To determine which amine has the smallest $pK_b$ value, we first need to understand the concept of basicity in terms of amines:
- The basicity of amines in an aqueous solution can be explained using the $K_b$ value, which represents the equilibrium constant for the base dissociation reaction.
- The smaller the $pK_b$ value, the stronger the base. This is because $pK_b = -\log(K_b)$, so a lower $pK_b$ means a higher $K_b$.
Now, let's analyze the given options based on these principles:
- $(CH_3)_2NH$ (Dimethylamine): Dimethylamine is a secondary amine. Secondary amines generally have stronger basicity due to two alkyl groups donating electron density to the nitrogen.
- $CH_3NH_2$ (Methylamine): Methylamine is a primary amine and has a high $pK_b$ value because there is only one alkyl group donating electron density to the nitrogen.
- $(CH_3)_3N$ (Trimethylamine): Although trimethylamine is a tertiary amine, the steric hindrance around the nitrogen reduces its basicity compared to secondary amines.
- $C_6H_5NH_2$ (Aniline): Aniline is an aromatic amine, and its basicity is significantly reduced because the lone pair of electrons on the nitrogen can delocalize into the benzene ring, making it less available for bonding with protons.
From the above analysis, we conclude that $(CH_3)_2NH$ has the smallest $pK_b$ value, indicating it is the strongest base among the given options.
Conclusion: The amine with the smallest $pK_b$ value is $(CH_3)_2NH$ (Dimethylamine).