Question

Barium crystallizes in a body-centered cubic structure. (a) How many Ba atoms are contained in each unit cell? (b) How many nearest neighbors does each Ba atom possess? (c) Estimate the length of the unit cell edge, \(a\), from the atomic radius of barium \((0.215 \mathrm{nm}) .\) (d) Estimate the density of Ba metal at this temperature.

Short answer

There are 2 barium atoms per unit cell in a body-centered cubic structure, and each barium atom has 8 nearest neighbors. The unit cell edge length (a) for barium is approximately 0.502 nm. The estimated density of barium at this temperature is approximately 3.61 g/cm³.

Step-by-step solution

Part (a): Number of Barium Atoms in the Unit Cell

In a body-centered cubic unit cell, there is one atom at each corner of the cube and one atom at the center. There are 8 corners, each containing 1/8 of an atom since it is shared by 8 unit cells. So, the total number of barium atoms in each unit cell can be calculated as: Atoms in corners = 8 corners * 1/8 atom per corner = 1 atom Atoms in center = 1 atom Total barium atoms = 1 atom (corners) + 1 atom (center) = 2 atoms Therefore, there are 2 barium atoms per unit cell.

Part (b): Number of Nearest Neighbors of Barium Atom

In a body-centered cubic structure, the atom at the center has 8 nearest neighbors at the corner of the cube, while an atom at the corner has 4 nearest neighbors along the edges connected to the corner and 1 center atom. So, each barium atom in the BCC structure has 8 nearest neighbors.

Part (c): Calculation of Unit Cell Edge Length (a)

The unit cell edge length (a) can be estimated from the atomic radius (r) of barium. In a body-centered cubic structure, the length of the diagonal of the cube is equal to 4 times the atomic radius. We can use Pythagorean theorem to relate the length of the diagonal to the unit cell edge length: Diagonal = \( \sqrt{3a^2} \) Given that Diagonal = 4r, we can rewrite the equation as: \( 3a^2 = (4r)^2 \) Now, we can solve for the unit cell edge length 'a' using the given atomic radius of barium (0.215 nm): \( a = \sqrt{\frac{(4 \times 0.215 \mathrm{nm})^2}{3}} \) \( a \approx 0.502 \mathrm{nm} \) Therefore, the unit cell edge length (a) for barium is approximately 0.502 nm.

Part (d): Estimation of Barium Density

To estimate the density of barium, we can use the following formula: Density = \( \frac{Mass}{Volume} \) First, we need to find the mass of the unit cell. The mass of a single barium atom can be calculated using its molar mass (137.33 g/mol) and Avogadro's number (6.022 x 10^23 atoms/mol): Mass of one Ba atom = \( \frac{137.33 \mathrm{g/mol}}{6.022\times10^{23} \mathrm{atoms/mol}} \approx 2.28 \times 10^{-22} \mathrm{g} \) Since there are 2 barium atoms in each unit cell, the mass of the unit cell is twice the mass of a single barium atom: Mass of unit cell = 2 x Mass of one Ba atom = 2 x \( 2.28 \times 10^{-22} \mathrm{g} \approx 4.56 \times 10^{-22} \mathrm{g} \) Next, we need to find the volume of the unit cell. Since the unit cell is a cube with edge length 'a', the volume is: Volume of unit cell = \( a^3 \approx (0.502 \mathrm{nm})^3 \) Converting nm to cm to keep the volume consistent with the mass unit: Volume of unit cell = \( (0.502 \times 10^{-7} \mathrm{cm})^3 \approx 1.26 \times 10^{-21} \mathrm{cm}^3 \) Now we can find the density: Density = \( \frac{4.56 \times 10^{-22} \mathrm{g}}{1.26 \times 10^{-21} \mathrm{cm}^3} \approx 3.61 \mathrm{g/cm^3} \) Therefore, the estimated density of barium at this temperature is approximately 3.61 g/cm³.

Key concepts

Atomic Radius

The atomic radius is a crucial factor when studying atomic structures, such as in the body-centered cubic (BCC) formation of barium. Atoms, including barium, have a size that can be represented as a radius. This defines the size of the sphere that conceptualizes an atom in a lattice. For barium, the atomic radius is 0.215 nm. This value is essential in estimating the geometry of the entire crystal lattice, affecting physical properties like the density and unit cell edge length. The atomic radius becomes particularly relevant when calculating parameters like the diagonal in a BCC unit cell. The relationship between the diagonal of the cube and the atomic radius can be expressed as the diagonal being 4 times the radius. Understanding the atomic radius helps in determining how various atoms are packed and how they interact in the periodic lattice.

Unit Cell

A unit cell is the smallest divisible portion of a crystal lattice that maintains the overall symmetry and properties of the entire crystal structure. In the context of barium's body-centered cubic (BCC) lattice, the unit cell plays a pivotal role in determining how barium atoms stack together.In a BCC structure:

• There is one atom at each of the eight corners.
• There is one atom located at the center of the cube.

This translates to a total of two equivalent atoms per unit cell, taking into account that corner atoms are shared among eight different unit cells. The unit cell is fundamental in calculating properties like the edge length \( a \) and the material's density. Understanding the composition and arrangement of a unit cell helps in elucidating how atoms are organized, directly affecting the crystal's macroscopic properties.

Density of Barium

Density is a measure of mass per unit volume, and in materials like barium in a body-centered cubic (BCC) structure, it is influenced significantly by the atomic arrangements within the unit cell. To calculate density, we use:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]First, calculate the mass of the barium atoms within a unit cell. This involves the molar mass of barium, which is 137.33 g/mol, divided by Avogadro's number, giving the mass of one barium atom. Since a BCC unit cell contains two barium atoms, we simply multiply the mass of one atom by two.Then, the unit cell volume is computed as the cube of the unit cell edge length. Converting all measurements to the same units (grams and cubic centimeters), we obtain the density. In this example, barium's density is found to be about 3.61 g/cm³, highlighting how atomic arrangement within a unit cell can dictate volumetric properties.

Nearest Neighbors

In solid-state physics, nearest neighbors refer to atoms in the immediate vicinity of a given atom, impacting the structural and mechanical properties of a material. In a body-centered cubic (BCC) structure like that of barium:

• The center atom in a unit cell is surrounded by 8 nearest neighbors, which are located at the corners of the cube.
• An atom situated at the corner of the cube shares its neighboring atoms with four other BCC cells, making it an intriguing structural feature.

The presence of nearest neighbors influences how atoms interact with each other, affecting the overall stability and density of the lattice. By understanding nearest neighbors, we gain insight into the material's structural attributes and potential applications, as it influences properties such as melting point, elasticity, and conductivity.