Problem 64 - Chemistry The Molecular Science

Question

You know that thallium chloride, TlCl, crystallizes in either a primitive cubic or a face-centered cubic lattice of \(\mathrm{Cl}^{-}\) ions with \(\mathrm{Tl}^{+}\) ions in the holes. If the density of the solid is \(7.00 \mathrm{~g} / \mathrm{cm}^{3}\) and the edge of the unit cell is \(3.85 \times 10^{-8} \mathrm{~cm},\) determine the unit cell geometry.

Step-by-Step Solution

Answer

TlCl crystallizes in a primitive cubic lattice.

Step 1: Determine the Volume of the Unit Cell

First, calculate the volume of the unit cell using the edge length provided. The formula for the volume of a cube is \( \text{Volume} = a^3 \), where \( a \) is the edge length of the cube. Given \( a = 3.85 \times 10^{-8} \text{ cm} \), we find:\[ \text{Volume} = (3.85 \times 10^{-8} \text{ cm})^3 \approx 5.70 \times 10^{-23} \text{ cm}^3 \]

Step 2: Calculate Mass of the Unit Cell

The mass of the unit cell can be found using the density formula, \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). Rearrange to find the mass as:\[ \text{Mass} = \text{Density} \times \text{Volume} \]Given that the density is \( 7.00 \text{ g/cm}^3 \), calculate:\[ \text{Mass} = 7.00 \text{ g/cm}^3 \times 5.70 \times 10^{-23} \text{ cm}^3 \approx 3.99 \times 10^{-22} \text{ g} \]

Step 3: Calculate the Number of Formula Units in the Unit Cell

First, determine the molar mass of TlCl using the atomic masses: Thallium \( \approx 204.38 \text{ g/mol} \), and Chlorine \( \approx 35.45 \text{ g/mol} \). Thus, the molar mass of \( \text{TlCl} \approx 239.83 \text{ g/mol} \).Using Avogadro's number \( 6.022 \times 10^{23} \text{ molecules/mol} \), we find the mass of one formula unit of TlCl:\[ \text{Mass per formula unit} = \frac{239.83 \text{ g/mol}}{6.022 \times 10^{23} \text{ units/mol}} \approx 3.98 \times 10^{-22} \text{ g} \]To find the number of formula units in the unit cell, divide the total mass of the unit cell by the mass per formula unit:\[ \text{Number of formula units} = \frac{3.99 \times 10^{-22} \text{ g}}{3.98 \times 10^{-22} \text{ g/unit}} \approx 1 \]

Step 4: Determine Unit Cell Geometry

Based on the possible crystal structures: - A primitive cubic (simple cubic) cell contains 1 atom per unit cell. - A face-centered cubic (FCC) cell contains 4 atoms per unit cell. Since the calculated number of formula units is 1, the TlCl crystal structure is consistent with a primitive cubic unit cell.

Key Concepts

Thallium Chloride

Thallium chloride, with the chemical formula TlCl, is a compound made from thallium ions (Tl⁺) and chloride ions (Cl⁻). These ions arrange themselves in a solid lattice structure when the compound forms a crystal. Understanding this structure is crucial because it affects many of the compound's physical properties, such as its density and how it interacts with light and electricity.
Thallium, being a heavy element, significantly contributes to the compound's overall mass. This can affect the density calculation of the crystal structure. In many applications, particularly in research and industrial use, knowing the exact crystal structure parameters can be beneficial for predicting how the material will behave in different conditions.

Density Calculation

The density of a material is a measure of how much mass it has per unit volume, typically expressed in grams per cubic centimeter (g/cm³). For thallium chloride, with a known density of 7.00 g/cm³, this property is crucial for determining the compound's unit cell geometry.
To calculate the mass of a unit cell, one can use the formula for density, which is mass divided by volume. Rearranging this formula allows us to find mass if we know the density and volume block:

• Formula: \( \text{Mass} = \text{Density} \times \text{Volume} \)

This value is pivotal for additional calculations, such as determining the number of formula units per unit cell and confirms the primitive cubic structure of TlCl.

Unit Cell Geometry

A unit cell is the smallest repeating structure or theme in a crystal lattice. It defines the crystal structure's symmetry and shape, which are fundamental for understanding how atoms are arranged.
In thallium chloride, identifying the geometry helps determine whether it crystallizes into a more basic cubic pattern like the primitive cubic or a more complex face-centered cubic (FCC). Geometry affects properties like mechanical strength and conductivity, making it a key feature for material scientists and engineers.

• Volume of the cube: \( \text{Volume} = a^3 \) where \( a \) is the edge length

Knowing the unit cell geometry provides insight into the material's fundamental properties and dictates how it behaves under various conditions.

Molar Mass

Molar mass, often presented in grams per mole (g/mol), is the mass of one mole of a substance. For TlCl, its molar mass is derived from adding the atomic masses of the elements thallium and chlorine.
Knowing this molar mass is important for several reasons:

• Essential for converting between the number of moles and the number of formula units or atoms.
• Helps in determining the mass of a single formula unit when combined with Avogadro's number, which is necessary for our calculation here.

In this context, understanding the molar mass aids in discerning how many formula units fit into a unit cell of TlCl, which impacts determining the crystal structure of the compound.

Primitive Cubic Lattice

The primitive cubic lattice, also known as a simple cubic lattice, is one of the basic types of crystal structures. It comprises a cube with atoms positioned at each of its eight corners, accounting for only one atom per unit cell in total.
In the case of thallium chloride, the discovery that the number of formula units in its unit cell equals one indicates a primitive cubic lattice. This specific arrangement reflects that each formula unit maps one-to-one with each theoretical atom in the simple cubic layout.

• Represents one of the simplest and least densely packed types of lattices.
• Often leads to different material properties compared to more complex lattice structures such as face-centered cubic or body-centered cubic.

Knowing that TlCl adopts this primitive structure informs us about its density and other physical attributes, distinguishing it from compounds with denser lattice types.