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Lattice Structures in Crystalline Solids

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Calculation of Ionic Radii

If we know the edge length of a unit cell of an ionic compound and the position of the ions in the cell, we can calculate ionic radii for the ions in the compound if we make assumptions about individual ionic shapes and contacts.

Example 5

Calculation of Ionic Radii
The edge length of the unit cell of LiCl (NaCl-like structure, FCC) is 0.514 nm or 5.14 Å. Assuming that the lithium ion is small enough so that the chloride ions are in contact, as in Figure 15, calculate the ionic radius for the chloride ion.
Note: The length unit angstrom, Å, is often used to represent atomic-scale dimensions and is equivalent to 10−10 m.
Solution
On the face of a LiCl unit cell, chloride ions contact each other across the diagonal of the face:
Three images are shown. The first shows a cube of alternating green and purple spheres. A smaller cube within that cube is outlined and a larger version of it appears next. This figure is a grey cube that appears to be made up of spheres. There are small spaces between each sphere. There is a right triangle outlined in this cube and a larger version of it appears next. This right triangle has two sides labeled “a,” and the hypotenuse, which spans two half-circles and one full one is labeled, “r, 2 r, and r.”
Drawing a right triangle on the face of the unit cell, we see that the length of the diagonal is equal to four chloride radii (one radius from each corner chloride and one diameter—which equals two radii—from the chloride ion in the center of the face), so d = 4r. From the Pythagorean theorem, we have:
a^2\;+\;a^2 = d^2
which yields:
(0.514\;\text{nm})^2\;+\;(0.514\;\text{nm})^2 = (4r)^2 = 16r^2
Solving this gives:
r = \sqrt{\frac{(0.514\;\text{nm})^2\;+\;(0.514\;\text{nm})^2}{16}} = 0.182\;\text{nm}\;(1.82\;\mathring{\text{A}})\;\text{for\;a\;Cl}^{-}\;\text{radius}.
Check Your Learning
The edge length of the unit cell of KCl (NaCl-like structure, FCC) is 6.28 Å. Assuming anion-cation contact along the cell edge, calculate the radius of the potassium ion. The radius of the chloride ion is 1.82 Å.

Answer:

The radius of the potassium ion is 1.33 Å.
It is important to realize that values for ionic radii calculated from the edge lengths of unit cells depend on numerous assumptions, such as a perfect spherical shape for ions, which are approximations at best. Hence, such calculated values are themselves approximate and comparisons cannot be pushed too far. Nevertheless, this method has proved useful for calculating ionic radii from experimental measurements such as X-ray crystallographic determinations.


To continue reading click on the link below:

https://opentextbc.ca/chemistry/chapter/10-6-lattice-structures-in-crystalline-solids/


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