Step 1: Understanding the Question:
The problem presents four different organic nitrogenous compounds and requires us to identify which one represents a secondary ($2^\circ$) amine.
Step 2: Key Formula or Approach:
Amines are classified structurally based on the number of organic carbon groups attached directly to the central nitrogen atom:
Primary ($1^\circ$) amine: Nitrogen is attached to exactly one alkyl or aryl carbon group (contains an $-\text{NH}_2$ functional group).
Secondary ($2^\circ$) amine: Nitrogen is attached to exactly two alkyl or aryl carbon groups (contains an $-\text{NH}-$ functional group).
Tertiary ($3^\circ$) amine: Nitrogen is attached to three alkyl or aryl carbon groups (contains a $-\ddot{\text{N}}-$ functional node with no hydrogen atoms directly bound to nitrogen).
Step 3: Detailed Explanation:
Let us analyze the chemical structure of each option:
(A) Hexane-1,6-diamine ($\text{H}_2\text{N-(CH}_2)_6\text{-NH}_2$): This molecule possesses two separate terminal functional groups, both of which are simple $-\text{NH}_2$ primary amine attachments.
(B) N,N-Dimethylbenzenamine ($\text{C}_6\text{H}_5\text{-N(CH}_3)_2$): Here, the nitrogen atom is bonded directly to one phenyl ring and two distinct methyl groups. Being attached to three carbon blocks makes it a tertiary ($3^\circ$) amine.
(C) N-methylbenzenamine ($\text{C}_6\text{H}_5\text{-NH-CH}_3$): In this structure, the core nitrogen atom is linked directly to one phenyl ring group and one methyl group chain, leaving exactly one single hydrogen atom bound to it. This satisfies the precise structural criteria for a secondary ($2^\circ$) amine.
(D) Prop-2-en-1-amine ($\text{CH}_2\text{=CH-CH}_2\text{-NH}_2$): The nitrogen belongs to a standard terminal $-\text{NH}_2$ configuration, making it a primary ($1^\circ$) amine.
Therefore, option (C) is the secondary amine.
Step 4: Final Answer:
The secondary amine is N-methylbenzenamine, which corresponds to option (C).