This question requires determining the number of unique carbon environments in 7-norbornanone using its proton-decoupled $^{13}C$ NMR spectrum. Proton decoupling simplifies the spectrum, making each unique carbon type appear as a single signal.
7-Norbornanone Structure
7-Norbornanone is bicyclo[2.2.1]heptan-7-one. It has a rigid bicyclic structure containing a ketone functional group at the C7 bridge position.
Symmetry and Equivalent Carbons
The molecule possesses a plane of symmetry ($C_s$) that passes through the carbonyl carbon (C7) and the C2 and C5 carbons. This symmetry element relates pairs of carbons:
- The plane equates the two bridgehead carbons: C1 $\leftrightarrow$ C4.
- The plane equates the methylene carbons on one side: C2 $\leftrightarrow$ C5.
- The plane equates the methylene carbons on the other side: C3 $\leftrightarrow$ C6.
Based on experimental $^{13}C$ NMR data, 7-norbornanone shows 3 distinct signals, implying further equivalence beyond the basic $C_s$ symmetry.
Signal Count Explanation
The 3 signals correspond to the following carbon groups:
- Signal 1: The carbonyl carbon (C7). This is chemically unique.
- Signal 2: The bridgehead carbons (C1 and C4). These are equivalent due to the $C_s$ symmetry.
- Signal 3: The four methylene carbons (C2, C3, C5, and C6). Experimental data suggests these carbons are also equivalent, yielding a single signal in the $^{13}C$ NMR spectrum.
Therefore, the proton-decoupled $^{13}C$ NMR spectrum of 7-norbornanone exhibits 3 signals.
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