Question

Potassium iodide has a unit cell similar to that of sodium chloride (Figure 9.21\() .\) The ionic radii of \(\mathrm{K}^{+}\) and \(\mathrm{I}^{-}\) are \(0.133 \mathrm{nm}\) and \(0.216 \mathrm{nm},\) respectively. How long is (a) one side of the cube? (b) the face diagonal of the cube?

Short answer

Answer: The length of one side of the potassium iodide unit cell is 0.698 nm, and the face diagonal is 0.987 nm.

Step-by-step solution

Determine the radii sum in the unit cell

Since the unit cell of potassium iodide is similar to that of sodium chloride, it means that it has a face-centered cubic lattice structure. In this lattice structure, the ion with a smaller radius (K+) is surrounded by six ions with larger radii (I-). The distance between a potassium ion (K+) and an iodide ion (I-) is equal to the sum of their ionic radii. So, we first calculate the sum of the ionic radii of K+ and I-: Sum of ionic radii = K+ radius + I- radius = \(0.133 \mathrm{nm} + 0.216 \mathrm{nm} = 0.349 \mathrm{nm}\)

Finding the length of one side of the cube (a)

In the face-centered cubic lattice, the side length (a) of the cube is equal to two times the distance between a potassium ion (K+) and an iodide ion (I-) (which also equals the sum of their ionic radii). Thus, we multiply the sum of the ionic radii by 2 to find the side length of the unit cell: Side length (a) = 2 × Sum of ionic radii = 2 × \(0.349 \mathrm{nm} = 0.698 \mathrm{nm}\)

Calculating the face diagonal of the cube

Now, we want to find the face diagonal of the cube. We can use Pythagorean theorem for this purpose. Let the face diagonal be d. In a face of the cube, we have a right triangle with two sides of equal length (a) and the face diagonal as the hypotenuse. Therefore, we have: \(d^2 = a^2 + a^2\) Now, we substitute the value of a and solve for d: \(d^2 = (0.698 \mathrm{nm})^2 +(0.698 \mathrm{nm})^2\) \(d^2 = 0.487204 + 0.487204 = 0.974408 \mathrm{nm^2}\) \(d = \sqrt{0.974408} = 0.987 \mathrm{nm}\) (approx.) So, the length of one side of the cube (a) is \(0.698 \mathrm{nm}\) and the face diagonal of the cube is \(0.987 \mathrm{nm}\).

Key concepts

Face-Centered Cubic Lattice

Understanding the structure of crystalline solids is essential in various fields of science and engineering. One common structure is the face-centered cubic (FCC) lattice, which is notable for its high packing efficiency. In this lattice, atoms, or in the case of ionic compounds, ions, sit at each of the eight corners of a cube, plus one at the center of each cube face.

For potassium iodide, which mimics the structure of sodium chloride, the potassium ions (K+) represent the smaller ions located at the cube corners, while the larger iodide ions (I-) sit at the face centers. This arrangement allows for a stable ionic structure with each ion surrounded and stabilized by ions of opposite charge.

When we visualize this, we can see that each side of the cube not only contains two ionic radii from the corner to the center but also spans the distance across the center ion to reach the opposite corner. This understanding is pivotal in solving problems related to the dimensions of the unit cell, as it directly relates to the lengths and diagonals of the cube.

Ionic Radii Calculation

Determining the size of ions in a solid allows us not only to predict the structure but also to calculate distances within that structure. Ionic radii calculation involves measuring the size of an ion within a crystal lattice. The radius is half the distance between the nuclei of neighboring ions of the same element in the lattice, or half the sum of the distances between the nuclei of different ions for compounds.

In our exercise, we're given the ionic radii for both potassium (K+) and iodide (I-). To find the lattice parameters, we use these radii since the distance between adjacent ions in the crystal lattice is equal to the sum of their respective radii. This fundamental concept leads to precise derivations of the cube's edge length and helps in further calculations of distances within the crystal, like the face diagonal.

Pythagorean Theorem in Crystallography

The Pythagorean theorem is a principle that extends beyond basic geometry, finding importance in crystallography. This theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In the context of crystallography, it's used to calculate distances between points in a crystal lattice that are not aligned along a straight axis, like the face diagonal of a cube. The exercise we've discussed exemplifies this application. Once we determine the edge-length 'a' of our FCC lattice cube, we can treat the face diagonal as the hypotenuse of a right-angled triangle where both legs are of length 'a'. Applying the Pythagorean theorem here showed us how to find the length of the diagonal across the face of the unit cell. This approach is crucial for understanding the geometric relationships within the lattice and for predicting the arrangement of ions or atoms in the solid.