Question

A binary solid \((A B)\) has a rock salt structure. If the edge length is \(400 \mathrm{pm}\), and radius of cation is \(80 \mathrm{pm}\) the radius of anion is: (a) \(100 \mathrm{pm}\) (b) \(120 \mathrm{pm}\) (c) \(250 \mathrm{pm}\) (d) \(325 \mathrm{pm}\)

Short answer

The radius of the anion is (b) \(120 \text{pm}\).

Step-by-step solution

Understanding the rock salt structure

A rock salt structure is a face-centered cubic lattice structure where cations occupy octahedral interstices and anions are at the corners and face centers. In such a structure, the edge length (a) of the cube is related to the radii of the cations (r+) and anions (r-) by the formula: \(a = 2 (r+ + r-)\).

Express the radius of the anion in terms of edge length and radius of cation

We can express the radius of the anion (r-) using the given edge length (a) and the known radius of the cation (r+). Plugging the values into the formula from Step 1 gives us: \(400 \text{pm} = 2 (80 \text{pm} + r-)\).

Solve for the radius of the anion

Solving the equation from Step 2 for \(r-\), we first simplify the equation: \(400 \text{pm} = 160 \text{pm} + 2r-\). Subtracting 160 pm from both sides gives us: \(240 \text{pm} = 2r-\). Dividing both sides by 2, we find the radius of anion: \(r- = 120 \text{pm}\).

Key concepts

Chemical Bonding

When we talk about chemical bonding, we’re discussing the force that holds atoms together in molecules and compounds. This force can be ionic, covalent, metallic, or hydrogen bonding. Ionic bonds are particularly pertinent when discussing rock salt structures like in the given exercise. In ionic bonding, one atom donates an electron to another, resulting in a structure where each ion has a full outer shell of electrons, leading to a stable electron configuration.

For example, table salt (NaCl) forms when sodium (Na) donates an electron to chlorine (Cl), creating Na+ and Cl- ions. These ions have opposite charges and attract each other, forming a solid crystalline structure. Understanding the nature of these ionic bonds is critical to grasp why these ions arrange themselves in particular patterns which is fundamental to the formation of the rock salt structure, as seen in our exercise.

Solid State Chemistry

The realm of solid state chemistry explores the structure, properties, and behavior of solid materials. A key area of interest is the study of crystal structures, like the one presented in our exercise — the rock salt structure. In these structures, the precise arrangement of ions and their interaction with each other govern the physical properties of the solid, such as melting point, hardness, and electrical conductivity.

In the rock salt structure, for instance, the ratio of the radius of anions to cations and their arrangement in a face-centered cubic lattice results in a tightly-packed, stable configuration that reflects the physical characteristics of salts. The calculation of ionic radii is an important aspect of solving problems in solid state chemistry, enabling us to predict the behavior of ionic solids under various conditions.

Ionic Compounds

Ionic compounds are substances formed by the chemical bonding between atoms with opposite charges, namely cations (positively charged) and anions (negatively charged). These compounds arise from the transfer of electrons, leading to a lattice of ions held together by strong electrostatic forces.

An interesting aspect of ionic compounds is their high melting and boiling points, which is a direct consequence of these strong forces. In the context of our textbook exercise, knowing that the binary solid AB forms a rock salt structure gives us insight that A is likely a metal that has lost electrons to become a cation and B is a nonmetal that has gained those electrons to become an anion, creating an ionic bond that results in the formation of the compound AB.

Crystal Lattice

The concept of a crystal lattice is integral in understanding solid state structures. It's a three-dimensional arrangement of atoms, molecules, or ions, spaced at specific distances from each other and repeating in a pattern. The lattice points in a crystal are not random but are based on the smallest repeating unit known as the unit cell.

In the rock salt structure, both anions and cations are arranged in such a way that they fit together into a face-centered cubic lattice, one of the most efficient and symmetrical ways ions can pack together. As shown in the provided exercise, cations fit into the spaces between anions, with the arrangement depending on the size of the ions. By knowing the length of the edge of the unit cell and the radius of one of the ions, we can calculate the radius of the other ion, as demonstrated in the steps of solving the exercise.