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Q.10 For the given compound $\ce{X}$, the total number of optically active stereoisomers is ____.

enter image description here

This is from JEE Advanced 2018 second question paper.

I thought the answer to be at least 32 because of exactly 5 stereocentres present. One of the double bond cannot show geometrical isomerism possibly.

But the answer given in the answer key is 7:

enter image description here

I did not find any plane of symmetry because of the different kind of double bonds.

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    $\begingroup$ Two issues. (1) The image is not clear enough; it's hard to tell exactly which bonds are wavy versus wedge. (2) You're misreading the question. The wedge bonds are fixed. You can change configuration only for the wavy bonds. $\endgroup$ – Zhe May 22 '18 at 17:21
  • $\begingroup$ I'm so sorry for the picture quality. I can clarify it, The long chain on the bottom of the cyclopentane system contain the 2 wavy bond S, all others are in dark bond. $\endgroup$ – Sarah jane May 22 '18 at 17:40
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    $\begingroup$ Well, then there are 3 configurations to worry about, for a maximum of $2^{3}=8$. It can't be 32, but 7 is certainly reasonable. I'll let you figure out why it's 7 and not 8. $\endgroup$ – Zhe May 22 '18 at 17:43
  • $\begingroup$ Okay, but where did you get the third stereogenic centre there? Those two in wavy and then? Does it show synanti about that sigma bond in cyclopentane. If it is 8 then it will be easy because there must be a plane of symmetry. $\endgroup$ – Sarah jane May 22 '18 at 17:54
  • $\begingroup$ There's no third stereogenic center. But if the double bond isomerism that generates a separate isomer, that isomer can be chiral. If it is chiral, it is optically active... $\endgroup$ – Zhe May 22 '18 at 18:07
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The question is tricky, because

  • the two ends of the wavy bond are both stereogenic units (chiral cyclopentane ring carbon atom, and the double bond), so there is some additional unintentional ambuguity, but it's not a big deal, because the legend at right explains the situation somewhat;
  • it in fact asks “How many … stereoisomers does this structure represent?” (because of some stereogenic units with unspecified configuration) instead of “how many … stereoisomers does it have?”.

The unspecified-configuration stereogenic units are two chiral carbons and one double bond, which give total 8 stereoisomers.

Fig.1 - stereoisomers

The first one (top left) has a plane of symmetry, the other 7 stereoisomers are optically active.

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  • $\begingroup$ (Just note that the question has been changed afterwards, original one did not include the answer key link/quotation :-| ) $\endgroup$ – mykhal Jan 16 at 15:54

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