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My teacher told me that trans-decalin (see below) is achiral due to the presence of both, a centre of symmetry and a plane of symmetry. But I could not spot the plane of symmetry untill now. Can someone point out the plane of symmetry in a diagram?

schematic representation of trans-decalin

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    $\begingroup$ It passes through the two central carbons, perpendicular to the molecule. $\endgroup$ – Ivan Neretin Oct 13 '16 at 8:22
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It's not easy to see from a diagram, because it distorts bonds and angles. I recommend building it with a balls-and-sticks model set. You can also use a molecular viewer to model it; there are a couple of open-source (or at least free) ones out there.

I have calculated the molecule on the DF-BP86/def2-SVP level of theory. The point group of the molecule is C2h. In the following model I have highlighted the plane of symmetry, and the rotational C2-axis. At their intersection is an inversion centre Ci.

depiction of trans-decalin with symmetry elements

(I needed to downscale this a lot, click on the image to get to a high resolution still. Images created with ChemCraft, assembled with GIMP.)

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  • $\begingroup$ Is it possible for one ring to have a boat conformation at room temperature? In this case the plane disappears. $\endgroup$ – porphyrin Oct 13 '16 at 16:46
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The question has already been answered nicely by Martin - マーチン♦ who pointed out that there is a mirror plane (i.e. a $\sigma = S_1$ element) and a center of inversion (i.e. an $S_2$ element), which both are already enough for a molecule to become achiral (existence of any $S_n$ element suffices for that). There is one important caveat though that often bugged me during stereochemistry lectures: Even if the ground state of a substance might not be achiral (for example if you placed some substituents on your decalin), then it can still be possible for the chemical to be achiral due to quick racemization when an achiral transition state with low energy is available (e.g. inversion of $\mathrm{NR_3},\mathrm{PR_3}$ systems or the $^1C_4\mapsto ^4C_1$ inversion of cyclohexanes). I think that is a little trap that is often missed in stereochemistry lectures.

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