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Every common textbook explains the method of the radii quotient between the cation and anion of a selected salt to determine to coordination number(s) in that salt. For example deduce "Inorganic Chemistry" (Huheey) the coordination number of BeS by following calculation:

Radii quotient $k = \dfrac{r_{\text{cation}}}{r_{\text{anion}}} = \dfrac{59 \text{ pm}}{170 \text{ pm}} \approx 0.35 $

This quotient $k$ indicates a tetrahedral coordination (coordination number for both ions = 6), because $k$ lays between the two limits 0,255 and 0,414 for a energetically favorable tetrahedral coordination (limits are derived through a simple mathematical problem with pythagoras).

Now my actual question or problem with this way to determine the coordination number: The used crystal radii $r_{\text{cation}}$ and $r_{\text{anion}}$ of Be2+ and S2- depend on their coordination; on their coordination number. For example are the crystal radii for Be2+:

  • Coordination Number = 6: $r_{\text{cation}} = 59 \text{pm}$
  • Coordination Number = 4: $r_{\text{cation}} = 41 \text{pm}$
  • Coordination Number = 3: $r_{\text{cation}} = 30 \text{pm}$

The difference between these values are tremendous and change the result of the $k$ calculation above, e.g. if I choose $r_{\text{cation}} = 30 \text{pm}$ then $k=\dfrac{30 \text{ pm}}{170 \text{ pm}} \approx 0.18$ and therefore BeS should be linear coordinated...

In Short: How can we use a method to determine the coordination of a ion in an (simple) ionic structure if the used radii in this method are again depended on the coordination?

Crystal radii data from: http://abulafia.mt.ic.ac.uk/

Edit: I am aware that this method has exceptions and is just simplified (there are many other factors that determine the coordination number)...

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