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What is the correct number of stereo isomers in case of the following compound, i.e. 3,6-dimethylpiperazine-2,5-dione?

3,6-dimethylpiperazine-2,5-dione

The answer key gives three, but I think there should be four. I recognised the two planes of symmetry, one through the methyl groups and the other through the keto groups.

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    $\begingroup$ The plane through methyl groups would project the keto group upon nitrogen, hence it is not a plane of symmetry. Ditto for the other one. But that's irrelevant; we are counting isomers here. What are your 4? $\endgroup$ Jan 29 '19 at 11:48
  • $\begingroup$ counting 2 nitrogen and 2 carbon as stereocentres, we hav total 4 stereocentres and the molecule does not hav plane of symmetry. but nitrogen goes inversion. so only two chiral centers so 4 $\endgroup$ Jan 29 '19 at 12:48
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    $\begingroup$ Good. But there is a catch: a certain symmetry element makes the chiral centers equivalent, so RS and SR are the same thing, which leaves only 3. $\endgroup$ Jan 29 '19 at 12:53
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    $\begingroup$ The nitrogen atoms should not be considered stereocenters, not because of rapid inversion but because amide nitrogen atoms are almost planar. $\endgroup$
    – Zhe
    Jan 29 '19 at 14:30
  • $\begingroup$ You should build 3D models of the compound. If you do you will see that two of the supposed stereoisomers are the same molecule. $\endgroup$
    – matt_black
    Jan 29 '19 at 16:09
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Three is the correct answer.

As Ivan pointed out, there are no planes of symmetry, for example:

enter image description here

Ivan is also correct - the "RS" is the same as the "SR". You can prove this to yourself most easily using molecular models, but here is how to see it on the page. You just need to rotate the molecule by 180° in the plane of the page:

enter image description here

So, despite having chiral centres, this compound is achiral because it is superposable with its mirror image. "But wait, I was told that a molecule needs to have an internal plane of symmetry to be a meso compound" I hear you say. Unfortunately, this is an incomplete truth that many sources repeat. To be a meso compound, it can alternatively have a centre of symmetry, also known as an inversion centre. This is simply one point in space through which any atom is reflected through - ie. each atom is diametrically opposite an equivalent atom at an equal distance from this point.

enter image description here

The other two stereoisomers are the (R,R) and the (S,S). These are non-superposable mirror images, and hence enantiomers. Note these isomers have no plane of symmetry or centre of symmetry.

enter image description here

Once again, this is easier to see with physical models.

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    $\begingroup$ For completeness sake: meso-compound - A term for the achiral member(s) of a set of diastereoisomers which also includes one or more chiral members. achirality (from chirality) - [...] If the object is superposable on its mirror image the object is described as being achiral. $\endgroup$ Jan 30 '19 at 10:34

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