Question

Solid \(\mathrm{AB}\) has a rock salt type structure. If the radius of the cation is \(200 \mathrm{pm}\), what is the maximum possible radius of the anion? (a) \(483.1 \mathrm{pm}\) (b) \(273.6 \mathrm{pm}\) (c) \(200 \mathrm{pm}\) (d) \(400 \mathrm{pm}\)

Short answer

The maximum possible radius of the anion given a cation radius of 200 pm in a rock salt type structure is 273.6 pm. Thus, the correct answer is (b) 273.6 pm.

Step-by-step solution

Determine the Coordination Number

In a rock salt structure, each ion, both the cation and the anion, has an octahedral coordination. This means each cation is surrounded by 6 anions and vice versa, which gives them a coordination number of 6.

Describe the Rock Salt Structure

The rock salt structure can be thought of as two interpenetrating face-centered cubic (fcc) lattices, one of anions and one of cations. The cation is at the center of the octahedron formed by six anions.

Apply the Radius Ratio Rule

In a rock salt structure, the edge of the unit cell is twice the sum of the cation and anion radii because the anions touch each other along the face diagonal. The face diagonal is \(4r_{\text{anion}}^2 = a^2 + a^2 = 2a^2\), where \(a\) is the edge length. For octahedral coordination, the radius ratio (\(r_{\text{cation}}/r_{\text{anion}}\)) typically ranges from 0.414 to 0.732.

Calculate the Maximum Anion Radius

Assuming the cation touches the anion along the edge of the unit cell, the maximum radius of the anion is found when the cation is just touching the anion at their closest approach along the diagonal, making the sum of their radii equals to \(\frac{a\sqrt{2}}{2}\). Given the cation radius \(r_{\text{cation}} = 200 \text{pm}\), we can set up the equation \(\frac{a\sqrt{2}}{2} = r_{\text{anion}} + 200 \text{pm}\).

Divide the Face Diagonal by 2

Since the face diagonal is 4 anion radii, we have \(2r_{\text{anion}} = \frac{a\sqrt{2}}{2}\). Therefore to solve for \(r_{\text{anion}}\), we divide the previous equation by 2.

Express Edge Length in Terms of the Anion Radius

From \(2r_{\text{anion}} + 2r_{\text{cation}} = a\sqrt{2}\), we can express \(a\) (the edge length) in terms of the anion radius: \(a =\sqrt{2}(r_{\text{anion}} + r_{\text{cation}})\).

Find the Maximum Anion Radius

Substitute \(r_{\text{cation}}\) into the expression for \(a\) and solve for \(r_{\text{anion}}\): \(\frac{a\sqrt{2}}{2} = r_{\text{anion}} + 200 \text{pm}\) and \(\frac{2r_{\text{anion}}\sqrt{2}}{2} = a\sqrt{2}\). After equating both expressions for \(a\sqrt{2}\) we find that \(r_{\text{anion}} = 273.2 \text{pm}\).

Key concepts

Coordination Number

The coordination number of an atom in a solid refers to the number of nearest neighbors it has. In the context of the rock salt structure chemistry, the coordination number is particularly significant for understanding the stability and packing of the ions in the solid.

For instance, in the rock salt structure, as illustrated in the solved exercise, each ion is surrounded by an octahedral arrangement of nearest neighbor ions. Specifically, this means that a cation is surrounded by six anions and each anion by six cations, giving both types of ions a coordination number of 6. This octahedral coordination is paramount as it influences the ionic radii and the potential layouts of ions within the crystal lattice.

Understanding the coordination number is essential for predicting the structural arrangement and for explaining the physical properties of the material.

Radius Ratio Rule

The radius ratio rule is pivotal in crystal chemistry for predicting the type of coordination polyhedron that will form around an ion. It relates to the ratio of the radii of cations to anions and dictates the arrangement of the ions within a crystal lattice to maximize the stability of the ionic solid.

For example, step three of the solved exercise uses the radius ratio rule to deduce that in a rock salt structure, the radius ratio for octahedral coordination ranges from 0.414 to 0.732. This range ensures that each cation can optimally fit within the voids created by the anions, and vice versa. The rule serves to rationalize why certain ionic solids, like rock salt, adopt specific structures and it guides us in calculating the maximum possible radius for an anion given the radius of a cation.

When applying this rule, one can determine whether an ion will form an octahedral, tetrahedral, or other types of coordination complex based on the size of the ions involved.

Face-Centered Cubic Lattice

A face-centered cubic (fcc) lattice is a crystal system where atoms are located at each of the corners and the centers of all the cube faces of the unit cell. When discussing the rock salt structure chemistry, it's essential to recognize that the structure is composed of two interpenetrating fcc lattices—one of anions and the other of cations.

In the fcc lattice arrangement, the unit cell's edge length and the ions' radii have a direct relationship, as evident from the exercise solution. This arrangement allows us to derive the maximum anion radius using the edge length and the known cation radius.

Furthermore, the fcc lattice is key in understanding the rock salt structure because it explains the high packing efficiency and density of the solid, which are important factors in determining the material's physical properties. The interstitial sites within the fcc lattice, where the cations reside, contribute to the crystal's overall stability and are crucial in biological processes, as well as in the manufacture of various materials.