Thermodynamics and Statistical
Mechanics
Blackbody Radiation
Thermo & Stat Mech Spring 2006 Class 21
1
Classical Energy Equipartition
  1   2   3
Z  Z1 Z 2 Z 3
ln Z  ln Z1  ln Z 2  ln Z 3
  ln Z1    ln Z 2    ln Z 3 
U
  
  

 
N
  V   V   V
U
   1   2   3
N
Thermo & Stat Mech - Spring 2006
Class 21
2
Classical Energy Equipartition
2

p
T ranslat ion (1 Dim)  
2m
Z  e

2
Rot at ion
L

2I
Spring P ot ent ial
  12 kx
All of form

Z  e
p2

2m

Z  e
L2

2I

2

Z  e

  az 2

Thermo & Stat Mech - Spring 2006
Class 21
dp
dL
  12 kx2
dx
dz
3
Classical Energy Equipartition

Z  e

  az 2

1
dz 

a


a

1

ln Z   ln   ln
  ln   ln
a
2
a
 ln Z
1
1
 

where  

2
kT
kT
 
2
Thermo & Stat Mech - Spring 2006
Class 21
4
Quantum Harmonic Oscillator
Averageexcitationenergy
U

1
    
where  
N
e
1
kT
Bose - Einstein view
energy

 avg. no.exitations
exitation
1
     
e
1
Thermo & Stat Mech - Spring 2006
Class 21
5
Energy of EM Waves in a Cavity
Calculate the number of standing wave states
in the frequency range between  and  + d.
Then Bose-Einstein statistics is used to
determine the number of photons in each state.
With that information the energy can be
calculated.
Thermo & Stat Mech - Spring 2006
Class 21
6
Density of States (Lec 17)
V 2
g (k )dk  
k dk
2
2

 
k
dk  d  
c
c
V
2
g ( )d  
 d
2 3
2 c
Thermo & Stat Mech - Spring 2006
Class 21
7
Density of States for Photons
g ( )d  
V
 d
2
2 c
  2 T wo polarizations
V
2
g ( )d  2 3  d
 c
2 3
Thermo & Stat Mech - Spring 2006
Class 21
8
Photon Energy
dU   f ( ) g ( )d
 1  V
2
     3 2  d
 e 1  c 
dU
 d
du 
 3 2  
V
c  e 1
3

Thermo & Stat Mech - Spring 2006
Class 21

9
Photon Energy
dU
 d
du 
 3 2  
V
c  e 1
Use,   
3


 d
8  d
du  3 3 2  
 3 3 
 c  e  1 h c e  1
3
Thermo & Stat Mech - Spring 2006
Class 21
3
10
Graph
Energy Density
Black Body Radiation
4.5E+18
4E+18
3.5E+18
3E+18
2.5E+18
2E+18
1.5E+18
1E+18
5E+17
0
5000
6000
7000
0
1
2
3
4
5
Photon Energy (eV)
Thermo & Stat Mech - Spring 2006
Class 21
11
Total Energy Density
8
u 3 3
hc

 d
3
8
x dx
4
 3 3 (kT )  x
1 h c
e 1
0
 e  
0

3


5 4

8
8 k  4
4 

u  3 3 (kT )
 
T
3 3 
hc
15  15h c 
4
16
-3 - 4
u  aT a  7.5510 J  m K
4
Thermo & Stat Mech - Spring 2006
Class 21
12
Energy Flux
1
Particleflux (Lec14)   n v
4
1
1
4
Energyflux e  uc  caT
4
4
5 4
1
2 k
4
e  T
  ca 
3 2
4
15h c
8
-2 -4
  5.67 10 W  m K
Thermo & Stat Mech - Spring 2006
Class 21
13
Human Eye and the Sun
It is often stated in textbooks that the peak
sensitivity of the human eye is at the
wavelength at which the energy radiated
by the sun is maximum.
Thermo & Stat Mech - Spring 2006
Class 21
14
Human Eye
Peak at 555 nm
Thermo & Stat Mech - Spring 2006
Class 21
15
A Problem?
The graph of the energy density as a
function of frequency, for a blackbody at
a temperature of 5800 K, has a maximum
at 1.41 eV. Photons of that energy have a
wavelength of 879 nm, which is out of the
visible spectrum, in the infrared. What is
wrong?
Thermo & Stat Mech - Spring 2006
Class 21
16
Solution
The energy density is plotted as a function
of energy, while the eye’s sensitivity is
plotted as a function of wavelength. If we
plot the energy density as a function of
wavelength, its peak does not occur at 879
nm. How can this be?
Thermo & Stat Mech - Spring 2006
Class 21
17
Change to Wavelength
8  d
du  3 3  
 R d
h c e 1
3


8

R  3 3  
h c e 1
3


du  R d  R d
d
R  R
d
Thermo & Stat Mech - Spring 2006
Class 21
18
Wavelength Spectrum
hc
d
hc
d
 2

R  R

d

d
3
3
hc 8 hc /   hc

8
 3 3  hc / 
R  3 3  
2
2
1 
hc e
h c e 1 
8 hc
R  5  hc / 
1
 e






Thermo & Stat Mech - Spring 2006
Class 21
19
Graph
Black Body Radiation
3500000
Energy Density
3000000
2500000
2000000
5000
1500000
5800
1000000
7000
500000
0
-500000 0
2E-07 4E-07 6E-07 8E-07 1E-06 1E-06
Wavelength (m)
Thermo & Stat Mech - Spring 2006
Class 21
20
Agreement
The maximum of the graph for 5800 K,
the temperature of the sun’s surface has a
maximum at 500 nm, in reasonable
agreement with the peak sensitivity of the
human eye.
Thermo & Stat Mech - Spring 2006
Class 21
21
Wien Displacement Law
P eak occurs at max . T o find max :
dR
0
d
hc
3
maxT 
 2.90 10 m  K
4.96k
Thermo & Stat Mech - Spring 2006
Class 21
22
Photon Gas
u  aT 4
U  aT 4V
 U 
3
CV  

4
aT
V

 T V
3
CV dT
dQ
T dT
S

 4aV 
0 T
0
0
T
T
T
4
2
3
S  4aV  T dT  aVT
0
3
T
T
Thermo & Stat Mech - Spring 2006
Class 21
T
23
Helmholtz Function
4
3
U  aT V
S  aVT
3
4
4
3
F  U  TS  aT V  T aVT
3
1 4
F   aT V
3
4
Thermo & Stat Mech - Spring 2006
Class 21
24
Pressure
4
3
U  aT V S  aVT
3
1 4
 F 
P  
  aT
 V T , N 3
4
1 4
F   aT V
3
1 U  1
P   u
3V  3
Thermo & Stat Mech - Spring 2006
Class 21
25