6
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$\ce{Be^2+}$ has an ionic radius of $\pu{45 pm}$, while $\ce{He}$ has a radius of only $\pu{31 pm}$. If they are isoelectronic (they both have two elections in the $\mathrm{1p}$ orbital), and $\ce{Be}$ has more protons to attract them, why doesn't $\ce{Be^2+}$ have a smaller radius?

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    $\begingroup$ He has VDW radisum of 140 pm, what particular radius of He are your referring to? There is more than one. $\endgroup$ – permeakra Feb 18 at 6:50
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    $\begingroup$ It's probably down to different definitions and/or measuring techniques. $\endgroup$ – TAR86 Feb 18 at 7:01
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You are mixing apples and oranges. Or, to be more precise, an ionic radius for $\ce{Be^2+}$ with coordination number (C.N.) 6 and van der Waals radius of $\ce{He}$. To make it clear I compiled data for van der Waals and covalent radii [1, p. 9-58] as well as ionic radii [1, p. 12-13]:

$$ \begin{array}{lccccc} \hline \text{Element} & R_\mathrm{vdW}/\pu{Å} & R_\mathrm{cov}/\pu{Å} & \text{Ion} & \text{C.N.} & R_\mathrm{i}/\pu{Å} \\ \hline \ce{Be} & 1.52 & 0.99 & \ce{Be^2+} & 4 & 0.27 \\ & & & & 6 & 0.45 \\ \ce{He} & 1.40 & 0.37 & & & \\ \hline \end{array} $$

Apparently, there is no experimental ionic radii for helium to compare with as helium, well, represents one of the most non-reactive elements. Also, all radii set are determined using various methods and cannot be directly compared. From the foreword [1. p. 9-58]:

There are many scales of van der Waals radii, but they are not fully consistent with one another. The van der Waals radii determined by Bondi [...] from x-ray diffraction data, crystal densities, gas kinetic collision cross sections, critical densities, and liquid- state properties are the most widely used values. [...]

The covalent radii tabulated here are recommendations for single covalent bonds, and they are based on a comprehensive evaluation of experimental data [...].

Ionic radii are estimated first and foremost from the experimental crystal structure data, averaged and supplemented by empirical and theoretical calculations (extrapolations, e.g. using Zachariasen method).

References

  1. Haynes, W. M.; Lide, D. R.; Bruno, T. J. CRC Handbook of Chemistry and Physics: A Ready-Reference Book of Chemical and Physical Data.; 2017; Vol. 97. ISBN 978-1-4987-5429-3.
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  • $\begingroup$ This is a bit over my head right now, but if I am understanding your basic point correctly, the variation is due to different types of radii and some measurement inconsistencies? $\endgroup$ – TrumpetDude Feb 20 at 8:12
  • $\begingroup$ @TrumpetDude Yep, this is pretty much it. $\endgroup$ – andselisk Feb 20 at 8:20
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    $\begingroup$ Different types, yes. It is much like comparing the height of one man with the height of house of another man, and concluding that the latter is higher. Sure, both parameters are called the same ("height"), but there is a catch. $\endgroup$ – Ivan Neretin Feb 20 at 8:21

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