Ch.12 - Solids and Modern Materials

Problem 130a

Chapter 12, Problem 130a

Silicon has the diamond structure with a unit cell edge length of 5.43 Å and eight atoms per unit cell. (a) How many silicon atoms are there in 1 cm³ of material?

Verified step by step guidance

First, convert the unit cell edge length from angstroms to centimeters. Since 1 Å = 1 x 10^-8 cm, multiply 5.43 Å by 1 x 10^-8 to get the edge length in centimeters.

Calculate the volume of the unit cell in cubic centimeters. Use the formula for the volume of a cube: V = a^3, where 'a' is the edge length in centimeters.

Determine the number of unit cells in 1 cm^3 of silicon. Divide 1 cm^3 by the volume of one unit cell (calculated in the previous step) to find the number of unit cells per cubic centimeter.

Since there are eight silicon atoms per unit cell, multiply the number of unit cells per cubic centimeter by 8 to find the total number of silicon atoms in 1 cm^3.

Ensure all units are consistent and verify each calculation step to ensure accuracy in determining the number of silicon atoms per cubic centimeter.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Unit Cell and Atomic Packing

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 silicon, which has a diamond cubic structure, each unit cell contains eight silicon atoms. Understanding the arrangement and number of atoms in a unit cell is crucial for calculating the density and number of atoms in a given volume.

Volume Conversion

To find the number of silicon atoms in 1 cm³ of material, it is essential to convert units appropriately. Since the unit cell edge length is given in angstroms (Å), which is a unit of length equal to 10^-10 meters, converting this to centimeters (1 cm = 10^8 Å) is necessary for accurate calculations. This conversion allows for the determination of how many unit cells fit into 1 cm³.

Density and Avogadro's Number

Density is defined as mass per unit volume and is a key factor in determining the number of atoms in a given volume. To find the total number of silicon atoms in 1 cm³, one can use the density of silicon and Avogadro's number (approximately 6.022 x 10²³ atoms/mol) to relate the mass of silicon in that volume to the number of atoms. This relationship is fundamental in solid-state chemistry and materials science.

Recommended video:

Guided course

01:56

Density Concepts

Related Practice

Textbook Question

(a) In polyvinyl chloride shown in Table 12.6, which bonds have the lowest average bond enthalpy?

Textbook Question

(b) When subjected to high pressure and heated, polyvinyl chloride converts to diamond. During this transformation which bonds are most likely to break first?

Textbook Question

(c) Employing the values of average bond enthalpy in Table 8.3, estimate the overall enthalpy change for converting PVC to diamond.

Textbook Question

Silicon has the diamond structure with a unit cell edge length of 5.43 Å and eight atoms per unit cell. (b) Suppose you dope that 1 cm3 sample of silicon with 1 ppm of phosphorus that will increase the conductivity by a factor of a million. How many milligrams of phosphorus are required?

Textbook Question

One method to synthesize ionic solids is by the heating of two reactants at high temperatures. Consider the reaction of FeO with TiO₂ to form FeTiO₃. Determine the amount of each of the two reactants to prepare 2.500 g FeTiO₃, assuming the reaction goes to completion. (a) Write a balanced chemical reaction. (c) Determine the moles of FeTiO₃.

Textbook Question

Silicon has the diamond structure with a unit cell edge length of 5.43 Å and eight atoms per unit cell. (a) How many silicon atoms are there in 1 cm3 of material?

Verified video answer for a similar problem:

Key Concepts

Unit Cell and Atomic Packing

Volume Conversion

Density and Avogadro's Number

Silicon has the diamond structure with a unit cell edge length of 5.43 Å and eight atoms per unit cell. (a) How many silicon atoms are there in 1 cm³ of material?