thermo & stat mech - spring 2006 class 21 1 thermodynamics and statistical mechanics blackbody...
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Thermo & Stat Mech - Spring 2006 Class 21
1
Thermodynamics and Statistical Mechanics
Blackbody Radiation
Thermo & Stat Mech - Spring 2006 Class 21
2
Classical Energy Equipartition
321
321
321
321
321
lnlnln
lnlnlnln
N
U
ZZZ
N
U
ZZZZ
ZZZZ
VVV
Thermo & Stat Mech - Spring 2006 Class 21
3
Classical Energy Equipartition
dzeZ
dxeZkx
dLeZI
L
dpeZm
p
az
kx
I
L
m
p
2
221
2
2
form of All
Potential Spring
2
Rotation
2
Dim) (1n Translatio
221
22
22
Thermo & Stat Mech - Spring 2006 Class 21
4
Classical Energy Equipartition
2
1
here w2
1ln
lnln2
1lnlnln
12
kT
kT
Zaa
Z
aadzeZ az
Thermo & Stat Mech - Spring 2006 Class 21
5
Quantum Harmonic Oscillator
1
1
exitations no. avg. exitation
energy
iewEinstein v-Bose
1 ere wh
1
energy excitation Average
e
kTeN
U
Thermo & Stat Mech - Spring 2006 Class 21
6
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
7
Density of States (Lec 17)
dc
Vdg
cddk
ck
dkkV
dkkg
232
22
2)(
2)(
Thermo & Stat Mech - Spring 2006 Class 21
8
Density of States for Photons
dc
Vdg
dc
Vdg
232
232
)(
onspolarizati Two 2
2)(
Thermo & Stat Mech - Spring 2006 Class 21
9
Photon Energy
1
1
1
)()(
23
3
223
ec
d
V
dUdu
dc
V
e
dgfdU
Thermo & Stat Mech - Spring 2006 Class 21
10
Photon Energy
1
8
1
Use,
1
3
33233
3
23
3
e
d
chec
ddu
ec
d
V
dUdu
Thermo & Stat Mech - Spring 2006 Class 21
11
Graph
Black Body Radiation
05E+171E+18
1.5E+182E+18
2.5E+183E+18
3.5E+184E+18
4.5E+18
0 1 2 3 4 5
Photon Energy (eV)
En
erg
y D
ensi
ty
5000
6000
7000
Thermo & Stat Mech - Spring 2006 Class 21
12
Total Energy Density
4-3-164
433
4544
33
0
34
330
3
33
KmJ 1055.7
15
8
15)(
8
1)(
8
1
8
aaTu
Tch
kkT
chu
e
dxxkT
che
d
chu
x
Thermo & Stat Mech - Spring 2006 Class 21
13
Energy Flux
4-2-8
23
454
4
Km W1067.5
15
2
4
1
4
1
4
1flux Energy
4
1 14) (Lecflux Particle
ch
kcaTe
caTuce
vn
Thermo & Stat Mech - Spring 2006 Class 21
14
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
16
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
17
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
18
Change to Wavelength
d
dRRdRdRdu
echR
dRe
d
chdu
1
8
1
8
3
33
3
33
Thermo & Stat Mech - Spring 2006 Class 21
19
Wavelength Spectrum
1
8
1
/8
1
8
/5
2/
3
332
3
33
2
hc
hc
e
hcR
hc
e
hc
ch
hc
echR
hc
d
dhc
d
dRR
Thermo & Stat Mech - Spring 2006 Class 21
20
Graph
Black Body Radiation
-500000
0
500000
1000000
1500000
2000000
2500000
3000000
3500000
0 2E-07 4E-07 6E-07 8E-07 1E-06 1E-06
Wavelength (m)
En
erg
y D
ensi
ty
5000
5800
7000
Thermo & Stat Mech - Spring 2006 Class 21
21
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
22
Wien Displacement Law
Km 1090.296.4
0
: find To .at occursPeak
3max
maxmax
k
hcT
d
dR
Thermo & Stat Mech - Spring 2006 Class 21
23
Photon Gas
3
0
2
0
3
00
3
44
3
44
4
4
aVTdTTaVS
T
dTTaV
T
dTC
T
QdS
VaTT
UC
VaTUaTu
T
TTV
T
VV
Thermo & Stat Mech - Spring 2006 Class 21
24
Helmholtz Function
VaTF
aVTTVaTTSUF
aVTSVaTU
4
34
34
3
13
43
4
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