(b) X-ray diffraction studies of buckminsterfullerene show that it has a face-centered cubic lattice of C 60 molecules. The length of an edge of the unit cell is 14.2 Å. Calculate the density of buckminsterfullerene.

Hey everyone, we're told when lent crystallizes, it adopts a face centered cubic crystal structure. The edge length of the unit cell is 495 PICO meters, calculate the radius in PICO meters and density and grams per cubic centimeters of a lead adam. So for a face centered cubic cell, our edge length is equal to two times the square root of two times our radius with edge length being denoted by a. We were told that our edge length was 495 m. So this is going to be equal to two times the square root of two times our radius. Now, when we divide both sides by two times the square root of two, we get a radius of 175 PICO meters and this is going to be one of our answers. Now let's go ahead and calculate for the density, we know that density is equal to our mass divided by our volume and this is going to be in grams per cubic centimeters. Now we know for the volume of our unit cell. This is equal to our edge length to the cubic power. But since this needs to be in centimeters, let's first go ahead and take our edge length And changes into cm. So taking our edge length of 495 km, we're going to use our dimensional analysis. Now we know that per one PICO meter, we have 10 to the negative 12 m and we know that her one centimeter, we have 10 to the negative two m. Now when we calculate this out and cancel out our units, we end up with an edge length of 4.95 times 10 to the negative eight centimeters. Now that we've converted this into cm. Let's go ahead and determine our volume. Taking that value of 4.95 times 10 to the negative eight cm, we can go ahead and cube this. This will get us a volume of 1.2129 times 10 to the negative 22 cubic centimeters. Now, let's go ahead and determine our mass in order to get our density. So to calculate our mass, We know that the molar mass of lead is 207.2 g per one mole. And we can easily find this in our periodic table. Now we know that per one mole we have 6. times 10 to the 23rd atoms, which is avocados number. And since we do have a face centered cubic cell, this is going to be four atoms per one unit cell. So when we calculate this out and cancel out all of our units, We end up with a value of one 0.3763 times 10 to the negative 21st grams per unit cell. Now that we've determined our mass, we can go ahead and determine our density which is mass divided by volume. So taking the value of 1.3763 times 10 to the negative 21st grams and dividing it by the volume that we calculated, which was 1.2129 times 10 to the negative cubic centimeters. And when we calculate this out, we end up with a density of 11.3 g per cubic centimeter, which is going to be our other final answer. Now, I hope that made sense and let us know if you have any questions.

Ch.12 - Solids and Modern Materials

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Brown 14th Edition

Ch.12 - Solids and Modern Materials

Problem 122b

Chapter 12, Problem 122b

(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.

Verified step by step guidance

Identify the type of lattice structure: Buckminsterfullerene has a face-centered cubic (FCC) lattice structure.

Determine the number of C60 molecules per unit cell in an FCC lattice: An FCC lattice has 4 atoms or molecules per unit cell.

Calculate the molar mass of a C60 molecule: The molar mass of carbon (C) is approximately 12.01 g/mol. Therefore, the molar mass of C60 is 60 times the molar mass of carbon, which is 60 * 12.01 g/mol.

Convert the edge length of the unit cell from Ångströms to centimeters: 1 Å = 1 x 10^-8 cm, so 14.2 Å = 14.2 x 10^-8 cm.

Calculate the density using the formula: \( \text{Density} = \frac{\text{Mass of molecules in unit cell}}{\text{Volume of unit cell}} \). Use the molar mass, Avogadro's number, and the converted edge length to find the mass and volume.

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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 overall symmetry and structure of the entire crystal. In the case of buckminsterfullerene, it has a face-centered cubic (FCC) lattice, meaning that the unit cell is cubic with atoms located at each corner and the center of each face. Understanding the unit cell is crucial for calculating properties like density.

Density Calculation

Density is defined as mass per unit volume and is typically expressed in grams per cubic centimeter (g/cm³). To calculate the density of a substance, one must know the mass of the unit cell and its volume. For buckminsterfullerene, the volume can be determined from the edge length of the unit cell, while the mass can be calculated based on the number of C60 molecules in the unit cell.

Molar Mass and Avogadro's Number

The molar mass of a substance is the mass of one mole of its entities, typically expressed in grams per mole (g/mol). For buckminsterfullerene (C60), the molar mass is approximately 720 g/mol. Avogadro's number (6.022 x 10²³) relates the number of molecules in a mole to its mass, allowing for the conversion between the mass of the unit cell and the number of molecules it contains, which is essential for density calculations.