Problem 71 - Chemistry: The Central Science

Question

Gold is a face-centered cubic structure that has a unit cell edge length of \(4.08 \AA\) (Figure 11.34). How many gold atoms are there in a sphere that is \(20 \mathrm{~nm}\) in diameter? Recall that the volume of a sphere is \(\frac{4}{3} \pi r^{3}\).

Step-by-Step Solution

Answer

The short answer is: First, calculate the sphere's volume to be \(V_{sphere} = \frac{4}{3}\pi (100 \ \text{Å})^{3}\). Next, compute the volume of the face-centered cubic unit cell, \(V_{unit\ cell} = (4.08 \ \text{Å})^{3}\). Divide the sphere's volume by the unit cell volume to determine the number of unit cells in the sphere. Finally, multiply the number of unit cells by 4 (since there are 4 gold atoms per unit cell) to obtain the total number of gold atoms.

Step 1: Calculate the volume of the sphere

Given the diameter of the sphere is 20 nm, we can find the radius r by dividing the diameter by 2: \(r = \frac{20 \ \text{nm}}{2} = 10 \ \text{nm}\) Now, convert the radius from nanometers to angstroms because the cell edge length is given in angstroms: 1 nm = 10 Å, so 10 nm = 100 Å. Now that the radius is in the same units as the cell edge length, we can calculate the volume of the sphere using the given formula: \(V_{sphere} = \frac{4}{3}\pi r^{3}\) \(V_{sphere} = \frac{4}{3}\pi (100 \ \text{Å})^{3}\) Now compute for the volume of the sphere.

Step 2: Determine the volume of the gold atoms in a unit cell

Gold has a face-centered cubic structure with a unit cell edge length of 4.08 Å. The volume V of a cube is equal to the edge length cubed: \(V_{unit\ cell} = (4.08 \ \text{Å})^{3}\) Compute for the volume of the unit cell.

Step 3: Calculate the number of unit cells in the sphere

Divide the volume of the sphere by the volume of the unit cell to determine the number of unit cells in the sphere: \(\text{Number of unit cells} = \frac{V_{sphere}}{V_{unit\ cell}}\)

Step 4: Determine the total number of gold atoms in the sphere

In a face-centered cubic structure, there are 4 gold atoms per unit cell. Multiply the number of unit cells by 4 to get the total number of gold atoms in the sphere: \(\text{Total number of gold atoms} = \text{Number of unit cells} \times 4\) Now, compute for the total number of gold atoms.

Key Concepts

Understanding the Unit Cell in a Face-Centered Cubic Structure

Picture a tiny cube, known as a unit cell, which repeats over and over to form the crystalline structure of gold. In a face-centered cubic (FCC) structure, each unit cell contains atoms at each of the cube's corners and one atom in the center of each face of the cube. This arrangement results in four atoms being part of each unit cell because each corner atom is shared by eight neighboring cubes, and each face-centered atom is shared with one adjoining unit cell. To figure out the volume of this unit cell, you use the formula for the volume of a cube, which is the edge length raised to the power of three. For gold, with an edge length of 4.08 Å, the unit cell volume is:
- \( V_{unit\_cell} = (4.08\ \text{Å})^{3} \).
Understanding the structure of the unit cell helps in calculating how gold atoms are packed in a given space, such as in a sphere.

Calculating the Volume of a Sphere

A sphere can be visualized as a perfectly-shaped three-dimensional object where every point on its surface is equidistant from its center. When given the diameter, as we have with the gold sphere at 20 nm, you find the radius by halving this value, thus getting 10 nm. The next step is converting this radius into angstroms for consistency with the unit cell measurement (since 1 nm = 10 Å, 10 nm = 100 Å). With this conversion, you apply the formula for the volume of a sphere:
- \( V_{sphere} = \frac{4}{3}\pi r^{3} \).
By plugging in the radius, 100 Å, you compute the sphere's volume:
- \( V_{sphere} = \frac{4}{3}\pi (100 \ \text{Å})^{3} \).
This calculation gives you the total volume available for packing gold atoms.

Gold Atoms in a Face-Centered Cubic Sphere

Gold is a well-known element that forms a face-centered cubic structure in its solid state. With 4 gold atoms per unit cell in this arrangement, these atoms tightly pack together to fill the space effectively. To determine how many gold atoms fit in a given sphere, you first calculate how many unit cells fit by dividing the sphere's volume by the unit cell's volume:

• Find the unit cell count by: \( \text{Number of unit cells} = \frac{V_{sphere}}{V_{unit\_cell}} \).
• Remember, each unit cell can accommodate 4 gold atoms.
• Therefore, the total number of gold atoms is: \( \text{Total number of gold atoms} = \text{Number of unit cells} \times 4 \).

This stepwise approach provides a clear picture of how gold atoms can fill a spherical space, a crucial concept for understanding material properties at the atomic scale. Understanding this arrangement helps in furthering insights into both chemistry and physics of materials.