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Chapter 12: Problem 71

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 \(\frac{4}{3} \pi r^{3}\).

Short Answer

Expert verified
There are approximately 247111 gold atoms in a sphere with a diameter of 20 nm, given that gold has a face-centered cubic structure with a unit cell edge length of 4.08 脜.

Step by step solution

01

Convert Edge Length to Nanometers

First, we need to convert the unit cell edge length from 脜 (angstroms) to nm (nanometers), as we need to work with the same unit of measurement. The conversion factor is 1 脜 = 0.1 nm. Thus, we have: Unit cell edge length in nm = \(4.08 脜 \times 0.1 nm/脜 = 0.408 nm\)
02

Calculate the Volume of the Unit Cell

The volume of a cube with edge length 'a' is given by the formula \(V = a^{3}\). Using the edge length in nm, we can find the volume of the unit cell: Volume of unit cell = \((0.408 nm)^{3} = 0.0678 nm^{3}\)
03

Calculate the Volume of the Sphere

The volume of a sphere with radius 'r' is given by the formula \(V = \frac{4}{3}\pi r^{3}\). To find the volume of the sphere with a diameter of 20 nm, we need to divide the diameter by 2 to determine the radius: Radius of sphere = \(20 nm / 2 = 10 nm\) Now we can calculate the volume of the sphere: Volume of sphere = \(\frac{4}{3}\pi (10 nm)^{3} = \frac{4}{3}\pi (1000 nm^{3}) = 4188.79 nm^{3}\)
04

Calculate the Number of Unit Cells in the Sphere

To find out how many unit cells fit into the sphere, divide the volume of the sphere by the volume of the unit cell: Number of unit cells = \(\frac{Volume\ of\ sphere}{Volume\ of\ unit\ cell} = \frac{4188.79 nm^{3}}{0.0678 nm^{3}} = 61777.7\)
05

Calculate the Number of Gold Atoms in the Sphere

Since there are 4 gold atoms in each unit cell, we can calculate the total number of gold atoms in the sphere by multiplying the number of unit cells by 4: Number of gold atoms = \(Number\ of\ unit\ cells \times 4 = 61777.7 \times 4 = 247110.8\) Because the number of atoms must be an integer, we will round this result to the nearest whole number: Number of gold atoms in the sphere 鈮� 247111

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