Understand the structure of tropolone

Tropolone is a seven-membered ring containing:

• three $\ce{C=C}$ bonds in the ring, and
• one carbonyl group $\ce{C=O}$.

It also exists as a resonance-stabilized system, and its aromaticity is discussed through delocalization over the ring.

Check Statement I: "Tropolone is an aromatic compound and has $8\pi$ electrons."

• Tropolone is indeed considered aromatic because it has a cyclic, planar, conjugated system with effective delocalization.
• However, the key point is the number of delocalized $\pi$ electrons involved in aromaticity.

Count the relevant electrons:

• three $\ce{C=C}$ bonds contribute $6\pi$ electrons.
• the carbonyl $\pi$ bond is not counted as an extra independent pair making the ring an $8\pi$ system in the simple sense stated.

Aromatic stabilization in tropolone is associated with a $6\pi$ aromatic system, not an $8\pi$ aromatic system.

Therefore, Statement I is false.

Check Statement II: " $\pi$ electrons of $\ce{>C=O}$ group in tropolone is involved in aromaticity."

In tropolone, the carbonyl group participates through resonance interaction with the ring. The $\pi$ system is conjugated, but the $\pi$ electrons of the carbonyl bond themselves are not counted as the aromatic sextet in the way the statement suggests.

Aromaticity is not described by saying that the carbonyl $\pi$ electrons directly contribute as part of an $8\pi$ aromatic circuit. Thus this statement is also taken as false in standard exam treatment.

Therefore, Statement II is false.

Evaluate options

• A: Statement I false, Statement II true $\rightarrow$ incorrect
• B: Both Statement I and Statement II are false $\rightarrow$ correct
• C: Statement I true, Statement II false $\rightarrow$ incorrect
• D: Both Statement I and Statement II are true $\rightarrow$ incorrect

Final answer

The correct option is: $\boxed{\text{B}}$