15. The Fermi-Dirac occupation probability is given by P
FD
= 1/
e
∆E/kT
+ 1
, and the Boltzmann occu-
pation probability is given by P
B
= e
−∆E/kT
. L et f be the fractional difference. Then
f =
P
B
− P
FD
P
B
=
e
−∆E/kT
−
1
e
∆E/kT
+ 1
e
−∆E/kT
.
Using a common denominator and a little algebra yields
f =
e
−∆E/kT
e
−∆E/kT
+ 1
.
The solution for e
−∆E/kT
is
e
−∆E/kT
=
f
1
− f
.
We take the natural logarithm of both sides and solve for T . The result is
T =
∆E
k ln
f
1
− f
.
(a) Letting f equal 0.01, we evaluate the expression for T :
T =
(1.00 eV)(1.60
× 10
−19
J/eV)
(1.38
× 10
−23
J/K) ln
0.010
1
− 0.010
= 2.5 × 10
3
K .
(b) We set f equal to 0.10 and evaluate the expression for T :
T =
(1.00 eV)(1.60
× 10
−19
J/eV)
(1.38
× 10
−23
J/K) ln
0.10
1
− 0.10
= 5.3 × 10
3
K .
Document Outline
- Main Menu
- Chapter 1 Measurement
- Chapter 2 Motion Along a Straight Line
- Chapter 3 Vectors
- Chapter 4 Motion in Two and Three Dimensions
- Chapter 5 Force and Motion I
- Chapter 6 Force and Motion II
- Chapter 7 Kinetic Energy and Work
- Chapter 8 Potential Energy and Conservation of Energy
- Chapter 9 Systems of Particles
- Chapter 10 Collisions
- Chapter 11 Rotation
- Chapter 12 Rolling, Torque, and Angular Momentum
- Chapter 13 Equilibrium and Elasticity
- Chapter 14 Gravitation
- Chapter 15 Fluids
- Chapter 16 Oscillations
- Chapter 17 Waves—I
- Chapter 18 Waves—II
- Chapter 19 Temperature, Heat, and the First Law of Thermodynamics
- Chapter 20 The Kinetic Theory of Gases
- Chapter 21 Entropy and the Second Law of Thermodynamics
- Chapter 22 Electric Charge
- Chapter 23 Electric Fields
- Chapter 24 Gauss’ Law
- Chapter 25 Electric Potential
- Chapter 26 Capacitance
- Chapter 27 Current and Resistance
- Chapter 28 Circuits
- Chapter 29 Magnetic Fields
- Chapter 30 Magnetic Fields Due to Currents
- Chapter 31 Induction and Inductance
- Chapter 32 Magnetism of Matter: Maxwell’s Equation
- Chapter 33 Electromagnetic Oscillations and Alternating Current
- Chapter 34 Electromagnetic Waves
- Chapter 35 Images
- Chapter 36 Interference
- Chapter 37 Diffraction
- Chapter 38 Special Theory of Relativity
- Chapter 39 Photons and Matter Waves
- Chapter 40 More About Matter Waves
- Chapter 41 All About Atoms
- Chapter 42 Conduction of Electricity in Solids
- 42.1 - 42.10
- 42.1
- 42.2
- 42.3
- 42.4
- 42.5
- 42.6
- 42.7
- 42.8
- 42.9
- 42.10
- 42.11 - 42.20
- 42.11
- 42.12
- 42.13
- 42.14
- 42.15
- 42.16
- 42.17
- 42.18
- 42.19
- 42.20
- 42.21 - 42.30
- 42.21
- 42.22
- 42.23
- 42.24
- 42.25
- 42.26
- 42.27
- 42.28
- 42.29
- 42.30
- 42.31 - 42.40
- 42.31
- 42.32
- 42.33
- 42.34
- 42.35
- 42.36
- 42.37
- 42.38
- 42.39
- 42.40
- 42.41 - 42.49
- 42.41
- 42.42
- 42.43
- 42.44
- 42.45
- 42.46
- 42.47
- 42.48
- 42.49
- 42.1 - 42.10
- Chapter 43 Nuclear Physics
- Chapter 44 Energy from the Nucleus
- Chapter 45 Quarks, Leptons, and the Big Bang
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