Question

The edge length of a CsCl unit cell (Problem 10.93) is \(412.3 \mathrm{pm}\). What is the length in picometers of the \(\mathrm{Cs}-\mathrm{Cl}\) bond? If the ionic radius of a Cl ion is \(181 \mathrm{pm}\), what is the ionic radius in picometers of a \(\mathrm{Cs}^{+}\) ion?

Short answer

Cs-Cl bond length is 206.15 pm; Cs+ radius is 25.15 pm.

Step-by-step solution

Identify the Structure

Recognize that CsCl crystallizes in a simple cubic lattice where a Cs+ ion is at the center and a Cl- ion is at the corner of the unit cell. Every unit cell has full coordination between the center ion and the corner ions.

Calculate Cs-Cl Distance

The Cs-Cl bond is measured from the center to the nearest corner of the cube. Hence, the Cs-Cl bond length is equal to half the edge length of the cube: \[ \text{Bond length} = \frac{a}{2} = \frac{412.3 \text{ pm}}{2} = 206.15 \text{ pm} \]

Calculate Cs+ Radius

Given that the ionic radius of Cl is 181 pm, the equation for the sum of ionic radii for complete touching in a unit cell is: \[ 181 \text{ pm} + r_{\text{Cs}^+} = 206.15 \text{ pm} \]. From this, the radius of Cs+ is: \[ r_{\text{Cs}^+} = 206.15 \text{ pm} - 181 \text{ pm} = 25.15 \text{ pm} \]

Key concepts

Simple Cubic Lattice

The simple cubic lattice is one of the basic types of crystal structures in solid materials. It's known for its straightforward and repeating pattern. In simpler terms, imagine a structure that looks like a box, and in the case of the CsCl crystal, it is filled with two alternating ions – one type at the corners and another type at the center of the box. This lattice is characterized by having each corner ion shared among eight unit cells, resulting in a relatively low packing efficiency compared to other crystal structures.
In the context of CsCl, the lattice is composed of cesium ions and chloride ions. Here, each cesium ion (Cs+) occupies the center of the lattice, while the chloride ions (Cl-) occupy the corners. This arrangement in a simple cubic lattice creates an equal allocation of ionic positions, leading to maximal coordination within the cell.

Ionic Radius

The ionic radius is a measure of an ion's size within a crystal lattice. It is indicative of how ions pack together while maintaining their structural integrity. In the CsCl crystal studied here, we are interested in the radii of the individual ions – Cs+ and Cl-.
Cesium ions and chloride ions have different sizes. The radius of the Cl- ion is given as 181 pm. This value, along with the bond length measured within the CsCl unit cell, enables us to derive the radius of the Cs+ ion. Knowing both ionic radii is crucial for understanding the space each ion occupies and how they influence the overall crystal structure.
When ions are packed together, their radii help determine parameters such as the cell's volume and shape. In this way, the finite size of each ion dictates how closely they can come to each other – a factor that significantly affects the compound's physical properties.

Cs-Cl Bond Length

In a CsCl lattice, the bond length between cesium and chlorine ions is an important parameter. The bond length is basically the distance between the centers of the cesium ion at the cube's center and the chlorine ion at the cube's corners. Because the cesium ion is situated in the middle of the unit cell and the chloride ions are positioned at the corners, the distance from the center to one of the corners provides the Cs-Cl bond length.
This distance is not just arbitrary; it's calculated to ensure that the forces between ions are balanced, maintaining the crystal's stability. For the CsCl lattice, the length of the Cs-Cl bond is half of the cube's edge length. Given the edge length of 412.3 pm, the bond length is therefore 206.15 pm. This measurement is critical for comprehending how the ions interact and stabilize the entire crystalline structure.

Coordination in Crystal Structures

Coordination in crystal structures refers to the number of direct neighbors surrounding an ion in a lattice. It's similar to counting how many friends someone has in a social circle, but in three-dimensional space with atoms. In the CsCl structure, each Cs+ ion is surrounded by eight Cl- ions, and conversely, each Cl- ion is surrounded by eight Cs+ ions. This results in a coordination number of eight, reflecting a highly interconnected and stable lattice.
This high coordination ensures that the electrostatic forces between the oppositely charged ions are maximized, leading to a solid structure. It's an essential feature that defines how tightly and strongly bonded the ions are within the lattice. Understanding coordination helps in predicting the properties of the crystal, such as its strength, hardness, and how it might interact with other materials.