Question

Iridium crystallizes in a face-centered cubic unit cell that has an edge length of 3.833 \(\dot{A}\). (a) Calculate the atomic radius of an iridium atom. (b) Calculate the density of iridium metal.

Short answer

The atomic radius of an iridium atom is approximately 1.365 Å, and the density of iridium metal is approximately 22.7 g/cm³.

Step-by-step solution

Finding the relationship between edge length and atomic radius

In a face-centered cubic unit cell, the atoms are located at the corners and the center of each face. We can consider a right-angled triangle formed by the center atom of a face, the atom at the corner, and the midpoint of the cube's edge. The hypotenuse of this triangle is equal to four times the atomic radius (two times the radius for each of the two atoms). The other two sides of the triangle are equal to half of the cube's edge length. Using the Pythagorean theorem, we can find the relationship between the edge length and the atomic radius. \(a^2 + a^2 = (4r)^2\) where a is the edge length and r is the atomic radius.

Calculate the atomic radius of an iridium atom

Now, we'll use the equation from step 1 to find the atomic radius of iridium. Given the edge length of the unit cell, a = 3.833 Å. \(a^2 + a^2 = (4r)^2\) \((3.833)^2 + (3.833)^2 = (4r)^2\) Solving for r: \(r = \frac{\sqrt{(3.833)^2 + (3.833)^2}}{4}\) \(r ≈ 1.365\, Å\) The atomic radius of an iridium atom is approximately 1.365 Å.

Calculate the number of atoms in an FCC unit cell

In a face-centered cubic unit cell, there are 8 corner atoms and 6 face-centered atoms. Each corner atom is shared among 8 unit cells, and each face-centered atom is shared with the adjacent unit cell. Number of atoms in an FCC unit cell = (8 corner atoms * 1/8) + (6 face-centered atoms * 1/2) = 1 + 3 = 4

Calculate the density of iridium metal

To find the density of iridium, we need to use the formula for density: \(density = \frac{mass}{volume}\) The mass of iridium in one unit cell is given by the product of the number of atoms in the unit cell and the atomic mass of iridium. The volume of a unit cell can be found by cubing the edge length (a). Given that the atomic mass of iridium (Ir) is 192.22 g/mol, and using the Avogadro's number (6.022 × 10^23 atoms/mol), we can find the mass of one iridium atom: \(mass_{atom} = \frac{192.22\, g/mol}{6.022 × 10^{23}\, atoms/mol} ≈ 3.191 × 10^{-22}\, g\) Now, find the mass of iridium in one unit cell: \(mass_{unit\, cell} = 4\, atoms/unit\, cell * 3.191 × 10^{-22}\, g/atom ≈ 1.276 × 10^{-21}\, g\) Next, find the volume of one unit cell: \(volume_{unit\, cell} = (3.833\, Å)^3 = 56.212\, Å^3\) (Note that 1 Å = 1 × 10^-8 cm) Convert the volume to cm³: \(volume_{unit\, cell} = 56.212\, Å^3 * (1 × 10^{-24}\, cm^3/Å^3) = 5.621 × 10^{-23}\, cm^3\) Now, use the mass and volume to calculate the density of iridium: \(density = \frac{mass_{unit\, cell}}{volume_{unit\, cell}} = \frac{1.276 × 10^{-21}\, g}{5.621 × 10^{-23}\, cm^3} ≈ 22.7\, g/cm^3\) The density of iridium metal is approximately 22.7 g/cm³.

Key concepts

Understanding the Face-Centered Cubic Unit Cell

When it comes to solid materials, particularly metals, the face-centered cubic (FCC) unit cell is a fundamental structure that is critical to understanding various properties of the material. In this type of unit cell, atoms are positioned at each corner and at the center of every cube face, resulting in a highly dense packed structure.

The beauty of the FCC structure is in its symmetry and efficiency in space utilization. This arrangement allows for the maximum number of atoms to be packed within a given volume, which is a key factor in the high density of certain metals. Moreover, the face-centered cubic structure affects the material's overall strength and ductility, making it a subject of great interest in materials science.

For visualization, imagine taking eight identical spheres and arranging them into the corners of a cube. Then, place six more identical spheres in the center of each face of the cube. This model represents an idealized FCC unit cell where the spheres signify the positions of the atoms. To fully appreciate this, it's often helpful to create a 3D model or diagram exposing the arrangement of atoms and how they relate spatially within the cell.

Atomic Radius Calculation in Metals

Determining the atomic radius of an atom within a crystal structure, such as face-centered cubic (FCC), is more than just academic exercise — it's crucial for understanding the properties of the material. The atomic radius is half of the distance between the nuclei of two adjacent, touching atoms, and calculating it requires ingenuity since atoms in a solid do not have clear-cut boundaries.

To arrive at this value, we often use geometrical relationships within the unit cell. For instance, we can construct a right-angled triangle within an FCC unit cell and apply the Pythagorean theorem, exploiting the relationship between the cell's edge length and atomic radii, as seen in the exercise. Such calculations aid in deciphering atomic sizes and how closely packed the atoms are within a structure, giving us insight into the metal's mechanical and thermal properties.

Furthermore, understanding the atomic radius can also shuffle up to a higher deck on the abstraction tower — it allows chemists and physicists to predict and explain trends in reactivity, bonding, and more across the periodic table. Hence, even though the calculation might seem contained within the unit cell's confines, its implications resonate throughout the vast chambers of material science and chemistry.

Density of Metals Calculation

Density is a significant physical property of materials, indicative of how tightly matter is packed within a given volume. For metals crystallizing in regular arrangements such as the FCC lattice, calculating density involves a multi-step process that leans on concepts from both geometry and chemistry.

Firstly, one must comprehend the atomic mass and Avogadro's number to determine the mass of a single atom. With this, you can then find the mass for the entire unit cell by accounting for the number of atoms present within it. The volume of the unit cell is derived from its edge length, a measurement that can be experimentally determined. With both mass and volume in hand, applying the density formula \(density = \frac{mass}{volume}\) clinches the requisite insight into the metal's density.

The result yields practical information, too. Engineers, for example, can select suitable metals for particular applications based on their density. Furthermore, comparing calculated densities to experimental values may expose discrepancies, opening dialogue for potential improvements in experimental techniques or theoretical models. In summary, the meticulous calculation of density from the atomic level up delivers crucial, foundational knowledge that can be used across a spectrum of scientific and engineering disciplines.