Total Number of Optically Active Stereoisomers for Compound X

To determine the total number of optically active stereoisomers for compound $\mathbf{X}$, we analyze the fixed and variable stereogenic units present in its structure.

1. Analysis of Stereogenic Units

• Fixed stereocenters and double bonds (represented by solid wedges/thick bonds):

  1. The two $-OH$ groups on the cyclopentane ring are fixed in their configurations (both represented by solid wedges).
  2. The upper side chain has:
     • A fixed double bond geometry ($E$-configuration / trans).
     • A fixed stereocenter at the carbon attached to the upper $-OH$ group (solid wedge).

• Variable stereogenic units (represented by wavy lines):

  1. The stereocenter at the cyclopentane ring carbon attached to the lower side chain (2 choices: $R$ or $S$).
  2. The double bond geometry in the lower side chain (2 choices: $E$ or $Z$).
  3. The stereocenter at the lower side chain carbon attached to the $-OH$ group (2 choices: $R$ or $S$).

2. Calculation of Total Stereoisomers

Since there are $3$ independent variable stereogenic units, the total number of stereoisomers generated is: $N_{\text{total}} = 2^3 = 8$

3. Determination of Optically Active Stereoisomers

The cyclopentane core containing the two identical ring $-OH$ groups with fixed, identical configurations possesses a potential internal plane of symmetry ($\sigma$) passing through the isolated ring $CH_2$ group and bisecting the bond between the two chain-bearing ring carbons.

• Meso Compound (Optically Inactive): Out of the $8$ possible configurations for the lower chain, exactly $1$ configuration matches the stereochemistry of the upper chain such that the molecule possesses a plane of symmetry. This single stereoisomer is a meso compound and is optically inactive.

• Optically Active Stereoisomers: The remaining stereoisomers lack any plane of symmetry or inversion center, making them chiral and optically active.

$\text{Total number of optically active stereoisomers} = N_{\text{total}} - N_{\text{meso}} = 8 - 1 = 7$