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How do I find out the total number of stereoisomers of 3,6-dimethylpiperazine-2,5-dione?
As far as I can see, the compound has two chiral carbons. Would the nitrogens make a difference? They seem chiral as well (three different groups and a lone pair).
My book says there are 3 total stereoisomers, but I expected more. It has considered only two chiral carbons, and nothing has been said about the nitrogens. Why?
I believe this is a sophomore organic chemistry text book question, so I won't elaborate possible stereoisomers involving lone pairs on nitrogen atoms. Nonetheless, 3,6-dimethylpiperazine-2,5-dione has 2 chiral carbon centers so that it should theoretically have $2^2 = 4$ stereoisomers. If you look at the molecule, it has center of symmetry (i):
Therefore, two of the isomers are identical and we call it meso-isomer. Thus, total isomers of this molecule are three: ($3R,6R$)-, ($3S,6S$)-, and ($3R,6S$)- (($3S,6R$)- is essentially ($3R,6S$)-).
