Question

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

Short answer

The NaCl-type RbI crystal structure has an edge length of 3.69 Å and a density of approximately 3.67 g/cm³. On the other hand, the high-pressure CsCl-type RbI crystal structure has an edge length of 5.21 Å and a density of approximately 4.57 g/cm³. The density of the CsCl-type RbI is higher than that of the NaCl-type structure.

Step-by-step solution

Calculate the length of the cubic unit cell edge using ionic radii

For the NaCl-type structure, the edge length (a) can be calculated by adding the ionic radii of Rubidium (Rb⁺) and Iodine (I⁻). The ionic radii of Rb⁺ is 1.52 Å and I⁻ is 2.17 Å. Edge length (a) = (Rb^+ radius + I^- radius) Plug in the values: a = 1.52 Å + 2.17 Å a = 3.69 Å

Estimate the density

To estimate the density, we will use the formula: Density = (Mass of one unit cell) / (Volume of one unit cell) Since, there are four RbI molecules in a unit cell(cell formula: Rb4I4) with the NaCl structure, the mass of one unit cell can be found by: Mass of one unit cell = (4 × (mass of one Rb atom+ mass of one I atom)) To get the mass of the atoms, we will use the molar mass of Rb (85.47 g/mol) and I (126.9 g/mol) and divide them by Avogadro's number (6.022 x 10^23 mol^(-1)) : _mass_of_one_Rb = (85.47 g/mol) / (6.022 x 10^23 mol^(-1)) _mass_of_one_I = (126.9 g/mol) / (6.022 x 10^23 mol^(-1)) Mass of one unit cell = 4 × (_mass_of_one_Rb + _mass_of_one_I) Next, the volume (V) of one unit cell can be calculated using (a^3). V = (3.69 Å)^3 Now, we can find the density: Density = Mass of one unit cell / V For CsCl-type structure (c & d):

Calculate the length of the cubic unit cell edge using ionic radii

For the CsCl-type structure, the edge length (a') can be calculated using the following formula, considering the face diagonals: a' = \(\sqrt{2}\)(Rb^+ radius + I^- radius) a' = \(\sqrt{2}\)(1.52 Å + 2.17 Å) a' = 5.21 Å

Estimate the density

To estimate the density, we will use the same formula as before: Density = (Mass of one unit cell) / (Volume of one unit cell) For the CsCl-type structure, there is one RbI molecule in a unit cell (cell formula: RbI): Mass of one unit cell = 1 × (mass of one Rb atom + mass of one I atom) Volume (V') of one unit cell can be calculated using (a'^3). V' = (5.21 Å)^3 Now, we can find the density: Density = Mass of one unit cell / V'

Compare the densities

Calculate the densities of NaCl-type and CsCl-type RbI crystal structures using the equations in Step 2 and Step 4. Then, compare them to see how they are related or if one is higher or lower than the other.

Key concepts

Cubic Unit Cell Edge Length

Understanding the cubic unit cell edge length is crucial when studying crystalline materials. It represents the distance between two parallel faces of a cube-shaped unit cell and is fundamental in determining the structure of a crystal. For example, in an NaCl-type structure, the edge length can be calculated by summing the ionic radii of cations and anions present in the crystal.

For RbI with NaCl-type structure at room temperature and pressure, we sum the radii of Rb⁺ and I⁻, giving us an edge length of 3.69 Å. Accurately calculating this edge length is the first step in estimating other properties of the crystal, such as density.

Crystal Structure Transformation

Crystal structure transformation occurs when a crystalline material changes its structure under varying conditions such as temperature and pressure. In our case, RbI transforms from NaCl-type to CsCl-type structure under high pressure. This transformation affects the arrangement of ions within the crystal lattice and, as a result, alters properties like density.

Understanding this can help predict how materials behave under different conditions, a concept essential in material science and engineering. When studying such transformations, calculations of unit cell dimensions for both structures become necessary to compare properties effectively.

Density Estimation of Crystalline Materials

Density estimation is a fundamental process in material science. For crystalline materials, the density can be calculated using the mass of the unit cell and its volume. The density gives insights into the material's properties, affecting its mechanical strength and stability.

For instance, our RbI crystal in the NaCl-type structure, the density is calculated by first finding the mass of the unit cell, which involves the molar mass of its constituent ions, Rb and I, and Avogadro's number. Subsequently, this mass is divided by the volume of the unit cell calculated as the cube of the edge length. This process is vital for understanding how the structure of a crystal relates to its density and overall characteristics.

NaCl-type Structure

A NaCl-type structure is one of the most common types of ionic crystal structures. It is characterized by each sodium ion being surrounded by six chloride ions and vice versa, in an octahedral coordination. This arrangement results in a face-centered cubic lattice for both sodium and chloride ions.

When analyzing RbI with a similar NaCl-type structure, we assume the same coordination and spacing, allowing us to use ionic radii to predict unit cell dimensions and further properties, like density, which is crucial in various practical applications, including predicting the behavior of the material under specific conditions.

CsCl-type Structure

The CsCl-type structure differs considerably from the NaCl-type. It manifests as a simple cubic structure where the chloride ion sits at the center of the cube, and the cesium ion is at the corner, or vice versa. This means every ion is surrounded by eight ions of the opposite charge, leading to a coordination number of eight.

This type of structure becomes relevant for RbI under high pressure, as the increased pressure causes a shift from the NaCl-type structure to the denser CsCl-type structure. The transition to this structure and the corresponding calculations of edge length and density can provide insights into the compound's behavior in different environments.