Count of Aromatic Molecules Among Given Structures

To determine the total number of aromatic compounds among the given structures, we apply Hückel's Rule for aromaticity. A compound is classified as aromatic if:

1. It is cyclic.
2. It is planar (or nearly planar) allowing continuous overlap of $p$-orbitals.
3. It has a completely conjugated $\pi$-electron system.
4. It contains $(4n + 2)$ $\pi$-electrons, where $n = 0, 1, 2, 3, \dots$

Additionally, any compound containing an aromatic moiety (such as a benzene ring) is classified as an aromatic compound.

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Detailed Analysis of Each Structure:

1. Cyclooctatetraene ($\text{C}_8\text{H}_8$):

   • It possesses $8\pi$ electrons ($4n$ electrons for $n=2$).
   • To avoid the instability of being antiaromatic, it adopts a non-planar (tub-shaped) conformation.
   • Classification: Non-aromatic

2. Cyclopropenyl anion ($\text{C}_3\text{H}_3^-$):

   • Contains $4\pi$ electrons ($2\pi$ from the double bond + $2\pi$ from the negative charge).
   • It is planar and fully conjugated, conforming to the $4n$ rule ($n=1$).
   • Classification: Antiaromatic

3. Cyclopropenyl cation ($\text{C}_3\text{H}_3^+$):

   • Contains $2\pi$ electrons.
   • Follows Hückel's rule for $n=0$: $4(0) + 2 = 2\pi$ electrons.
   • It is cyclic, planar, and fully conjugated.
   • Classification: Aromatic

4. 1,3-Cyclohexadiene:

   • Contains two $sp^3$-hybridized carbon atoms in the ring, interrupting continuous conjugation around the cycle.
   • Classification: Non-aromatic

5. Cycloheptatrienyl cation (Tropylium cation, $\text{C}_7\text{H}_7^+$):

   • Contains $6\pi$ electrons across 7 $sp^2$-hybridized carbon atoms.
   • Follows Hückel's rule for $n=1$: $4(1) + 2 = 6\pi$ electrons.
   • It is cyclic, planar, and fully conjugated.
   • Classification: Aromatic

6. Cyclopentadienyl cation ($\text{C}_5\text{H}_5^+$):

   • Contains $4\pi$ electrons in a planar, fully conjugated ring.
   • Follows the $4n$ rule ($n=1$).
   • Classification: Antiaromatic

7. Cyclopentadienyl anion ($\text{C}_5\text{H}_5^-$):

   • Contains $6\pi$ electrons ($4\pi$ from double bonds + $2\pi$ from the lone pair/negative charge).
   • Follows Hückel's rule for $n=1$: $4(1) + 2 = 6\pi$ electrons.
   • It is cyclic, planar, and fully conjugated.
   • Classification: Aromatic

8. 1,2-Dihydronaphthalene:

   • Consists of a benzene ring fused to a partially unsaturated 6-membered ring.
   • Since it contains a intact $6\pi$-electron benzene ring system, the molecule is a benzenoid aromatic compound.
   • Classification: Aromatic

9. Phenanthrene ($\text{C}_{14}\text{H}_{10}$):

   • Consists of three fused benzene rings with a completely conjugated system containing $14\pi$ electrons.
   • Follows Hückel's rule for $n=3$: $4(3) + 2 = 14\pi$ electrons.
   • Classification: Aromatic

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Conclusion:

The aromatic compounds among the given structures are:

1. Cyclopropenyl cation
2. Cycloheptatrienyl cation
3. Cyclopentadienyl anion
4. 1,2-Dihydronaphthalene
5. Phenanthrene

The total number of aromatic compounds is 5.