   # Titanium metal has a body-centered cubic unit cell. The density of titanium is 4.50 g/cm 3 . Calculate the edge length of the unit cell and a value for the atomic radius of titanium. (Hint: In a body-centered arrangement of spheres, the spheres touch across the body diagonal.) ### Chemistry: An Atoms First Approach

2nd Edition
Steven S. Zumdahl + 1 other
Publisher: Cengage Learning
ISBN: 9781305079243

#### Solutions

Chapter
Section ### Chemistry: An Atoms First Approach

2nd Edition
Steven S. Zumdahl + 1 other
Publisher: Cengage Learning
ISBN: 9781305079243
Chapter 9, Problem 57E
Textbook Problem
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## Titanium metal has a body-centered cubic unit cell. The density of titanium is 4.50 g/cm3. Calculate the edge length of the unit cell and a value for the atomic radius of titanium. (Hint: In a body-centered arrangement of spheres, the spheres touch across the body diagonal.)

Interpretation Introduction

Interpretation:

The edge length of the unit cell of Titanium has to be determined.

Concept introduction:

In packing of atoms in a crystal structure, the atoms are imagined as spheres. The two major types of close packing of the spheres in the crystal are – hexagonal close packing and cubic close packing.

In body-centered cubic unit cell, each of the six corners is occupied by every single atom. Center of the cube is occupied by one atom.

Each atom in the corner is shared by eight unit cells and a single atom in the center of the cube remains unshared. Thus the number of atoms per unit cell in BCC unit cell is,

8×18atomsincorners+1atomatthecenter=1+1=2atoms       The edge length of one unit cell is given bya=4R3where  a=edgelength of unit cellR=radiusofatom.

### Explanation of Solution

Explanation

Calculate the mass of unit cell of Ti.

Average mass of one Ti atom=atomicmassofTiAvogadronumber=47.9g6.022×1023=7.95×10-23gEachunit cell contains2Ti atoms. therefore,Massofaunitcell=2×7.95×10-23g  =15.9×10-23g

Each unit cell contains 2 Ti atoms. Therefore two times the average mass of one Ti atom gives mass of a unit cell.

Calculate the volume of unit cell of Ti.

knowndata:mass=15.9×10-23gdensity=4

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