Number of Optically Active Phenolic Isomers of C8H10O2

To find the total number of possible isomers of the compound with molecular formula $\text{C}_8\text{H}_{10}\text{O}_2$ that satisfy the given conditions, we analyze the information step-by-step:

1. Degree of Unsaturation (DU)

The degree of unsaturation for a molecule with formula $\text{C}_8\text{H}_{10}\text{O}_2$ is given by: $\text{DU} = C + 1 - \frac{H}{2} = 8 + 1 - \frac{10}{2} = 4$

A degree of unsaturation equal to $4$ corresponds to a benzene ring ($3$ double bonds $+ 1$ ring). Consequently, the remaining side chains/substituents attached to the benzene ring must be fully saturated (no additional rings or double/triple bonds).

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2. Functional Group Identification

• Reaction with Neutral $\text{FeCl}_3$: The formation of a pink/violet color with neutral $\text{FeCl}_3$ confirms the presence of a phenolic hydroxyl group ($-\text{OH}$ directly attached to the benzene ring).

• Optical Activity: The compound rotates plane-polarized light, which means it must be optically active and contain a chiral (asymmetric) carbon center.

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3. Finding the Chiral Center

To create an asymmetric carbon atom using the remaining formula ($\text{C}_2\text{H}_5\text{O}$ attached to a $-\text{C}_6\text{H}_4\text{OH}$ ring), the carbon atom must be bonded to four different groups:

1. $-\text{H}$
2. $-\text{OH}$
3. $-\text{CH}_3$
4. $-\text{C}_6\text{H}_4\text{OH}$ (hydroxyphenyl ring)

This gives the side chain structure as $-\text{CH(OH)CH}_3$ (1-hydroxyethyl group).

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4. Counting the Isomers

A. Positional Isomers

The $-\text{CH(OH)CH}_3$ group can be attached at three different positions relative to the phenolic $-\text{OH}$ group on the benzene ring:

1. ortho-isomer: 1-(2-hydroxyphenyl)ethanol
2. meta-isomer: 1-(3-hydroxyphenyl)ethanol
3. para-isomer: 1-(4-hydroxyphenyl)ethanol

B. Stereoisomers (Enantiomers)

Each of these $3$ positional isomers possesses $1$ chiral carbon atom ($\text{C}^*$), which gives rise to $2$ optically active enantiomeric configurations: $(R)$ and $(S)$.

$\text{Total optically active isomers} = 3 \text{ positional isomers} \times 2 \text{ enantiomers} = 6$

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Final Answer

The total number of all possible isomers for this compound is 6.