Problem 91 - General Chemistry

Question

Tungsten has a body-centered cubic lattice with all atoms at the lattice points. The edge length of the unit cell is \(316.5 \mathrm{pm}\). The atomic mass of tungsten is \(183.8\) amu. Calculate its density.

Step-by-Step Solution

Answer

The density of tungsten is approximately 19.29 g/cm³.

Step 1: Identify the Unit Cell Type

Tungsten has a body-centered cubic (BCC) lattice. This means there are atoms at each corner of a cube and one atom in the center of the cube.

Step 2: Calculate the Number of Atoms per Unit Cell

In a BCC lattice, each corner atom is shared by 8 neighboring unit cells, and the center atom belongs entirely to the unit cell. Therefore, the number of atoms per unit cell is \[8 \times \frac{1}{8} + 1 = 2 \text{ atoms}\]

Step 3: Calculate the Volume of the Unit Cell

The volume of a cube can be calculated using the formula for the volume in terms of its side length \(a\): \[ V = a^3 \]Here, the edge length \(a\) is given as 316.5 pm:\[ V = (316.5 \times 10^{-12} \text{ m})^3 = 3.166 \times 10^{-29} \text{ m}^3\]

Step 4: Calculate the Mass of One Mole of Tungsten Atoms

The atomic mass of tungsten is given as 183.8 amu. Using Avogadro’s number \(6.022 \times 10^{23} / \text{mol}\), the mass of one mole of tungsten atoms:\[183.8 \text{ g/mol}\]

Step 5: Calculate the Mass of Atoms in One Unit Cell

Since there are 2 atoms per unit cell, the mass of atoms in one unit cell is: \[\frac{183.8 \text{ g/mol}}{6.022 \times 10^{23} \text{ atoms/mol}} \times 2 \approx 6.098 \times 10^{-22} \text{ g}\]

Step 6: Calculate the Density of Tungsten

Density \( \rho \) is defined as mass per unit volume. Using the mass of the unit cell and its volume:\[ \rho = \frac{6.098 \times 10^{-22} \text{ g}}{3.166 \times 10^{-29} \text{ m}^3} \approx 19286 \text{ kg/m}^3\]

Step 7: Convert the Density to the Correct Units

The density is given here in \( \text{kg/m}^3 \). Ensuring all conversions:\[ \approx 19.29 \text{ g/cm}^3\]

Key Concepts

Body-Centered Cubic Lattice

A body-centered cubic (BCC) lattice is one of the common arrangements of atoms in metals. In this structure, each unit cell contains atoms at all corners and a single atom at the center. This central atom ensures better atomic packing and facilitates the stable structure of many metals. Imagine the cube as a small building block, where each block shares its corner atoms with seven other blocks. Key points about BCC lattice:

• *Corner atoms* are shared among multiple unit cells.
• Each BCC unit cell effectively contains *2 atoms* (one from the central location and one "averaged" from the corner atoms).

Visualizing these layers and how they stack is crucial for understanding why certain properties, like density or hardness, can be observed in materials.

Atomic Mass

Atomic mass is the weight of an atom measured in atomic mass units (amu), reflecting the average mass of atoms in an element, accounting for isotopes. This intrinsic property is crucial for calculating the density and mass of elements in any chemistry or physics problem. For tungsten, given its dense configuration, the atomic mass is significant at *183.8 amu*. Understanding atomic mass:

• Atoms have varying numbers of neutrons, so atomic mass accounts for these differences.
• It is a *weighted average* of all naturally occurring isotopes.
• Essential for figuring out grams per mole, which helps in calculations like density and molar conversions.

Unit Cell

The unit cell represents the smallest repeating structure of a mineral or metal lattice. It's like a tiny building block that, when repeated in space, creates the entire lattice structure. In a BCC lattice configuration, this cube-shaped unit cell is particularly defined by its edge length, in the case of tungsten, being *316.5 pm*. Essentials of unit cells:

• Understanding the *edge length* helps in determining the volume of the cell, crucial for density calculations.
• Variations in unit cell types (e.g., face-centered cubic, BCC) impact the material's properties.
• Calculating how many atoms pretend to be inside the unit cell is key to precise mass measurements.

Avogadro's Number

Avogadro's number, approximately *6.022 x 10^{23} atoms/mol*, is the counting unit for atoms at the molecular scale. This constant is essential in bridging the micro-world of atoms and the macro-world of grams and moles, converting atomic mass units to a usable scale for laboratory experiments. Why Avogadro's number is crucial:

• It connects *atomic scale* measurements to laboratory-scale quantities, making it an indispensable tool in stoichiometry.
• Facilitates conversion between the number of atoms and the *mass of samples* through molar mass.
• Enables the computation of complementary properties, like density, by providing a *conversion factor* between the atomic and macroscopic scales.