X-ray diffraction studies of buckminsterfullerene show that it has a face-centered cubic lattice of C60 molecules. The length of an edge of the unit cell is 14.2 Å. Calculate the density of buckminsterfullerene.

Verified step by step guidance

Step 1: Understand the structure of the face-centered cubic (FCC) lattice. In an FCC lattice, there are 4 atoms per unit cell. For buckminsterfullerene, these are C60 molecules.

Step 2: Calculate the volume of the unit cell. The edge length of the unit cell is given as 14.2 Å. Convert this length into centimeters (1 Å = 1 x 10^-8 cm) and then calculate the volume using the formula: \( V = a^3 \), where \( a \) is the edge length.

Step 3: Determine the mass of one unit cell. First, find the molar mass of C60. Carbon has an atomic mass of approximately 12.01 g/mol, so the molar mass of C60 is \( 60 \times 12.01 \) g/mol. Convert this to grams per molecule using Avogadro's number (\( 6.022 \times 10^{23} \) molecules/mol). Multiply by the number of molecules per unit cell (4) to find the mass of the unit cell.

Step 4: Calculate the density of buckminsterfullerene. Density is defined as mass per unit volume. Use the mass of the unit cell calculated in Step 3 and the volume from Step 2 to find the density using the formula: \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \).

Step 5: Review the units and ensure that the density is expressed in g/cm³, which is the standard unit for density in chemistry.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Unit Cell and Lattice Structure

A unit cell is the smallest repeating unit in a crystal lattice that reflects the symmetry and structure of the entire crystal. In the case of buckminsterfullerene (C60), it forms a face-centered cubic lattice, meaning that the unit cell has atoms at each corner and the center of each face. Understanding the unit cell is crucial for calculating properties like density, as it defines the arrangement and number of molecules in the crystal.

Density Calculation

Density is defined as mass per unit volume and is calculated using the formula: density = mass/volume. For crystalline solids, the mass can be determined from the number of molecules in the unit cell and their molar mass, while the volume is derived from the dimensions of the unit cell. In this case, knowing the edge length of the unit cell allows for the calculation of its volume, which is essential for determining the density of buckminsterfullerene.

Molar Mass of Buckminsterfullerene

The molar mass of a substance is the mass of one mole of its entities, typically expressed in grams per mole. For buckminsterfullerene (C60), the molar mass can be calculated by summing the atomic masses of its constituent carbon atoms. This value is necessary for determining the total mass of the molecules within the unit cell, which is a key component in the density calculation.

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Related Practice

Textbook Question

In their study of X-ray diffraction, William and Lawrence Bragg determined that the relationship among the wavelength of the radiation 1l2, the angle at which the radiation is diffracted 1u2, and the distance between planes of atoms in the crystal that cause the diffraction (d) is given by nl = 2d sin u. X rays from a copper X-ray tube that have a wavelength of 1.54 Å are diffracted at an angle of 14.22 degrees by crystalline silicon. Using the Bragg equation, calculate the distance between the planes of atoms responsible for diffraction in this crystal, assuming n = 1 (first-order diffraction).

Textbook Question

Germanium has the same structure as silicon, but the unit cell size is different because Ge and Si atoms are not the same size. If you were to repeat the experiment described in the previous problem but replace the Si crystal with a Ge crystal, would you expect the X rays to be diffracted at a larger or smaller angle u?

Textbook Question

(a) The density of diamond is 3.5 g/cm³, and that of graphite is 2.3 g/cm³. Based on the structure of buckminsterfullerene, what would you expect its density to be relative to these other forms of carbon?

Textbook Question

(b) X-ray diffraction studies of buckminsterfullerene show that it has a face-centered cubic lattice of C₆₀ molecules. The length of an edge of the unit cell is 14.2 Å. Calculate the density of buckminsterfullerene.