Question

At room temperature and pressure RbCl crystallizes with the NaCl-type structure. (a) Use ionic radii to predict the length of the cubic unit cell edge. (b) Use this value to estimate the density. (c) At high temperature and pressure, the structure transforms to one with a CsCl-type structure. Use ionic radii to predict the length of the cubic unit cell edge for the high- pressure form of RbCl. (d) Use this value to estimate the density. How does this density compare with the density you calculated in part (b)?

Short answer

The edge length of the NaCl-type RbCl unit cell is 666 pm, and the density is 1.63 g/cm^3. For the CsCl-type RbCl, the edge length is 576.78 pm, and the density is 2.50 g/cm^3. The CsCl-type RbCl structure has a higher density than the NaCl-type RbCl.

Step-by-step solution

(a) Determine the length of the cubic unit cell edge for NaCl-type RbCl

First, we need to find the sum of the ionic radii of the Rb+ ion and the Cl- ion. The ionic radii can be found in a chemistry textbook or online. For Rb+ ion, the ionic radius is approximately 152 pm, and for the Cl- ion, the ionic radius is approximately 181 pm. The sum of these two radii would be the distance between the Rb+ ion and the Cl- ion in the crystal lattice. Sum of ionic radii = Rb+ radius + Cl- radius = 152 pm + 181 pm = 333 pm In NaCl-type structure, the edge length of the unit cell is twice the distance between the two ions in the crystal lattice since there are ions on each corner and there will be ions in each face of the cube. So, Edge length (a) = 2 × sum of ionic radii = 2 × 333 pm = 666 pm

(b) Estimate the density of NaCl-type RbCl

To calculate the density, first find the volume of the unit cell: Volume = a^3 = (666 pm)^3 = (6.66 × 10^{-8} cm)^3 = 2.96 × 10^{-22} cm^3 Since there are 4 RbCl formula units in the NaCl-type unit cell, find the mass of one RbCl unit: Mass of RbCl = mass of Rb + mass of Cl = (85.47 g/mol) + (35.45 g/mol) = 120.92 g/mol Now, calculate the density: Density = (mass × number of formula units) / volume = (120.92 g/mol × 4) / (2.96 × 10^{-22} cm^3) = 1.63 g/cm^3

(c) Determine the length of the cubic unit cell edge for CsCl-type RbCl

In the CsCl-type structure, the unit cell edge length is equal to the diagonal of a cube with side length equal to the sum of the ionic radii: a = √(3 × (sum of ionic radii)^2) = √(3 × (333 pm)^2) = 576.78 pm

(d) Estimate the density of CsCl-type RbCl and compare

Calculate the volume of the CsCl-type unit cell: Volume = a^3 = (576.78 pm)^3 = (5.7678 × 10^{-8} cm)^3 = 1.92 × 10^{-22} cm^3 Since there is only 1 RbCl formula unit in the CsCl-type unit cell, calculate the density: Density = (mass × number of formula units) / volume = (120.92 g/mol × 1) / (1.92 × 10^{-22} cm^3) = 2.50 g/cm^3 Now compare the density of the CsCl-type structure with the density of the NaCl-type structure: The density of the CsCl-type RbCl (2.50 g/cm^3) is higher than the density of the NaCl-type RbCl (1.63 g/cm^3). This means that RbCl in the CsCl-type structure is more densely packed than in the NaCl-type structure.

Key concepts

Ionic Radii

Ionic radii are essential in determining the structure and properties of ionic compounds. The ionic radius is the measure from the center of an ion to the outermost electrons. Different ions have different radii based on their charge and electron configuration.
The size of an ion affects how closely ions will pack together in a crystal lattice. This is why knowing the ionic radii is crucial for predicting crystal structures.
For instance, in RbCl, we look at the radii of the Rb\(^+\) ion, which is approximately 152 pm, and the Cl\(^-\) ion, around 181 pm. Adding them gives us a total distance of 333 pm that separates these ions in the crystal. This helps us understand how the ions will align in the unit cell to ensure the overall stability of the structure.

Unit Cell Edge Length

The unit cell edge length is a fundamental feature of crystalline materials. It refers to the length of the edges of the smallest repeating unit of the structure, known as the unit cell. This dimension is influenced directly by the ionic radii of the ions that compose the crystal.
In NaCl-type structures like RbCl, the edge length is often twice the sum of the ionic radii because the ionic arrangement repeats over the edges. Therefore, for RbCl with a unit cell edge length calculated at 666 pm, it shows how each Rb\(^+\) and Cl\(^-\) are positioned in relation to each other within the unit cell.

• The manipulation of these values can affect various properties, including the compound's density and stability.
• A thorough understanding of unit cell dimensions is vital for various applications in material science and engineering.

Density Calculation

Density is a critical property measured in g/cm\(^3\) that describes how much mass is contained in a unit volume of a substance. In crystalline structures, density can differ significantly based on how atoms or ions are packed in the unit cell.
To calculate density, you need to divide the mass of the compound by the volume of the unit cell. For RbCl in the NaCl-type structure, we have calculated its density to be approximately 1.63 g/cm\(^3\).
However, when RbCl adopts the CsCl-type structure under high pressure and temperature, its density rises to 2.50 g/cm\(^3\) due to a more compact arrangement of the Rb\(^+\) and Cl\(^-\) ions within the unit cell.
This comparison highlights how crystalline arrangements impact physical characteristics like density.

NaCl and CsCl Structures

Crystalline structures like NaCl and CsCl represent different ways ions can be packed in a solid. The NaCl structure is known for its face-centered cubic form where each ion is surrounded by six oppositely charged ions. This gives it a coordination number of 6.
On the other hand, CsCl exhibits a body-centered cubic form with each ion at the center of a cube of eight oppositely charged ions, which corresponds to a coordination number of 8.
These differences in structure greatly affect physical properties, including density.

• NaCl-type structures usually have lower densities due to less tightly packed ions.
• In contrast, CsCl structures are denser since the ions occupy more of the available space.

Understanding these structural variations not only helps in predicting density but also in designing materials for specific applications.