Chemistry: An Atoms-Focused Approach
Chemistry: An Atoms-Focused Approach
14th Edition
ISBN: 9780393600681
Author: Gilbert
Publisher: W. W. Norton & Company
Question
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Chapter 18, Problem 18.125QA
Interpretation Introduction

To find:

a)  The density of bcc iron, assuming the radius of an iron atom is  126 pm.

b) The density of hexagonal iron, given a unit cell volume 5.414 ×10-23 cm3

c) The density of Si substituted Fe crystal.

Expert Solution & Answer
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Answer to Problem 18.125QA

Solution:

a) The density of bcc iron is 7.53 g/cm3.

b) The density of hexagonal iron is 3.42 g/cm3.

c) The density of silicon substituted iron crystal is 3.38 g/cm3.

Explanation of Solution

1) Concept:

The unit cell of the iron is given as bcc unit cell. The edge length can be calculated for the given unit cell as the radius of the Fe is known, by the relation  4r= 3l. From the edge length of the unit cell, we can calculate the unit cell volume. We will calculate the mass of the unit cell from the number of atoms present in unit cell. From volume and mass of the unit cell, we will calculate the density of the unit cell.

2) Formula:

i)  4r= 3l

i) Vunit cell=l3 cm3     

ii) density= massunit cellVunit cell

3) Given:

i) Atomic radii of Fe=126 pm=126 pm × 1.0 ×10-10cm1 pm= 1.26 ×10-8cm

ii) Molar mass of Fe=55.845 g/mol

iii) Number of atoms in unit cell =2

iv) Avogadro’s number 1 mol =6.023 ×1023 atoms

v) Volume of the hexagonal unit of Fe=5.414 ×10-23 cm3

vi) Molar mass of Si=28.08 g/mol

4) Calculations:

a) Calculate the density of bcc iron, assuming the radius of an iron atom is  126 pm.

Since the unit cell of Fe in this case is bcc, there are two atoms of Fe present in the unit cell.

2 atoms ×1 mol6.022 ×1023atom  × 55.845 g1 mol =1.855 ×10-22g

Mass of the unit cell is 1.855 ×10-22g.

Calculate the edge length from atomic radius.

l= 4r3= 4 × 1.26 ×10-8cm3=2.9098 ×10-8cm

Volume of unit cell

Vunit cell=l3 cm3= 2.9098 ×10-83 cm3=2.4638 ×10-23 cm3

Density of the bcc iron

density, d= massunit cellVunit cell

d= 1.855 ×10-22 g2.4638 ×10-23 cm3

d=7.53 g/cm3

Hence, the density of the Fe in bcc unit cell is 7.53 g/cm3.

b) Calculate the density of hexagonal iron, given a unit cell volume 5.414 ×10-23 cm3.

Since the unit cell of Fe in this case is bcc, there are two atoms of Fe present in the unit cell.

2 atoms ×1 mol6.022 ×1023atom  × 55.845 g1 mol =1.855 ×10-22g

Mass of the unit cell is 1.855 ×10-22g.

Density of the hexagonal iron

density, d= massunit cellVunit cell

d= 1.855 ×10-22 g5.414 ×10-23 cm3

d=3.42g/cm3

Hence, the density of Fe in hexagonal form is 3.42 g/cm3.

c) Calculate the density of Si substituted Fe crystal.

It is given that 4% by mass of Si is substituted for Fe. This means that from the total mass of the Fe unit cell, 4% has been replaced by Si.

The mass of Fe remaining in the unit cell is calculated as

mass of Fe remaining=2 atoms Fe unit cell×55.845gmol- 4 ×28.0855100gmol

mass of Fe remaining=111.69-1.12342 g/mol

mass of Fe remaining=110.56658 g/mol

Mass of iron in unit cell is

110.57 g1 mol ×1 mol6.022 ×1023 atoms=1.83 ×10-22 g

Density of the system

density= massunit cellVunit cell

density= 1.83 ×10-22 g5.414 ×10-23 cm3=3.38 g/cm3

Hence the density of such a crystal is 3.38 g/cm3.

Conclusion:

The density of the Fe atom in bcc unit cell is calculated from the edge length and volume of the unit cell.

