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(C) The equilibrium constant for the formation of a hydrate depends on the electrophilicity of the carbonyl carbon. $1$. Electron-withdrawing groups $

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$I < II < III < IV$

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(C) The equilibrium constant for the formation of a hydrate depends on the electrophilicity of the carbonyl carbon. $1$. Electron-withdrawing groups $(EWG)$ increase the positive charge on the carbonyl carbon,making it more susceptible to nucleophilic attack by $H_2O$,thus increasing the equilibrium constant for hydrate formation. $2$. Electron-donating groups $(EDG)$ decrease the positive charge on the carbonyl carbon,making it less susceptible to nucleophilic attack,thus decreasing the equilibrium constant. $3$. Comparing the substituents: - $-NO_2$ (in $IV$) is a strong $EWG$,which increases the electrophilicity significantly. - $-H$ (in $III$) is the reference. - $-NH_2$ is an $EDG$. In $I$ ($6$-position),the resonance effect is more pronounced compared to $II$ ($7$-position) due to better conjugation with the carbonyl group,making $I$ less reactive than $II$. $4$. Therefore,the increasing order of reactivity (and thus equilibrium constant) is: $I < II < III < IV$.

Arrange the following compounds in the increasing order of their equilibrium constants for the formation of a hydrate:
$(I)$ $6$-amino-$1$-tetralone
$(II)$ $7$-amino-$1$-tetralone
$(III)$ $1$-tetralone
$(IV)$ $6$-nitro-$1$-tetralone

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Arrange the following compounds in the increasing order of their equilibrium constants for the formation of a hydrate: $(I)$ $6$-amino-$1$-tetralone$(II)$ $7$-amino-$1$-tetralone$(III)$ $1$-tetralone$(IV)$ $6$-nitro-$1$-tetralone