The display below shows four atoms packed together in a closest-packed arrangement. This particular geometry is tetrahedral. In the very center of this tetrahdron is a hole. This hole is said to be a tetrahedral hole because the hole is surrounded by four atoms (the coordination number is four).
Take note that the size of this hole is relatively small. If an atom is sufficiently small, it can fit into this hole. Click on the "Show Atom in Hole" to insert a small atom (colored red) in the hole. (You may need to rotate the image to clearly see the red atom in the tetrahedral hole.)
What is the largest atom that can fit into the hole without pushing the outer (blue) atoms apart?
The radius of the outer atoms is designated r. The radius of the hole (this is the radius of the largest sphere that can fit in the hole) is represented by rhole.
What is the ratio rhole/r?
Derive this ratio using the tetrahedral geometry of the outer (blue) atoms and the laws of trigonometry.
Hint: The law of cosines will be particularly useful: C2 = A2 + B2 - 2 A B cos c . In addition, recall that the bond angle for a molecule with perfect tetrahedral geometry (for example, CH4) is 109.47o (this is the angle whose cosine is -1/3).
When you have completed your derivation, you may check your answer.
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