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Chapter 18 Solutions

Chemistry: An Atoms-Focused Approach

Ch. 18 - Prob. 18.11VPCh. 18 - Prob. 18.12VPCh. 18 - Prob. 18.13VPCh. 18 - Prob. 18.14VPCh. 18 - Prob. 18.15VPCh. 18 - Prob. 18.16VPCh. 18 - Prob. 18.17VPCh. 18 - Prob. 18.18VPCh. 18 - Prob. 18.19QACh. 18 - Prob. 18.20QACh. 18 - Prob. 18.21QACh. 18 - Prob. 18.22QACh. 18 - Prob. 18.23QACh. 18 - Prob. 18.24QACh. 18 - Prob. 18.25QACh. 18 - Prob. 18.26QACh. 18 - Prob. 18.27QACh. 18 - Prob. 18.28QACh. 18 - Prob. 18.29QACh. 18 - Prob. 18.30QACh. 18 - Prob. 18.31QACh. 18 - Prob. 18.32QACh. 18 - Prob. 18.33QACh. 18 - Prob. 18.34QACh. 18 - Prob. 18.35QACh. 18 - Prob. 18.36QACh. 18 - Prob. 18.37QACh. 18 - Prob. 18.38QACh. 18 - Prob. 18.39QACh. 18 - Prob. 18.40QACh. 18 - Prob. 18.41QACh. 18 - Prob. 18.42QACh. 18 - Prob. 18.43QACh. 18 - Prob. 18.44QACh. 18 - Prob. 18.45QACh. 18 - Prob. 18.46QACh. 18 - Prob. 18.47QACh. 18 - Prob. 18.48QACh. 18 - Prob. 18.49QACh. 18 - Prob. 18.50QACh. 18 - Prob. 18.51QACh. 18 - Prob. 18.52QACh. 18 - Prob. 18.53QACh. 18 - Prob. 18.54QACh. 18 - Prob. 18.55QACh. 18 - Prob. 18.56QACh. 18 - Prob. 18.57QACh. 18 - Prob. 18.58QACh. 18 - Prob. 18.59QACh. 18 - Prob. 18.60QACh. 18 - Prob. 18.61QACh. 18 - Prob. 18.62QACh. 18 - Prob. 18.63QACh. 18 - Prob. 18.64QACh. 18 - Prob. 18.65QACh. 18 - Prob. 18.66QACh. 18 - Prob. 18.67QACh. 18 - Prob. 18.68QACh. 18 - Prob. 18.69QACh. 18 - Prob. 18.70QACh. 18 - Prob. 18.71QACh. 18 - Prob. 18.72QACh. 18 - Prob. 18.73QACh. 18 - Prob. 18.74QACh. 18 - Prob. 18.75QACh. 18 - Prob. 18.76QACh. 18 - Prob. 18.77QACh. 18 - Prob. 18.78QACh. 18 - Prob. 18.79QACh. 18 - Prob. 18.80QACh. 18 - Prob. 18.81QACh. 18 - Prob. 18.82QACh. 18 - Prob. 18.83QACh. 18 - Prob. 18.84QACh. 18 - Prob. 18.85QACh. 18 - Prob. 18.86QACh. 18 - Prob. 18.87QACh. 18 - Prob. 18.88QACh. 18 - Prob. 18.89QACh. 18 - Prob. 18.90QACh. 18 - Prob. 18.91QACh. 18 - Prob. 18.92QACh. 18 - Prob. 18.93QACh. 18 - Prob. 18.94QACh. 18 - Prob. 18.95QACh. 18 - Prob. 18.96QACh. 18 - Prob. 18.97QACh. 18 - Prob. 18.98QACh. 18 - Prob. 18.99QACh. 18 - Prob. 18.100QACh. 18 - Prob. 18.101QACh. 18 - Prob. 18.102QACh. 18 - Prob. 18.103QACh. 18 - Prob. 18.104QACh. 18 - Prob. 18.105QACh. 18 - Prob. 18.106QACh. 18 - Prob. 18.107QACh. 18 - Prob. 18.108QACh. 18 - Prob. 18.109QACh. 18 - Prob. 18.110QACh. 18 - Prob. 18.111QACh. 18 - Prob. 18.112QACh. 18 - Prob. 18.113QACh. 18 - Prob. 18.114QACh. 18 - Prob. 18.115QACh. 18 - Prob. 18.116QACh. 18 - Prob. 18.117QACh. 18 - Prob. 18.118QACh. 18 - Prob. 18.119QACh. 18 - Prob. 18.120QACh. 18 - Prob. 18.121QACh. 18 - Prob. 18.122QACh. 18 - Prob. 18.123QACh. 18 - Prob. 18.124QACh. 18 - Prob. 18.125QACh. 18 - Prob. 18.126QACh. 18 - Prob. 18.127QACh. 18 - Prob. 18.128QACh. 18 - Prob. 18.129QACh. 18 - Prob. 18.130QACh. 18 - Prob. 18.131QACh. 18 - Prob. 18.132QACh. 18 - Prob. 18.133QA
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