Problem 59 - Chemistry An Atoms First Approach

Question

The radius of gold is \(144 \mathrm{pm},\) and the density is \(19.32 \mathrm{g} / \mathrm{cm}^{3}\) Does elemental gold have a face-centered cubic structure or a body-centered cubic structure?

Step-by-Step Solution

Answer

The calculated densities for both FCC and BCC structures were compared with the given density of gold (19.32 g/cm³), and the one yielding a density close to the given value was chosen as the most likely structure for elemental gold. In this case, the face-centered cubic (FCC) structure was found to be the correct structure for elemental gold, as its calculated density matched closely with the given density.

Step 1: Convert the given radius to meters

Convert 144 pm to meters by using the conversion factor 1 pm = 1 × 10⁻¹² m. Given radius, r = 144 pm = 144 × 10⁻¹² m

Step 2: Calculate the molar mass of gold

Determine the molar mass of gold (M_gold) from the periodic table. Gold (Au) has an atomic number of 79 and a molar mass of 197 g/mol.

Step 3: Calculate the number of gold atoms per unit cell (n)

Because both FCC and BCC unit cells consist of one lattice point per primitive cell, n for both FCC and BCC structures can be calculated using the following formulas: n_FCC = 4 (4 atoms per unit cell in FCC) n_BCC = 2 (2 atoms per unit cell in BCC)

Step 4: Determine the volume of the unit cell

First, calculate the volume of one gold atom from its radius: V_atom = (4/3)πr³ Now, determine the atomic packing factor (APF) for both FCC and BCC structures. The APFs for FCC and BCC structures are well-known and are as follows: APF_FCC = 0.74 APF_BCC = 0.68

Step 5: Calculate the density of gold for FCC and BCC structures

Calculate the density (ρ) using the formula: ρ = (n × M) / (N_A × V_cell) Where, n is the number of atoms per unit cell (determined in step 3), M is the molar mass (from step 2), N_A is Avogadro's number (6.022 × 10²³ atoms/mol), and V_cell is the volume of the unit cell. Calculate the density for both FCC and BCC structures and compare it with the given density of gold (19.32 g/cm³). If one of the structures yields a density close to the given value, it is most likely the correct structure for elemental gold.

Step 6: Compare the results and determine the structure

Compare the calculated densities (ρ_FCC and ρ_BCC) with the given density of gold (19.32 g/cm³). If one of the calculated densities is very close to the given density, it should be considered as the most likely structure for elemental gold. If both calculated densities are not close to the given density, more calculations might be needed to determine the correct structure.

Key Concepts

Face-Centered Cubic (FCC)

The face-centered cubic (FCC) structure is a type of crystal lattice where atoms are located at each of the corners and the centers of all the cube faces of the unit cell. This arrangement is known for its high packing efficiency.

• In an FCC structure, each corner atom is shared among eight adjacent unit cells, and each face-centered atom is shared between two.
• This results in a total of 4 atoms in each FCC unit cell (1/8 of an atom from each of the 8 corners plus half of an atom from each of the 6 faces).
• FCC structures are quite common and are found in many metals, including aluminum, copper, and gold.

The FCC packing allows for a high degree of structural stability and is often associated with metals that exhibit high ductility because the closely packed atoms can slide past one another easily when a force is applied.

Body-Centered Cubic (BCC)

The body-centered cubic (BCC) structure is another type of crystal lattice. Unlike FCC, BCC has atoms at each corner of the cube and a single atom in the very center of the cube. This configuration provides certain unique properties.

• In BCC, each corner atom is shared among eight neighboring unit cells, similar to the FCC.
• This results in a total of 2 atoms per BCC unit cell, with the central atom contributing fully and the corner atoms contributing a fraction.
• Though less densely packed than FCC structures, BCC is common amongst metals like iron and chromium.

The BCC structure, while not as tightly packed as FCC, is still stable and often found in metals that require strength, hardness, and are typically less ductile compared to FCC metals.

Atomic Packing Factor

The atomic packing factor (APF) is an important concept in determining how efficiently atoms pack in a crystal structure. It measures the fraction of volume occupied by atoms in a given unit cell.

• For FCC, the APF is 0.74, indicating that 74% of the unit cell's volume is taken up by the atoms. This is the highest possible packing efficiency in these types of structures.
• In contrast, the APF for BCC is lower at 0.68, meaning 68% of the unit cell's volume is filled.
• Both FCC and BCC are contrasted with other structures, such as the simple cubic, which has a much lower APF of 0.52.

The APF essentially allows us to predict properties such as density and stability of the material, with higher packing factors often resulting in denser materials.

Density Calculation

Calculating the density of a crystal structure involves understanding how atoms are packed within the unit cell. Density is calculated by finding the mass of atoms in the unit cell and dividing it by the unit cell's volume. The formula used is:\[\rho = \frac{n \times M}{N_A \times V_{cell}}\]where,

• \( n \) is the number of atoms per unit cell,
• \( M \) is the molar mass of the element,
• \( N_A \) is Avogadro's number, and
• \( V_{cell} \) is the volume of the unit cell.

To determine the crystal structure of an element like gold, we compare the calculated density with the experimentally measured density. If our calculated density using the FCC or BCC structure aligns closely with the measured value, it suggests that structure is likely correct. In the case of gold, the density calculated using FCC parameters is a closer match, indicating gold likely has an FCC structure.