INO-13

Coordination from radius ratio

CN jumps at the critical radius ratios. A hard-sphere snapshot.

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Crystal chemistryPauling CN

Governing equation

\mathrm{CN}=f(r_+/r_-)

$r_+ \in \left[0.2, 1.8\right]$

where

r_+: Cation radius (Å)
r_-: Anion radius (Å)
CN: Coordination number (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\mathrm{CN}=f(r_+/r_-)

$r_+ \in \left[0.2, 1.8\right]$

$r_+ = 0.2$ $CN = 3$

Along the curve

②

Domain of the variable — blue

r_+_{\min} = 0.2,\quad r_+_{\max} = 1.8

$r_+ \in \left[0.2, 1.8\right]$

$r_+ = 0.2$ $r_+ = 0.2$

Along the curve

③

Response curve — green

\mathrm{CN}=f(r_+/r_-)

$r_+ \in \left[0.2, 1.8\right]$

$r_+ = 0.2$ $CN = 3$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}CN}{\mathrm{d}r_+}

$r_+ \in \left[0.2, 1.8\right]$

$r_+ = 0.21$ $CN^{\prime} = 0$

Along the curve

Lecture brief

Historical brief

Pauling and Allred–Rochow electronegativity, Kapustinskii lattice energy, CFSE and Goldschmidt radii organise the periodic solid. The sheets predict bond character and crystal packing. This sheet (INO-13 — Coordination from radius ratio) is the form associated with Pauling CN. Working symbols: $r_+$ , $r_-$ $\rightarrow$ $CN$ . Cations sit in the largest hole they can rattle in: triangular, tetrahedral, octahedral, cubic.

Purpose

Purpose: compute

CN

from

r_+

r_-

in Inorganic chemistry via

\mathrm{CN}=f(r_+/r_-)

CN jumps at the critical radius ratios. A hard-sphere snapshot. Use it when a real inorganic chemistry question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

r_+ = 0.950\,\mathrm{Å}

r_- = 1.810\,\mathrm{Å}

, the governing relation

\mathrm{CN}=f(r_+/r_-)

yields

CN = 8\,\mathrm{—}

. Same radii as the radius-ratio sheet; output predicted CN. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Cation radius r_+0.95 ÅAnion radius r_-1.81 Å

Outputs

Coordination number CN8 —

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Narration of this film

Same radii as the radius-ratio sheet; output predicted CN.

Cations sit in the largest hole they can rattle in: triangular, tetrahedral, octahedral, cubic.

Reading speed

Watch on YouTube