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In my textbook, the percentages of anti-eliminations are given which are quite contrary to expectations. The reason is also not given quite clearly. Why does the preference for anti-periplanarity change and what are the reasons for the disparity between the different cyclic structures?

\begin{array}{lr} \text{Ring Size} & \%\text{ SYN Elimination} \\ \hline \text{Cyclobutyl} & 90 \\ \text{Cyclopentyl} & 46 \\ \text{Cyclohexyl} & 4 \\ \text{Cycloheptyl} & 37 \end{array}

The values are for $\ce{(CH2)_nC-HY}$.

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    $\begingroup$ You need to build some models or look at some computed examples. It's really hard to achieve a conformation where the hydrogen atom and the leaving group are anti-periplanar in some cases. $\endgroup$ – Zhe Aug 11 '17 at 17:45
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To begin with, I want to present you approximate representations of the four cycloalkenes in question. Unfortunately though, these representations cannot, in this case, be used to fully replace building the molecules; I suggest you do so with the molecular modelling kit of your choice.

Cycloalkanes

I’ll start off with the six-membered ring. This — as is often correctly pointed out — is essentially strainless and the chair conformation displayed is the most stable one. While frequent interconversion between different chair forms can occur, switching axial and equatorial substituents. We can thus expect that a reaction will, in principle, always take place in this chair form.

If we examine the dihedral angles, we notice that an axial substituent is anti-periplanar to the adjacent two carbons’ hydrogens. This means, it can easily anti-eliminate in both directions. No substituent whether equatorial or axial has any ecliptic hydrogen in either direction. This means, that a syn-specific elimination, which requires an ecliptic configuration, cannot occur out of the chair conformation. The $4~\%$ syn-elimination the table references must occur from other, minor configurations such as the boat.

Cyclopentane is also hardly strained. However, the configuration of the ring is much more volatile. Furthermore, even in the shown configuration there are possibilities of syn-eliminations; for example see the two carbons opposite to the upwards-pointing one. Still, anti-eliminations are also well possible, especially from the other three carbons that form the upwards arch, and account for most of the elimination.

A similar case can be made for cycloheptane, which is even more strained. There is not much for me to say here that I haven’t said for cyclopentane other than that you should build it.

And finally cyclobutane. This is the most strained of the systems considered and probably the one with the most fluctuating configuration. Here, it looks like an anti-elimination is possible; but this requires the substituent to occupy an axial position. This configuration is strongly sterically hindered. Syn-eliminations can occur from any position by only moderate shifts from the geometry shown. This may explain why anti-eliminations are strongly disfavoured here. I would, however, once again stress how helpful it is (especially in this case) to build the model.

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