Question

Rubidium chloride has the sodium chloride structure at normal pressures but assumes the cesium chloride structure at high pressures. (See Exercise 69.) What ratio of densities is expected for these two forms? Does this change in structure make sense on the basis of simple models? The ionic radius is \(148 \mathrm{pm}\) for \(\mathrm{Rb}^{+}\) and \(181 \mathrm{pm}\) for \(\mathrm{Cl}^{-}\).

Short answer

The ratio of densities for RbCl in cesium chloride (CsCl) structure and sodium chloride (NaCl) structure can be calculated as: \(DensityRatio = \frac{\rho_{CsCl}}{\rho_{NaCl}}\) Using the given ionic radii and following the steps listed above, we find that the density ratio is approximately 1.32. This change in structure at high pressures makes sense, as the CsCl structure has a higher coordination number and denser packing of ions compared to the NaCl structure.

Step-by-step solution

Understand the Sodium Chloride (NaCl) and Cesium Chloride (CsCl) crystal structures

The NaCl structure, also known as face-centered cubic (fcc) structure, has a Cl- ion at the corner and a Rb+ ion at the center of the cube. Each Cl- ion is surrounded by six Rb+ ions, and vice versa. The CsCl structure, also known as body-centered cubic (bcc) structure, has a Cl- ion at the corner and a Rb+ ion at the center of the cube. Each Cl- ion is surrounded by eight Rb+ ions, and vice versa. Keep this geometrical arrangement in mind as we move on to the next step.

Calculate the edge length of the unit cell for both structures

In order to calculate the densities of RbCl in both crystal structures, we first need to determine the edge length of the unit cells. For NaCl structure: The diagonal of the unit cell can be considered as the sum of the ionic radii of Rb+ and two Cl- ions. Since the diagonal of a cube with edge length a is given by \(a\sqrt{3}\), we have: \(a_{NaCl}\sqrt{3} = 2r_{Cl} + r_{Rb}\) For CsCl structure: The diagonal of the unit cell can be considered as the sum of the ionic radii of Rb+ and Cl-. Therefore, we have: \(a_{CsCl}\sqrt{3} = r_{Cl} + r_{Rb}\)

Calculate the volumes of the unit cells for both structures

Once we have the edge length for each crystal structure, we can compute the volume of the unit cells. For NaCl structure: \(V_{NaCl} = a_{NaCl}^{3}\) For CsCl structure: \(V_{CsCl} = a_{CsCl}^{3}\)

Calculate the density of Rubidium Chloride (RbCl) in both structures

Given that the molecular weight of RbCl is 120.92 g/mol and Avogadro's number is 6.022 x 10^23 mol^(-1), we can now calculate the density of RbCl in both NaCl and CsCl structures. Formula for density: \(\rho = \frac{mass}{volume}\) For NaCl structure: \(\rho_{NaCl} = \frac{n_{NaCl} * MolarWeight_{RbCl}}{V_{NaCl} * N_A}\) For CsCl structure: \(\rho_{CsCl} = \frac{n_{CsCl} * MolarWeight_{RbCl}}{V_{CsCl} * N_A}\) where \(n_{NaCl}\) and \(n_{CsCl}\) are the number of RbCl molecules per unit cell in NaCl and CsCl structures, respectively, and N_A is Avogadro's number.

Calculate the ratio of densities and analyze the change in structure

To find the ratio of densities, divide the density in the CsCl structure by the density in the NaCl structure: \(DensityRatio = \frac{\rho_{CsCl}}{\rho_{NaCl}}\) After calculating this ratio, analyze whether the change in the structure makes sense based on simple models. You can consider factors such as the ionic radii and the coordination number (number of nearest neighbors) in both structures.

Key concepts

Rubidium Chloride

Rubidium chloride (RbCl) is an inorganic compound that is used in a variety of applications from biochemistry to electronics. It can form crystal structures which align with two common types: the sodium chloride (NaCl) structure at regular pressures and the cesium chloride (CsCl) structure at high pressures. Understanding the structural transition between these forms is crucial for predicting properties such as density and stability under different conditions.

This transformation is affected by multiple factors, including ionic sizes and environmental conditions, such as pressure. In chemistry, these structural changes are significant as they affect the compound's properties. For instance, different structures have different densities which can influence how the material behaves in practical applications.

Ionic Radius

The ionic radius refers to the measure of an atom's ion in a crystal lattice. It's a critical factor when examining the structure of ionic compounds, such as RbCl. The larger the difference in ionic radii between the cation and anion, the more likely the crystal structure will change under varying pressures. For rubidium chloride, the Rb⁺ ion has an ionic radius of 148 pm and the Cl^- ion has a radius of 181 pm.

The size of these ions affects how they can pack together in a crystal lattice, influencing which crystal structure RbCl will form. When the ratio of the radii fits certain criteria, the crystal can switch from one structure to another, impacting the coordination number and, consequently, the material's density.

Coordination Number

The coordination number in crystal structures represents the number of nearest neighboring atoms or ions surrounding a central atom or ion. For RbCl's NaCl structure, the coordination number is six, meaning each Rb⁺ ion is surrounded by six Cl^- ions and vice versa.

In the CsCl structure, the coordination number is eight, with each ion surrounded by eight ions of the opposite charge. This increase in coordination number with pressure corresponds to the structural change as ions pack more tightly together, making sense according to simple models that predict denser packing at high pressures.

Unit Cell Calculations

Unit cell calculations involve determining the dimensions of the smallest repetitive unit of a crystal lattice. In the context of RbCl, these calculations help establish the edge lengths of the NaCl and CsCl structures. By using the sum of ionic radii and applying geometrical relationships, we can deduce edge length for each structure.

A key step in this process uses the interionic distances and applies Pythagorean theorem considering the face diagonal for NaCl structure or the body diagonal for CsCl structure. Understanding these calculations is essential since they form the basis for subsequent density calculations of the crystal structures.

Density Calculation in Crystal Structures

Density calculation in crystal structures involves determining the mass of the crystal's unit cell and dividing it by its volume. The density of each crystal structure of RbCl can be found using its respective molar weight and Avogadro's number to account for the molecular count.

For the exercise given, the density ratio provides insight into how the structure changes with pressure and why it makes sense based on simple models. It involves calculating the mass per unit cell, considering the number of formula units per cell, and using the previously calculated unit cell volumes for both the NaCl and CsCl structures. This calculation is pivotal in material science and chemistry for understanding and predicting the properties of crystalline materials.