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enter image description here

I always thought that a molecule couldn't be chiral if it had a plane of symmetry. Doesn't the chiral molecule here have symmetry diagonally across the OHs? The achiral one doesn't look symmetrical to me, so I also don't know how it is achiral.

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    $\begingroup$ Both of your structures (Fischer projections) are chiral. Read this to help you understand such structures. chemistry.stackexchange.com/questions/48995/… $\endgroup$ – user55119 Feb 13 '18 at 2:15
  • $\begingroup$ The one on the left has point symmetry (in the Fischer projection, not in reality), but point symmetry anyway is not relevant for chirality. $\endgroup$ – Karl Feb 13 '18 at 6:49
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Regarding the first molecule: I understand that the simplistic Fishcher projection - with all its bond depicted by solid lines - can be very confusing at first glance in determining optical activity. You may believe it to be optically inactive, hoping for a center of symmetry to exist, but it is not so.

You must keep in mind how the molecule actually exists in 3d! Recall the rules of Fisher projection, the horizontal bonds project outward, while the vertical bonds extend inward. Hence, the molecule would actually look like this:

enter image description here

Now, I hope you can tell, there is neither any center of symmetry nor any plane of symmetry here. Hence, the molecule is chiral.

PS: Your second molecule is chiral by similar logic, as also mentioned in the comments, and as you yourself noted.

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