IN CUBIC CRYSTAL OF cScL(d=3.97gm/cc) the eight corners are occupied by Cl- with a Cs+ at centre and vice versa.calcolate distance between neighbouring Cs+ and Cl-. what is radius ratio of two ions?

at wt. Cs+132.92 and Cl=35.45gm

To find the distance between neighboring Cs⁺ and Cl^- ions in a cubic crystal structure of CsCl, we first need to understand the arrangement of the ions in the crystal lattice. In a cubic structure, the Cs⁺ ion is located at the center of the cube, while the Cl^- ions occupy the corners. Let's break this down step by step.

Step 1: Understanding the Crystal Structure

In a cubic unit cell of CsCl, the Cl^- ions are positioned at each of the eight corners of the cube, and the Cs⁺ ion is at the center. The arrangement can be visualized as follows:

• 8 Cl^- ions at the corners of the cube
• 1 Cs⁺ ion at the center of the cube

Step 2: Calculating the Distance Between Ions

The distance between the Cs⁺ ion at the center and a Cl^- ion at a corner can be calculated using the geometry of the cube. The length of the edge of the cube (a) can be derived from the density of the crystal.

Finding the Edge Length (a)

The density (d) of the crystal is given as 3.97 g/cm³. The formula for density is:

d = mass/volume

For the CsCl unit cell, the mass can be calculated as follows:

• Mass of Cs⁺ = 132.92 g/mol
• Mass of Cl^- = 35.45 g/mol
• Total mass of the unit cell = 132.92 + 35.45 = 168.37 g/mol

Since there are 6.022 x 10²³ formula units in a mole, the mass of one unit cell in grams is:

Mass of unit cell = 168.37 g/mol / (6.022 x 10²³ units/mol) = 2.80 x 10^-22 g

The volume of the unit cell can be expressed as a³, where a is the edge length. Therefore, we can rearrange the density formula:

3.97 g/cm³ = (2.80 x 10^-22 g) / (a³)

Solving for a³ gives:

a³ = (2.80 x 10^-22 g) / (3.97 g/cm³) = 7.05 x 10^-23 cm³

Taking the cube root:

a ≈ 4.22 x 10^-8 cm = 4.22 Å

Calculating the Distance

The distance from the center of the cube (Cs⁺) to a corner (Cl^-) can be calculated using the formula for the diagonal of a cube:

Distance = √(a² + a² + a²) = √(3a²) = a√3

Substituting the value of a:

Distance = 4.22 Å × √3 ≈ 7.31 Å

Step 3: Radius Ratio of Ions

To find the radius ratio of the ions, we need to know the ionic radii of Cs⁺ and Cl^-. The typical ionic radii are approximately:

• Radius of Cs⁺ ≈ 1.67 Å
• Radius of Cl^- ≈ 1.81 Å

The radius ratio (r_Cs / r_Cl) can be calculated as follows:

Radius ratio = r_Cs / r_Cl = 1.67 Å / 1.81 Å ≈ 0.92

Summary of Findings

In summary, the distance between neighboring Cs⁺ and Cl^- ions in the cubic crystal of CsCl is approximately 7.31 Å, and the radius ratio of Cs⁺ to Cl^- is about 0.92. This information is crucial for understanding the stability and properties of the ionic compound CsCl.