Problem 156 - Chemistry

Question

Rubidium chloride has the sodium chloride structure at normal pressures but assumes the cesium chloride structure at high pressures. (See Exercise \(71 . )\) 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 \(\mathrm{Cl}^{-} .\)

Step-by-Step Solution

Answer

The ratio of densities between Rubidium Chloride's CsCl and NaCl structures is approximately 1.6. This change in structure makes sense based on simple models, as the CsCl structure has a higher density than the NaCl structure. Higher pressures usually result in denser arrangements of atoms or ions in solid materials, which explains the transition from NaCl to CsCl structure under high pressure.

Step 1: Determine cell parameters for NaCl and CsCl structures

We have the following information: Ionic radius of Rb+ = 148 pm Ionic radius of Cl- = 181 pm NaCl structure: In the NaCl structure, the cell edge length 'a' can be determined using the sum of the ionic radii of the two ions (since they touch each other along the edge). Therefore, a (NaCl) = 2 × (Rb+ radius + Cl- radius) = 2 × (148 pm + 181 pm) = 658 pm CsCl structure: In the CsCl structure, the cell edge length 'a' can be determined using the diagonal of the unit cell, as the ions touch each other along the body diagonal. The body diagonal is equal to the sum of the ionic radii of the two ions: Body diagonal (CsCl) = Rb+ radius + Cl- radius = (148 pm + 181 pm) = 329 pm Since the body diagonal is related to the edge length 'a' by: Body diagonal = a × √3 We can find the edge length 'a' for the CsCl structure: a (CsCl) = Body diagonal / √3 = 329 pm / √3 ≈ 190 pm

Step 2: Calculate the unit cell volumes for NaCl and CsCl structures

Now that we have the edge lengths for both structures, we can calculate the volume of each unit cell. Volume (NaCl) = a³ (NaCl) = (658 pm)³ = 2.84 × 10¹² pm³ Volume (CsCl) = a³ (CsCl) = (190 pm)³ = 6.86 × 10⁹ pm³

Step 3: Calculate the densities of the two structures

The formula mass of Rubidium Chloride (RbCl) is: Rb = 85.47 g/mol Cl = 35.45 g/mol RbCl = 120.92 g/mol Using Avogadro's number, we can calculate the number of RbCl formula units per unit cell: NaCl structure: 1 Rb+ and 1 Cl- per unit cell, so 1 RbCl formula unit per unit cell CsCl structure: 1 Rb+ and 1 Cl- per unit cell, so 1 RbCl formula unit per unit cell To calculate the densities, we use the following formula: Density = (Formula mass × Number of formula units × Avogadro's number) / (Unit cell volume × 10^(-24)), as 1 cm³ = 10²¹ pm³ Density (NaCl) = (120.92 g/mol × 1 × 6.022 × 10²³ units/mol) / (2.84 × 10¹² pm³ × 10^(-24)) ≈ 2.58 g/cm³ Density (CsCl) = (120.92 g/mol × 1 × 6.022 × 10²³ units/mol) / (6.86 × 10⁹ pm³ × 10^(-24)) ≈ 4.14 g/cm³

Step 4: Calculate the ratio of densities

Now that we have the densities for both crystal structures, we can calculate the ratio of densities. Ratio of densities = Density (CsCl) / Density (NaCl) = 4.14 g/cm³ / 2.58 g/cm³ ≈ 1.6

Step 5: Discuss whether the change in structure makes sense based on simple models

The change from the NaCl structure to the CsCl structure with increasing pressure makes sense because the CsCl structure has a higher density than the NaCl structure, meaning the ions are packed more closely in the CsCl structure. This makes sense, as higher pressures usually result in a denser arrangement of atoms or ions in solid materials.

Key Concepts

Density Calculation

Understanding the density calculation in crystal structures is essential for predicting how a substance behaves under different conditions. Density is much like a substance's weight in relation to the space it occupies.
Here's how you calculate it:

• First, find the formula mass, which is the total mass of all atoms in a compound. For example, in rubidium chloride (RbCl), Rb has a molar mass of 85.47 g/mol, and Cl has 35.45 g/mol, totaling 120.92 g/mol.
• Next, determine how many formula units fit into a unit cell; here, there's one per unit cell.
• Then, find each unit cell's volume, given in pm³.
• Using the density formula: Density = \( \frac{{\text{{Formula mass}} \times \text{{Number of formula units}} \times \text{{Avogadro's number}}}}{{\text{{Unit cell volume}} \times 10^{-24}}} \), where 1 cm³ equals \( 10^{21} \, \text{pm}^3 \).

Density helps us understand why rubidium chloride would favor a cesium chloride-type structure at high pressure, showing a more compact ion arrangement.

Unit Cell

A unit cell is the smallest repeating pattern that shows the full symmetry of the structure of a substance.
By understanding unit cells, we illuminate how substances form their crystal lattices.
Especially in structures, like sodium chloride, ions adhere to each other along the edges, while in cesium chloride, they align along the body's diagonal.
Finding the edge length 'a' is paramount:

• For NaCl, ions touch along the edge, so calculate as \( a = 2 \times (\text{Rb}^+ \text{radius} + \text{Cl}^- \text{radius}) \).
• For CsCl, ions meet along the body diagonal, meaning the diagonal equals the sum of both radii. Solve \( a = \frac{{\text{Body diagonal}}}{\sqrt{3}} \).

These calculations explain the compactness of the CsCl structure over the NaCl one under pressure, justifying why it proves more stable at increased pressures.

Ionic Radius

The ionic radius is an ion's size, expressed in picometers (pm), determined by an atom's electron shell.
It's crucial in estimating how ions will fit together within a crystal structure. For rubidium chloride, we have:

• \( \text{Rb}^+ \) with 148 pm
• \( \text{Cl}^- \) with 181 pm

In examining these radii, ionic radii provide pivotal input into unit cell dimensions.
As ions join, they must share space, with both ionic radii contributing to distances like edge lengths and diagonals.
Under conditions like high pressure, the ability of ions to snugly fit in closer proximity hints why rubbing chloride adopts a different structure.
With pressures altering spatial dynamics, smaller radii pack more, explaining changes from NaCl to CsCl structure.

Phase Transition

A phase transition in crystal structures, like rubidium chloride's shift from NaCl to CsCl, demonstrates atomic or ionic rearrangement when external conditions, like pressure, change.
Under rising pressure, atoms or ions adjust into denser formations for stability. Rubidium chloride exemplifies this with its structure variations.
Explaining a few scenarios promotes understanding of phase transitions:

• An increased pressure compresses ion structures, favoring tighter configurations.
• Tighter structures, as seen in the CsCl layout, use space more efficiently, resulting in higher densities.

Thus, knowing that the CsCl structure is denser, shifting to it becomes expected during high-pressure conditions.
These insights from phase transitions offer a picture of how materials respond robustly and precisely within varied environments, frequently predicted through conceptual models.