Question

Silver metal crystallizes in a face-centered cubic unit cell with an edge length of \(408 \mathrm{pm}\). The molar mass of silver is \(107.9 \mathrm{~g} / \mathrm{mol}\), and its density is \(11.50 \mathrm{~g} / \mathrm{cm}^{3}\). Use these data to calculate a value for Avogadro's number.

Short answer

The calculated Avogadro's number is approximately \(5.532 \times 10^{23}\), which is slightly off from the true value \(6.022 \times 10^{23}\) due to likely rounding or approximation errors.

Step-by-step solution

Calculate the Volume of the Unit Cell

Given that the edge length of the face-centered cubic (FCC) unit cell is 408 pm, first, convert it into centimeters: \(408 \text{ pm} = 408 \times 10^{-12} \text{ cm}\). The volume \(V\) of the cubic unit cell is calculated by cubing the edge length: \[ V = (408 \times 10^{-12} \text{ cm})^3 = 6.784 \times 10^{-23} \text{ cm}^3 \]

Determine the Mass of the Unit Cell

In an FCC unit cell, there are 4 atoms of silver. Using the density formula, \( \text{density} = \frac{\text{mass}}{\text{volume}} \), rearrange it to find mass: \[ \text{mass} = \text{density} \times \text{volume} = 11.50 \text{ g/cm}^3 \times 6.784 \times 10^{-23} \text{ cm}^3 = 7.801 \times 10^{-22} \text{ g} \]

Find the Mass of One Silver Atom

Since there are 4 atoms in each unit cell, the mass of a single silver atom is:\[ \text{mass of one silver atom} = \frac{7.801 \times 10^{-22} \text{ g}}{4} = 1.950 \times 10^{-22} \text{ g} \]

Calculate Avogadro's Number

Use the definition of molar mass: the mass of one mole of silver (107.9 g/mol) contains Avogadro's number of atoms. Therefore, \[ \text{Avogadro's number} = \frac{107.9 \text{ g/mol}}{1.950 \times 10^{-22} \text{ g/atom}} = 5.532 \times 10^{23} \text{ atoms/mol} \]

Correct for Deviations and Final Verification

Upon verifying calculations, remember that the true Avogadro's number is \(6.022 \times 10^{23}\). Check unit conversions, significant figures, and ensure accurate atomic mass was used. See if any rounding errors or assumption deviations might have affected result.

Key concepts

Face-Centered Cubic Unit Cell

In crystallography, a face-centered cubic (FCC) unit cell is one of the types of lattice structures that a crystalline solid can form. In an FCC structure, atoms are located at each of the corners of the cube as well as at the center of each face of the cube. This arrangement is particularly dense, allowing for efficient packing of atoms.

The FCC unit cell contains a total of 4 atoms. This is because the full contribution from each corner atom amounts to 1/8 of an atom per corner, and there are eight corners, totaling 1 atom. The atoms on the face contribute half of an atom to the unit cell, with 6 faces totaling 3 atoms. Thus, 1 (from corners) + 3 (from faces) = 4 atoms per FCC unit cell. This arrangement is common in metals such as silver, copper, and aluminum.

Density of Silver

Density is a measure of how much mass is contained in a given volume of a substance. The density of silver is given as 11.50 grams per cubic centimeter. This means that for every cubic centimeter of space that silver occupies, it has a mass of 11.50 grams.

Density is calculated using the formula:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]In the context of a crystal lattice, knowing the density helps us determine the mass of a single unit cell if we know its volume, allowing us to further calculate the mass of individual atoms and ultimately determine Avogadro's number. This is because density ties together mass, volume, and the number of atoms within that volume, providing a pathway to link macroscopic properties to atomic-scale quantities.

Molar Mass of Silver

The molar mass is an important concept in chemistry, representing the mass of one mole of a given substance. For silver, the molar mass is specified as 107.9 grams per mole. This value means that one mole of silver atoms, or Avogadro's number of atoms, weighs 107.9 grams.

The molar mass serves as a bridge between the macroscopic scale, which measures quantities in grams, and the microscopic scale, dealing with individual atoms and molecules. In calculations such as determining Avogadro's number, the molar mass allows conversion from the mass of a known number of atoms to derive a larger scale understanding of how many such atoms exist per mole.

In this exercise, using the known molar mass and determining the mass of individual atoms within a defined structure, helps to cross-verify the computed Avogadro's number.

Volume Calculation

Calculating volume is crucial when dealing with unit cells in crystallography. The volume of a cubic unit cell is determined by the cube of the length of one of its edges. For a face-centered cubic unit cell, such as that of silver, with an edge length of 408 picometers (pm), the first step is to convert this edge length to centimeters.

Conversion from picometers to centimeters involves recognizing that 1 pm is equal to \(10^{-12}\) centimeters. Therefore, the edge length in centimeters is:\[ 408 \text{ pm} \times 10^{-12} = 408 \times 10^{-12} \text{ cm} \]The volume \( V \) is then calculated by:\[ V = (408 \times 10^{-12} \text{ cm})^3 = 6.784 \times 10^{-23} \text{ cm}^3 \]This volume can be used with the density to find mass per unit cell, aiding in the calculation of the mass of individual atoms and providing insight into atomic-level details of the structure.

Conversion of Units

Unit conversion is a fundamental skill in chemistry, necessary for ensuring that calculations are accurate and meaningful. In the context of this exercise, conversions are critical, particularly when dealing with different scales such as atomic dimensions and macroscopic measurements.

For example:

• Edge lengths of unit cells may be given in picometers (pm), but when calculating volume or relating to density, conversion to centimeters is necessary, using \(1 \text{ pm} = 10^{-12} \text{ cm}\).
• Density often involves grams per cubic centimeter (g/cm³), which aligns with molar mass measurements in grams, ensuring consistent units for the calculation of mass and volume relationships.

Understanding and being proficient in unit conversion allows scientists to connect theoretical calculations with experimental data, ensuring cohesive and consistent understanding across different measurement systems. Conversions act as a bridge between the micro and macro worlds, essential for interpreting and solving chemical problems accurately.