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

15

Human Eye

Peak at 555 nm

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

Thermo & Stat Mech - Spring 2006 Class 21

25

Pressure

uV

UP

aTV

FP

VaTFaVTSVaTU

NT

3

1

3

1

3

13

1

3

4

4

,

434