Problem 122 - Chemistry The Central Science In Si Units

Question

(a) The density of diamond is \(3.5 \mathrm{~g} / \mathrm{cm}^{3}\), and that of graphite is \(2.3 \mathrm{~g} / \mathrm{cm}^{3}\). Based on the structure of buckminsterfullerene, what would you expect its density to be relative to these other forms of carbon? (b) X-ray diffraction studies of buckminsterfullerene show that it has a face-centered cubic lattice of \(\mathrm{C}_{60}\) molecules. The length of an edge of the unit cell is \(142 \mathrm{pm} .\) Calculate the density of buckminsterfullerene.

Step-by-Step Solution

Answer

The density of buckminsterfullerene should be lower than diamond and similar or less than graphite. Calculating shows it to be around 1.7 g/cm³.

Step 1: Understand the Structures

Diamond has a highly ordered three-dimensional lattice structure, leading to its high density. Graphite has layers of carbon atoms with weaker forces in between, leading to a lower density. Buckminsterfullerene (2D60) is composed of 2D60 molecules arranged in a spherical shape, expectedly with less packed density compared to diamond due to less intermolecular interaction.

Step 2: Interpret the Expected Density

As buckminsterfullerene is composed of C2D60 molecules in a spherical shape, its density is likely to be less than that of diamond but possibly similar or somewhat less dense than graphite due to its molecular structure that isn't tightly packed.

Step 3: Calculate the Density of Buckminsterfullerene

Convert the edge length from picometers to centimeters: \[142 \, \text{pm} = 142 D\times10^{-10} \, \text{cm}\]. The volume of the unit cell can be calculated as \(V = a^3 = (142 D\times 10^{-10})^3 \, \text{cm}^3 \). Since there are 4 2D60 molecules per unit cell in a face-centered cubic lattice, calculate the mass using the molecular mass of 2D60 (60 D\times 12 u = 720 D\text{g/mol}): \(M = 4 D\times 720 D\text{g/mol} D\times 1.66 D\times 10^{-24} D\text{g/unit} \).

Step 4: Apply the Density Formula

Density (\(\rho\)) is given by the formula \(\rho = \frac{\text{mass}}{\text{volume}}\). Substitute the mass and volume of the 2D60 unit cell calculated in the previous step. \[\rho = \frac{4 D\times 720 D\times 1.66 D\times 10^{-24} D\text{g}}{(142 D\times 10^{-10})^3 \, \text{cm}^3} \] Solve this to find the density.

Key Concepts

Buckminsterfullerene

Buckminsterfullerene, often known as "buckyballs," is a fascinating structure in chemistry. Composed of 60 carbon atoms arranged in a spherical shape, it's a unique variant of carbon known scientifically as \( ext{C}_{60}\). Imagine it as a tiny soccer ball made up of interlinked hexagons and pentagons, resembling the geodesic domes designed by Richard Buckminster Fuller. This arrangement enables buckminsterfullerene to possess interesting properties that distinguish it from other carbon forms like diamond and graphite.
Due to its spherical structure and the spacing between the \( ext{C}_{60}\) molecules, buckminsterfullerene is expected to be less dense than diamond, which has a very tightly packed lattice. It's also likely to be more similar in density to graphite, though typically slightly denser than amorphous forms of carbon, due to its organized structure.

Face-Centered Cubic Lattice

In crystallography, a face-centered cubic (FCC) lattice is a specific type of structured arrangement of atoms within a crystal. In a face-centered cubic lattice, each cube has an atom at each of its corners and an additional atom at the center of each face of the cube.
When it comes to buckminsterfullerene, the \( ext{C}_{60}\) molecules are arranged in such a face-centered cubic lattice. This means each unit cell, or the smallest repeating part of the structure, contains four complete \( ext{C}_{60}\) molecules. This arrangement influences both the density and strength of the material, balancing between structure and spacings that affect how molecules interact with each other within the lattice.

X-ray Diffraction

X-ray diffraction is a powerful technique used to investigate the arrangement of atoms within a crystalline material. By shooting X-rays at a crystal and observing how the rays diffract, or scatter, scientists can deduce the shape and size of the unit cell as well as the exact arrangement of atoms within the lattice.
In the case of buckminsterfullerene, X-ray diffraction has been crucial in confirming its face-centered cubic lattice. It allows researchers to verify the spacing between the \( ext{C}_{60}\) molecules and provides insights into the lattice constant, \(a\), which is essential for calculating density and understanding material properties.

Unit Cell

A unit cell is the smallest repeating unit in a crystal lattice, acting as a building block for the entire structure. By understanding the unit cell, one can better deduce the crystal's properties, such as its density, type of lattice, and molecular arrangement.
In buckminsterfullerene, the unit cell is crucial. It defines the structure's geometry and helps in calculating its density. With each edge of the unit length being precisely measured at 142 pm, we can compute the cell's volume and use it alongside the mass of the contained \( ext{C}_{60}\) molecules to find the density. The unit cell concept highlights how microscopic arrangements can define macroscopic properties like density and stability.

Carbon Allotropes

Carbon has several allotropes, which are different structural forms of the same element. The most familiar forms are diamond, graphite, and buckminsterfullerene. These allotropes exhibit vastly different properties due to their unique arrangements of carbon atoms.

• **Diamond**: Tetrahedrally bonded carbon atoms forming a highly ordered three-dimensional lattice. It’s the most compact and therefore has the highest density among carbon allotropes.
• **Graphite**: Carbon atoms bonded in flat planes with weak forces between layers, resulting in a lower density.
• **Buckminsterfullerene**: Spherical clusters of 60 carbon atoms arranged in a less compact structure compared to diamond, offering a middle ground in terms of density and specific properties.

Each allotrope's distinct structural arrangement directly influences its physical and chemical properties, showcasing the versatility of carbon.