Ionic sizes and crystal structures (Pauling's rule #1)

Susannah Dorfman, Michigan State University
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Initial Publication Date: May 28, 2024

Summary

Students derive the radius ratio rule of ionic coordination (Pauling's 1st rule) empirically using balls of different sizes followed by a mathematical derivation.

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Context

Audience

Undergraduate required course in mineralogy

Skills and concepts that students must have mastered

High school trigonometry

How the activity is situated in the course

I teach this as a 50-minute lecture class activity during our unit on bonding and mineral chemistry.

Goals

Content/concepts goals for this activity

ionic sizes, bonding coordination, mineral structures

Higher order thinking skills goals for this activity

3D visualization, 3D penetration, development of empirical model, analysis of precision and accuracy

Other skills goals for this activity

working in groups

Description and Teaching Materials

Materials:

  • Balloons and spheres of various sizes
  • Rulers/tape measures

Lesson plan:

  • In previous classes, students have been introduced to the concept of bonding coordination and the bonding configuration for common cations in rock-forming minerals, but without a theoretical basis. Class opens with a few lecture slides reviewing this information and introducing a statement of Pauling's first rule, that ratios of ionic radii determine bonding coordination.
  • Students are divided into groups. Each group is assigned a different sized sphere, e.g. a marble or a softball. These spheres represent cations. Balloons are distributed to all the groups. The class inflates the balloons to represent ~spherical anions, attempting to make all anions the same size.
  • Each group measures their cation's radius and the anion radius and determines the cation/anion radius ratio. We discuss the precision of these measurements. Class results are tabulated.
  • Each group works together to determine the maximum number of anions that can fit around their cation, in contact with the cation (a chemical bond). Class results are tabulated.
    - Results are compared to a table with radius ratios derived for 3-12 bonds around a cation. We check whether our measurements were within precision of the correct ratios and discuss sources of uncertainty in the measurement (e.g. are all balloons the same size?).

Polyhedron and minimum radius ratio for each coordination number
C.N. Polyhedron Radius ratio
3 triangular 0.155
4 tetrahedron 0.225
6 octahedron 0.414
7 capped octahedron 0.592
8 square antiprism (anticube) 0.645
8 cube 0.732
9 triaugmented triangular prism 0.732
12 cuboctahedron 1.00

  • Instructor provides a cross-section of one example coordination polyhedron, with important features labelled. I usually use triangular since it fits in the 2D plane: CN3_diagram.pdf (Acrobat (PDF) 48kB May26 24). Students work in groups to use geometry to derive the correct radius ratio. At the end of class, confirm that the result matches the tabulated list of ratios.

Teaching Notes and Tips

  • Once the students have worked out appropriate coordination for the spheres, take a moment to show everyone how small the tetrahedral cation is. Remind everyone this represents silicon, the most abundant cation in Earth's crust. It's shockingly small compared to the anions!
  • We refer to the Shannon ionic radii in class and in homework. Students are expected to be able to find reliable values for ionic radii in textbook and online resources.
  • Deriving an expression for a ratio can be a leap for students who are rusty or anxious about math. A few check ins at the board with hints on the setup help a lot.

Assessment

Following this activity, students complete homework exercises on Pauling's rules and bonding. Relevant questions expect students can:

  1. Identify the anions, cations, and bonding coordination in an image of a crystal structure.
  2. Look up appropriate ionic radii and compute the radius ratio.
  3. Predict coordination number for a pair of ions based on the radius ratio rule.