Problem 62 - Chemistry The Central Science

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 high-pressure form of RbI. (d) Use this value to estimate the density. How does this density compare with the density you calculated in part (b)?

Step-by-Step Solution

Answer

The edge lengths and densities of RbI in NaCl-type and CsCl-type structures were calculated using ionic radii. For the NaCl-type structure, the edge length was found to be 7.44 Å, and the density was 3.48 g/cm³. For the high-pressure CsCl-type structure, the edge length was 5.24 Å, and the density was 4.24 g/cm³. Comparing the densities, we see that the density of RbI increases when it transforms from NaCl-type to CsCl-type structure under high pressure.

Step 1: Ionic radii of Rb+ and I-

We will first find the ionic radii for the Rb+ and I- ions. From a table of ionic radii, we find the ionic radius of Rb+ is 1.52 Å and the ionic radius of I- is 2.20 Å.

Step 2: Calculate the edge length for NaCl-type structure

In a NaCl-type structure, the edge length (a) is equal to twice the sum of the radii of the cation and anion, i.e., a = 2(Rb+ + I-). Thus, we can calculate the edge length as: a = 2(1.52 Å + 2.20 Å) = 7.44 Å (b) Density of NaCl-type structure

Step 3: Calculate molar mass and number of formula units per unit cell

First, we need to determine the molar mass of RbI. From the periodic table, we find that the molar mass of Rb is 85.47 g/mol, and the molar mass of I is 126.9 g/mol. So the molar mass of RbI is 85.47 + 126.9 = 212.37 g/mol. In a NaCl-type structure, there are 4 formula units of RbI per unit cell.

Step 4: Calculate density

Now, we can calculate the density of RbI in the NaCl-type structure using the formula: Density = (mass of RbI in unit cell) / (volume of unit cell) Density = (4 formula units/cell * 212.37 g/mol) / (7.44 Å)^3 Convert Å to cm: 1 Å = 10^-8 cm, so (7.44 Å)^3 = (7.44 × 10^-8 cm)^3 Density = (4 × 212.37 g/mol) / (7.44 × 10^-8 cm)^3 ≈ 3.48 g/cm³ (c) High-pressure CsCl-type structure edge length

Step 5: Calculate the edge length for CsCl-type structure

In a CsCl-type structure, the edge length (a) is equal to √2 times the sum of the radii of the cation and anion, i.e., a = √2(Rb+ + I-). Thus, we can calculate the edge length as: a = √2(1.52 Å + 2.20 Å) ≈ 5.24 Å (d) Density of CsCl-type structure

Step 6: Calculate density

Now, we can calculate the density of RbI in the CsCl-type structure using the same formula as before: Density = (mass of RbI in unit cell) / (volume of unit cell) Density = (1 formula unit/cell * 212.37 g/mol) / (5.24 Å)^3 Convert Å to cm: 1 Å = 10^-8 cm, so (5.24 Å)^3 = (5.24 × 10^-8 cm)^3 Density = (1 × 212.37 g/mol) / (5.24 × 10^-8 cm)^3 ≈ 4.24 g/cm³ Lastly, we are asked to compare the densities calculated in parts (b) and (d).

Step 7: Comparison of densities

The density of RbI in the NaCl-type structure is 3.48 g/cm³, while the density of RbI in the CsCl-type structure is 4.24 g/cm³. This shows that the density of RbI increases when it changes from the NaCl-type structure to the CsCl-type structure under high pressure.

Key Concepts

NaCl-type structure

The NaCl-type structure is a common crystal structure formed by ionic compounds, named after the mineral halite or rock salt (NaCl). This structure is also known as the rock salt structure. It's characterized by each ion in the crystal being surrounded by six ions of opposite charge, forming an octahedral coordination.

In the case of RbI crystallizing in the NaCl-type structure, both the Rb\(^+\) cations and the I\(^-\) anions are positioned at the vertices of a cubic lattice. The geometry is such that the edge length of the cubic unit cell (\(a\)) is determined by twice the sum of the ionic radii of the cation and anion. For RbI:

• Ionic radius of Rb\(^+\) = 1.52 Å
• Ionic radius of I\(^-\) = 2.20 Å

Therefore, the edge length \(a\) is calculated as \(a = 2\times (1.52 + 2.20) = 7.44\, \text{Å}\). The NaCl-type structure provides an efficient packing of the ions that is typical for many alkali halides.

CsCl-type structure

The CsCl-type structure is another classic ionic crystal structure, notably different from the NaCl-type, and it is typically favored under higher pressure conditions. In this type of structure, each cation is surrounded by eight anions at the corners of a cube, forming a cubic coordination. Similarly, each anion is encased by eight cations.

For RbI in a CsCl-type structure, the configuration shifts to having just one formula unit (Rb\(^+\) and I\(^-\)) per unit cell, unlike the NaCl variant. The edge length of the cubic unit cell \(a\) is given by \(a = \sqrt{2}(\text{Rb}^+ + \text{I}^-)\). Using the ionic radii:

• Ionic radius of Rb\(^+\) = 1.52 Å
• Ionic radius of I\(^-\) = 2.20 Å

The computation results in \(a = \sqrt{2} \times (1.52 + 2.20) \approx 5.24\, \text{Å}\). This structural shift from NaCl to CsCl under high pressure leads to a more compact arrangement, impacting the density of the crystal.

Ionic radii

Ionic radii are critical measurements that describe the effective size of an ion in a crystal lattice. The radius affects how ions pack together and form stable structures. Accurate knowledge of ionic sizes is essential in predicting crystallographic arrangements and calculating unit cell parameters for various compounds.

For example, in the structures of RbI:

• The radius of Rb\(^+\) is 1.52 Å
• The radius of I\(^-\) is 2.20 Å

These values allow for precise calculation of the unit cell edge lengths for both NaCl and CsCl-type structures. When combining these radii, different packing efficiencies arise based on the structure type, influencing material properties such as density and stability.

Density calculation

Density is an important physical property of materials, defined as mass per unit volume. When calculating the density of ionic compounds within a crystal lattice, we are typically concerned with the arrangement of ions within the unit cell and the mass of the formula units present.

For the NaCl-type structure of RbI:

• There are 4 formula units per unit cell.
• The unit cell edge is 7.44 Å.
• Density is calculated by dividing the total mass of the formula units by the volume of the cell: \((4 \times 212.37\, \text{g/mol})/(7.44 \times 10^{-8} \text{cm})^3\) yielding \(3.48\, \text{g/cm}^3\).

For the CsCl-type structure, the unit cell contains just 1 formula unit with an edge length of 5.24 Å. The density calculation follows a similar formula, resulting in \(4.24\, \text{g/cm}^3\).

This increase in density in the CsCl-type structure reflects its more compact ionic arrangement, a common occurrence at higher pressures as atoms are pushed closer together within the lattice.