Potassium iodide has a unit cell similar to that of sodium...

Potassium iodide has a unit cell similar to that of sodium chloride (Figure 9.18). The ionic radii of $\mathrm{K}^{+}$and $\mathrm{I}^{-}$are 0.133 nm and 0.216 nm , respectively. How long is
(a) one side of the cube?
(b) the face diagonal of the cube?

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Key Concepts

Relationship of Ionic Radii to Unit Cell Dimensions

In crystals with a rock salt structure, the unit cell edge length is determined by the fact that a cation and an anion are in contact along the edge of the cube. Thus, the edge length equals twice the sum of the radii of the oppositely charged ions. This relationship is essential for translating microscopic ion sizes into macroscopic unit cell parameters in ionic solids.

Cube Geometry and Face Diagonals

The face diagonal of a cube is a geometric feature calculated using the Pythagorean theorem. In a cube with edge length a, the face diagonal is given by a?2. This concept is key for determining distances within the unit cell along non-axial directions, and it is frequently used in crystallography and solid state chemistry.

Rock Salt Structure

This structure, exemplified by the NaCl-type lattice, is a common ionic arrangement in crystals where each ion is octahedrally coordinated by six oppositely charged ions. In such a lattice, the cations and anions alternate positions in a face?centered cubic arrangement, and the geometry allows the distance between the centers of adjacent oppositely charged ions (which touch along the cube edge) to be expressed as the sum of their ionic radii.

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