/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 28 What is the minimum number of at... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 28

What is the minimum number of atoms that could be contained in the unit cell of an element with a face-centered cubic lattice? \((\mathbf{a}) 1,(\mathbf{b}) 2,(\) (c) 3 (d) 4 (e) 5.

Short Answer

Expert verified
In a face-centered cubic (FCC) lattice, there are 8 corner atoms and 6 face atoms. The corner atoms contribute 1 atom to the unit cell, while the face atoms contribute 3 atoms to the unit cell. Therefore, the minimum number of atoms in the unit cell of an element with a face-centered cubic lattice is \(1 + 3 = 4\). The correct answer is (d) 4.

Step by step solution

01

The face-centered cubic lattice structure

An FCC lattice has atoms positioned at every corner of the cube and at the center of each face. Each unit cell in an FCC lattice has 8 corner atoms and 6 face atoms. However, since the corner atoms are shared by 8 adjacent unit cells, and the face atoms are shared by 2 adjacent unit cells, we need to determine their contributions to the unit cell.
02

Calculate the contribution of the corner atoms

Each corner atom is shared by 8 unit cells. So, the contribution of a corner atom to a single unit cell is \( \frac{1}{8}\). Since there are 8 corner atoms, their total contribution to the unit cell is: \[8 \times \frac{1}{8} = 1\]
03

Calculate the contribution of the face atoms

Each face atom is shared by 2 unit cells. So, the contribution of a face atom to a single unit cell is \( \frac{1}{2}\). Since there are 6 face atoms, their total contribution to the unit cell is: \[6 \times \frac{1}{2} = 3\]
04

Determine the total number of atoms in the unit cell

Now we need to add the contributions from both the corner atoms and the face atoms to find the total number of atoms in the unit cell: \[1 + 3 = 4\] So, the minimum number of atoms that could be contained in the unit cell of an element with a face-centered cubic lattice is 4. The correct answer is (d) 4.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Unit Cell
The unit cell is the smallest structural component of a crystal lattice. It acts as the building block, repeating in three-dimensional space to form the entire crystal. In a face-centered cubic (FCC) lattice, the unit cell is a cube where the atoms are organized at specific points:
  • At each of the eight corners
  • At the center of each of the six faces
This arrangement is characteristic of the FCC lattice and is crucial for calculating how many atoms are actually within the unit cell. By understanding how these atoms are shared between neighboring unit cells, we gain insight into the makeup of materials at the atomic level.
Corner Atoms and Their Contribution
In a crystal structure, corner atoms are located at the vertices of the unit cell. A key understanding is that these atoms are not exclusive to a single cell but are shared by eight adjacent unit cells. Consequently, the contribution of a single corner atom to one unit cell is only \( \frac{1}{8} \) of an atom.
Given there are eight such corner atoms in a face-centered cubic unit cell:
  • The cumulative contribution of all corner atoms is calculated by multiplying:\[ 8 \times \frac{1}{8} = 1 \]atom per unit cell.
With this understanding, we see that these corner atoms collectively contribute one whole atom to the structure of the unit cell.
The Role of Face Atoms
Face atoms are positioned at the centers of each cube face in a face-centered cubic lattice. In contrast to corner atoms, each face atom is shared between only two unit cells. This means the contribution of each face atom is half an atom to a single unit cell. There are six face atoms in an FCC unit cell leading to:
  • The overall contribution calculated by:\[ 6 \times \frac{1}{2} = 3 \]atoms to the unit cell.
Thus, these face atoms contribute significantly to the total atom count, playing a vital role in the stability and properties of the crystal.
Comprehending the Crystal Lattice Structure
A crystal lattice structure refers to the orderly arrangement of atoms in a crystalline solid. The face-centered cubic (FCC) lattice is a specific type of structure where cubes repeat in a symmetrical pattern, providing both stability and high packing density. Each unit cell connects seamlessly to its neighbors, forming a continuous three-dimensional grid.
In the FCC structure:
  • Atoms are arranged efficiently, reducing empty space.
  • It contributes to the material's properties, such as ductility and strength.
Understanding the way units like FCC lattices compose a crystal helps explain the behavior and characteristics of different crystalline materials, which is crucial in materials science and engineering.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The rutile and fluorite structures, shown here (anions are colored green), are two of the most common structure types of ionic compounds where the cation to anion ratio is 1: 2 . (a) For \(\mathrm{CaF}_{2}\) and \(\mathrm{ZnF}_{2}\) use ionic radii, \(\mathrm{Ca}^{2+}(r=114 \mathrm{pm})\), \(\mathrm{Zn}^{2+}(r=88 \mathrm{pm})\), and \(\mathrm{F}^{-}(r=119 \mathrm{pm})\), to predict which compound is more likely to crystallize with the fluorite structure and which with the rutile structure. (b) What are the coordination numbers of the cations and anions in each of these structures?

Potassium metal (atomic weight \(39.10 \mathrm{~g} / \mathrm{mol}\) ) adopts a body-centered cubic structure with a density of \(0.856 \mathrm{~g} / \mathrm{cm}^{3}\). (a) Use this information and Avogadro's number \((6.022 \times\) \(10^{23}\) ) to estimate the atomic radius of potassium. (b) If potassium didn't react so vigorously, it could float on water. Use the answer from part (a) to estimate the density of \(\mathrm{K}\) if its structure were that of a cubic close-packed metal. Would it still float on water?

Calcium crystallizes in a face-centered cubic unit cell at room temperature that has an edge length of \(558.8 \mathrm{pm}\). (a) Calculate the atomic radius of a calcium atom. (b) Calculate the density of Ca metal at this temperature.

Silicon is the fundamental component of integrated circuits. Si has the same structure as diamond. (a) Is Si a molecular, metallic, ionic, or covalent- network solid? (b) Silicon readily reacts to form silicon dioxide, \(\mathrm{SiO}_{2,}\) which is quite hard and is insoluble in water. Is \(\mathrm{SiO}_{2}\) most likely a molecular, metallic, ionic, or covalent- network solid?

What molecular structural features cause high-density polyethylene to be denser than low-density polyethylene?
See all solutions

Recommended explanations on Chemistry Textbooks

Nuclear Chemistry

Read Explanation

Organic Chemistry

Read Explanation

Chemical Analysis

Read Explanation

Inorganic Chemistry

Read Explanation

Chemistry Branches

Read Explanation

Chemical Reactions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.