ 1-\text{Bromo}-3-\text{methylbutane}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{CH}_3\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ |\\ \text{CH}_3-\text{CH}_2-\text{CH}-\text{CH}_2-\text{Br}\\ 1-\text{Bromo}-2-\text{methylbutane}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \text{CH}_3\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ |\\ \text{CH}_3-\text{C}-\text{CH}_2-\text{Br}\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ | \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{CH}_3\\1-\text{Bromo}-2,2-\text{methylpropane}$
The steric hindrance to the nucleophile in the $S_N2$ mechanism increases with a decrease in the distance of the substituents from the atom containing the leaving group. Further, the steric hindrance increases with an increase in the number of substituents. Therefore, the increasing order of steric hindrances in the given compounds is as below:
$1-$Bromobutane $< 1-$Bromo$-3-$methylbutane $< 1-$Bromo$-2-$methylbutane $< 1-$Bromo$-2, 2-$dimethylpropane
Hence, the increasing order of reactivity of the given compounds towards $S_N2$ displacement is:
$1-$Bromo$-2, 2-$dimethylpropane $< 1-$Bromo$-2-$methylbutane $< 1-$Bromo$-3-$ methylbutane $< 1-$ Bromobutane.
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