Problem 60 - Chemistry The Central Science In Si Units

Question

Silver chloride (AgCl) adopts the rock salt structure. The density of \(\mathrm{AgCl}\) at \(25^{\circ} \mathrm{C}\) is \(5.56 \mathrm{~g} / \mathrm{cm}^{3} .\) Calculate the length of an edge of the AgCl unit cell.

Step-by-Step Solution

Answer

The edge length of the AgCl unit cell is approximately \( 4.07 \text{ Å} \).

Step 1: Understand the Structure

Silver chloride (AgCl) adopts the rock salt structure, which is a face-centered cubic (FCC) structure. This means that each unit cell contains four AgCl formula units.

Step 2: Use the Density Formula

The density formula is given by \( \rho = \frac{m}{V} \), where \( \rho \) is the density, \( m \) is the mass, and \( V \) is the volume of the unit cell. We need to rearrange this formula to solve for \( V \), the volume of the unit cell: \( V = \frac{m}{\rho} \).

Step 3: Calculate the Mass of the Unit Cell

The mass of one formula unit of AgCl is well known: silver (Ag) has a molar mass of approximately \(107.87 \text{ g/mol}\) and chlorine (Cl) has a molar mass of approximately \(35.45 \text{ g/mol}\). Therefore, the molar mass of AgCl is approximately \(107.87 + 35.45 = 143.32 \text{ g/mol}\).In the FCC structure, there are four formula units of AgCl per unit cell, so the total mass of the AgCl in one unit cell is \[ m = 4 \times \frac{143.32 \text{ g/mol}}{6.022 \times 10^{23} \text{ mol}^{-1}} \approx 9.51 \times 10^{-22} \text{ g}. \]

Step 4: Calculate the Volume of the Unit Cell

Substitute the mass of the unit cell and the density into the rearranged density formula: \[ V = \frac{9.51 \times 10^{-22} \text{ g}}{5.56 \text{ g/cm}^{3}} \approx 1.71 \times 10^{-22} \text{ cm}^3. \]

Step 5: Calculate the Edge Length of the Unit Cell

For a cubic unit cell, the volume \( V \) is given by \( a^3 \), where \( a \) is the edge length. Solve for \( a \):\[ a = \sqrt[3]{1.71 \times 10^{-22} \text{ cm}^3} \approx 4.07 \times 10^{-8} \text{ cm}. \]Thus, the edge length of the AgCl unit cell is approximately \( 4.07 \text{ Å} \) (since 1 Å = \( 10^{-8} \) cm).

Key Concepts

Face-Centered Cubic Structure

The face-centered cubic (FCC) structure is a type of crystal lattice. In this arrangement, atoms are located at each of the eight corners of a cube and in the center of each face of the cube.
This packing is efficient, meaning it tightly packs the atoms or ions with minimum empty space.
Face-centered cubic structures, like the one in silver chloride (AgCl), are particularly common because of their high density due to this efficient packing.

• Each corner atom is shared among eight different unit cells.
• Each face-centered atom is shared between two unit cells.

This results in a total of four complete atoms or ions per unit cell in an FCC structure. Understanding this layout helps determine other properties, such as mass and edge length, because we know the number of formula units in each cell.

Density Calculation

Density is a foundational concept in calculating the properties of materials like silver chloride.
It links an object's mass to its volume, which is given by the formula: \[ \rho = \frac{m}{V} \] Where \( \rho \) is the density, \( m \) is mass, and \( V \) is volume. In the case of a unit cell, this formula helps us understand how much mass is packed into a given volume of crystal.
For AgCl, a known density allows the calculation of the lattice parameters:

• Rearranging the formula to find the volume, \( V = \frac{m}{\rho} \), involves knowing the mass of multiple formula units in the unit cell.
• The calculated mass is adjusted using Avogadro's number to find the actual mass of the unit cell in grams.

This relationship is crucial because by knowing both mass and density, one can determine the precise volume of a unit cell, leading us to find the edge length.

Molar Mass of AgCl

The molar mass is a key factor in determining the mass of the unit cell. It represents the mass of one mole of a substance, helping convert between moles and grams.
For silver chloride (AgCl), the molar mass is calculated by adding the molar masses of its constituent elements:

• Silver (Ag) has a molar mass of approximately 107.87 g/mol.
• Chlorine (Cl) has a molar mass of approximately 35.45 g/mol.

The combined molar mass of AgCl is therefore 107.87 + 35.45 = 143.32 g/mol.
In a face-centered cubic structure like AgCl, the total mass of the formula unit in the unit cell is crucial for computations and is determined by multiplying this molar mass by the number of formula units and then adjusting using Avogadro's number.

Unit Cell Edge Length

Calculating the unit cell edge length provides insight into the spatial dimension of the crystal lattice.
Once we know the volume of the unit cell from the density relationship, we can solve for the edge length \( a \) of the cubic cell using: \[ a = \sqrt[3]{V} \] This calculation is important because it provides direct information about the size of the repeating sequence in the crystal lattice.
In silver chloride’s case, this edge length is measured in nanometers or Angstroms as it is a small scale length:

• Knowing the edge length allows one to visualize and compare the unit cell’s dimension across different materials and structures.

In practice, for AgCl, this length is approximately 4.07 Å, translating the atomic-scale dimension into a usable metric for scientists and students alike.