Question

Tantalum \(\left(\mathrm{Ta} ; d=16.634 \mathrm{~g} / \mathrm{cm}^{3}\right.\) and \(\left.\mathscr{H}=180.9479 \mathrm{~g} / \mathrm{mol}\right)\) has a body-centered cubic structure with a unit-cell edge length of 3.3058 A. Use these data to calculate Avogadro's number.

Short answer

6.022 x 10^23 atoms/mol

Step-by-step solution

- Convert unit-cell edge length

Convert the unit-cell edge length from angstroms to centimeters. 1 angstrom is equal to 1 x 10^-8 centimeters. So, the edge length is 3.3058 Å = 3.3058 x 10^-8 cm.

- Calculate the volume of the unit cell

The volume of a cube is given by the cube of the edge length. Therefore, the volume of the unit cell is .

- Determine the mass of the unit cell

First, calculate the mass of one tantalum atom by dividing the molar mass by Avogadro's number: . Then, determine the number of atoms per unit cell. For a body-centered cubic unit cell, there are two atoms per unit cell.

- Calculate the mass of the unit cell

The mass of the unit cell is the density of tantalum ( g/cm³) multiplied by the volume of the unit cell. Use the formula: mass = density × volume.

- Use the mass of the unit cell to find Avogadro's number

Using the mass of the unit cell and the fact that there are two atoms per unit cell, we can set up the equation to solve for Avogadro's number.

Key concepts

tantalum

Tantalum is a rare and highly durable metal. It's symbolized as \(\text{Ta}\) in the periodic table and belongs to the transition metal category. Tantalum is commonly used in electronic components, surgical instruments, and even in making alloys due to its high melting point and resistance to corrosion.
Tantalum has a density of 16.634 g/cm³, which makes it one of the densest elements. This high density is a key property to remember when performing calculations related to its physical properties.
The molar mass of tantalum is given as 180.9479 g/mol, which tells us how much one mole of tantalum atoms weighs.

body-centered cubic structure

In a body-centered cubic (BCC) structure, atoms are arranged in a very specific way.
This structure consists of one atom at each corner of the cube and one atom in the very center of the cube. Effectively, there are 2 atoms per BCC unit cell.
Here are some notable features of BCC:

• It’s a highly efficient arrangement that provides good density.
• Each corner atom is shared among eight adjacent unit cells, contributing one-eighth of an atom per unit cell.
• The central atom belongs entirely to that particular unit cell.
• Metals with BCC structures typically exhibit high tensile strength and hardness.

Examples of elements with BCC structures include iron (at certain temperatures), chromium, and of course, tantalum.

unit cell volume

The unit cell volume is the space occupied by one unit cell of the crystal structure.
Since a unit cell in a BCC structure is cubic, its volume is the cube of its edge length.
Given the edge length of tantalum's unit cell is 3.3058 Å, you can convert this to centimeters using the conversion factor: 1 Å = 1 x 10^-8 cm.
This gives us:
3.3058 Å = 3.3058 x 10^-8 cm.
Now, calculate the volume \(V\):
\[ V = \text{edge length}^3 \]
\[ V = (3.3058 \times 10^{-8} \text{ cm})^3 \]
This leads to an extremely small volume, which is typical for unit cells.

density

Density is defined as mass per unit volume. In this problem, we are given the density of tantalum as 16.634 g/cm³.
Density (\( \rho \)) is a critical property because it will be used to connect the mass and the volume of the unit cell.
The relation is given by:
\[ \text{mass} = \text{density} \times \text{volume} \]
Hence, once you have the volume of the unit cell from the previous section, you can multiply it by the density to find the mass of the unit cell.
This mass value is crucial for calculating Avogadro's number, which links the microscopic world (the unit cell) to the macroscopic world (the molar mass).

molar mass

Molar mass denotes the mass of one mole of atoms, usually in grams per mole (g/mol).
For tantalum, the molar mass is 180.9479 g/mol.
Knowing the molar mass is vital to determine the mass of a single tantalum atom. Using Avogadro's number (\( N_A \)) which we want to determine, we use the relation:
\[ \text{Mass of one Ta atom} = \frac{\text{Molar mass}}{N_A} \]
Where:

• Molar mass is in g/mol
• N_A is Avogadro’s number, typically about \ (6.022 \times 10^{23})\ atoms/mol

This mass will then be used in conjunction with the mass of the unit cell from the density to volume relationship to solve for Avogadro’s number.