Physics for Scientists and Engineers
Physics for Scientists and Engineers
6th Edition
ISBN: 9781429281843
Author: Tipler
Publisher: MAC HIGHER
Question
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Chapter 38, Problem 31P

(a)

To determine

The free-electron density in gold.

(a)

Expert Solution
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Answer to Problem 31P

The free-electron density in goldis 5.90×1028e/m3 .

Explanation of Solution

Given:

The density of gold is ρ=19.3×103kg/m3 .

The atomic mass is m=196.97g/mol .

Formula used:

The expression for free-electron density is given by

  ne=ρNANatomm

Here, NA is Avogadro’s constant and Natom is number of electron per atom.

Calculation:

The free electron density of goldis calculated as,

  ne=( 19.3× 10 3 kg/ m 3 )( 6.022× 10 23 atoms/ mol )( 1e 1atom )( 196.97g/ mol )=( 19.3× 10 3 kg/ m 3 )( 6.022× 10 23 atoms/ mol )( 1e 1atom )( 196.97g/ mol )( 10 3 kg/ mol 1g/ mol )=5.90×1028e/m3

Conclusion:

Therefore, the free electron density in gold is 5.90×1028e/m3 .

(b)

To determine

The Fermi energy for gold.

(b)

Expert Solution
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Answer to Problem 31P

The Fermi energy for goldis 5.50eV .

Explanation of Solution

Given:

The Fermi speed for goldis vF=1.39×106m/s .

Formula used:

The expression for Fermi energy is given by,

  EF=12mevF2

Here, me is the mass of the electron.

Calculation:

The Fermi energy for gold is calculated as,

  EF=12(9.109× 10 31kg)(1.39× 10 6m/s)2=(8.7997× 10 19J)( 1eV 1.602× 10 19 J)=5.4929eV5.50eV

Conclusion:

Therefore, the Fermi energy for gold is 5.50eV .

(c)

To determine

The factor between Fermi energy and kT energy.

(c)

Expert Solution
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Answer to Problem 31P

The factor between Fermi energy and kT energyis 212 .

Explanation of Solution

Given:

The kT energy at room temperature is kT=0.026eV .

Formula used:

The expression for required factor is given by,

  f=EFkT

Calculation:

The required factor is calculated as,

  f=5.50eV0.026eV=211.5212

Conclusion:

Therefore, the factor by between Fermi energy and kT energy is 212 .

(d)

To determine

The difference between Fermi energy and kT energy.

(d)

Expert Solution
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Explanation of Solution

Introduction:

The difference between higher and lower energy level that is occupied by the charged particle of material at 0K is called Fermi energy.

kT energy of the average conduction electron at any temperature T and it is always less than or equal to Fermi energy.

At absolute zero, the energy available at conduction electron in a higher energy state is termed as Fermi energy. It is higher than or equal to kT energy because no two electrons can occupy the same higher energy state simultaneously.

When the electron does not obey the exclusion principle, the energy of average conduction electrons at any temperature T is called kT energy.

Conclusion:

Therefore, the Fermi energy is always greater than or equal to kT energy.

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