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Aight, let's keep the train moving. We just figured out why ceramics are so strong (that Coulombic force). Now let's figure out how they're built.

Math Topic 2: Predicting a Ceramic's Structure (Radius Ratio)

Remember we said ionic bonds are nondirectional? This means the ions act like sticky spheres, and they just want to pack together as tightly as possible. But there's a rule: an ion only wants to touch ions of the opposite charge.

So, the question is: How many negative anions can you physically pack around one positive cation?

This number has a special name: Coordination Number (CN).

The answer depends entirely on how big the ions are relative to each other.

The Breakdown (Line by Line):

This isn't really a formula, it's a ratio. We call it the Radius Ratio, r/R.

\(Radius Ratio = \frac{r}{R}\)

  • r (lowercase): The radius of the smaller ion (usually the positive cation).
  • R (uppercase): The radius of the larger ion (usually the negative anion).

You look up the radii of your two ions in a table (like in Appendix 2 of your book), you do this one simple division, and the number you get tells you the Coordination Number.

How to Use the Radius Ratio:

You calculate your r/R value, and then you look it up in a chart like Table 2.1 from your book.

Here's a simplified version of that chart:

If your calculated r/R is... Then the maximum number of neighbors (CN) is... And the shape they make is a...
< 0.155 2 Line
0.155 - 0.225 3 Triangle
0.225 - 0.414 4 Tetrahedron
0.414 - 0.732 6 Octahedron
0.732 - 1.000 8 Cube

So, what does this actually tell us in real life?

This simple ratio is a powerful design tool. It predicts the fundamental, repeating building block of a ceramic's crystal structure.

Imagine you're trying to design a new ceramic. You can pick a cation and an anion from the periodic table, look up their radii, calculate the r/R, and instantly predict the geometry of its basic structure.

This geometry is paramount (vocab alert: means more important than anything else) because it determines the material's density, how it interacts with light, and even its mechanical properties. It all comes back to that simple ratio of sizes.

Quantitative Check-in #2:

Let's do a real example your professor could ask. You want to make a ceramic called Magnesium Oxide (MgO).

  1. Go to Appendix 2 in your book (or I can just give you the numbers). Find the ionic radius for the Magnesium ion (Mg²⁺) and the Oxygen ion (O²⁻).
  2. Identify which one is r (smaller) and which is R (larger).
  3. Calculate the Radius Ratio r/R.
  4. Using the table above, predict the Coordination Number (CN) for Magnesium in this ceramic.

The ionic radii you need are:

  • Radius of Mg²⁺ = 0.078 nm
  • Radius of O²⁻ = 0.132 nm