PHY 341/641 Thermodynamics and Statistical Physics Lecture...

4/20/2012 PHY 341/641 Spring 2012 -- Lecture 33 1

PHY 341/641 Thermodynamics and Statistical Physics

Lecture 33

Analysis of classical gases and liquids (STP Chapt. 8)

• Virial coefficients • Radial distribution functions


-- student presentations 4/30, 5/2 (need to pick topics)


www.wfu.edu/physics/sps/spszone52012conf/welcome.html

Part of SPS zone 5 conference April 20-21, 2012

Received 1997 Nobel Prize with Steven Chu and Claude Cohen-Tannoudji “for development of methods to cool and trap atoms with laser light”

http://www.wfu.edu/physics/sps/spszone52012conf/welcome.html


∫ −=

∂∂

+=

∂∂

−=

∂∂

−=

+=−−=−=

UNNC

NT

C

NT

C

NT

CIG

CIGN

erdrdrdV

Z

VZ

NV

NkTPV

VF

VNkT

VFP

NVTFNVTFZkTZkTZkTNVTF

β32

31

3

,

,,

1

ln1

),,(),,( lnlnln),,(

:particles ginteractinfor energy Free

Estimate of partition function for interacting gas or fluid; virial expansion:


Model simulations of ZC

( )∑<

−=ji

N uU 2121 ),,(:potentialsn interactio pairwise Isotropic

rrrrr

−

=

612

4)(

:potential Jones-Lennard

rrruLJ

σσε

><∞

= for 0

for )(

:potential core-Hard

σσ

rr

ruHC u HC(

r)

r σ


( )∫

∫−−−−−

−

=

=

)()()()(32

31

3

32

31

3

11323121

1:potentialpair For

NNrurururuNN

UNNC

erdrdrdV

erdrdrdV

Z

ββββ

β

( )( )( ) ( )( )( ) ( )( )

23131233

23

13

31223

13

2

11323123

23

13

)(

621

211

11111

1

:Define

fffrdrdrdV

NNNfrdrdVNN

ffffrdrdrdV

Z

ef

NNNNC

ruij

ij

∫ ∫

∫−−

+−

+=

++++=

−≡

−

−β


VN

TBρTρB

VZ

NV

NkTPV

NT

C

≡

+++≈

∂∂

+=

ρ where

)()(1

ln1

:state ofequation ofexpansion Virial""

32

2

,

( ) ( ) ( )( )( ) ( )( ) ( )( )

( )( )∫

∫∫

∫∫

−−=⇒

−−≈

−

−∂∂

−−

=−

−

−−

−

121)(

12

12

1

12

12

1:)( of Evaluation

12

1212

12

123

2

123

2

,12

3

123

1223

13

2

2

ru

ru

NT

ru

ru

erdTB

erderdV

NNV

erdV

NNfrdrdVNN

TB

β

ββ

β

ρ


( )( )

( )( )( )1113

2 )( : wellsquareFor

32 )( :sphere hardFor

121)(

: )( of Evaluation

33

2

3

2

123

2

2

12

−−−=

=

−−= ∫ −

ReTB

TB

erdTB

TB

ru

βε

β

πσ

πσ

∫∞

−−

−−=

−

=

0

11423

2

612

12)(

4)(

:potential Jones-Lennard

612 xx

LJ

edxxTB

rrru

εβπσ

σσε


−

−−

1612114

2 xxexεβ

1/(4εβ) = 1

1/(4εβ) = 10


∫∞

−−

−−=

0

1142

32 1

2)( 612 xxedxxTB εβ

πσ

1/(4εβ)

Note: Correlates well with experimental measurements on N2, Ne, Ar, and He (less well)


Radial distribution function g(r)

∫ −= UNNC erdrdrd

VZ β3

23

131

:functionpartition theon tocontributin correlatio thedefined have wePreviously

∫ ∫

= − U

NNC erdrdV

rdrdZ β33

32

31

3 1:rewritten becan This


∫ ∫

= − U

NNC erdrdV

rdrdZ β33

32

31

3 1

The quantity in () is related to the “pair distribution function” g(r1,r2) – given a particle at r1 what is the probability of finding a second particle at r2.

( )

VN

erdrdVZ

NNg

erdrdV

rdrdZ

UNN

C

UNNC

≡

−=

=

∫

∫ ∫−

−

ρ

ρ β

β

where

1),(

1

33

321

2

33

32

31

3

rr


( )

∫

∫

≈−=

−=

−= −

NNrdrgr

gg

erdrdVZ

NNg UNN

C

1)(4

then)(),( If

1),(

2

2121

33

321

2

πρ

ρ β

rrrr

rr

Radial distribution function – continued --

Measurement of g(r) by neutron scattering:


Experiment measures neutron scattering signal as a function of the change in neutron wave number Q:

( ) ( )∫∞

−+=0

sin1)(41)( drQrrrgQ

QS πρ


Estimate of g(r) from theory:

For systems described by pair potentials in the dilute limit:

)()( ruerg β−≈

1/(4εβ) = 1

1/(4εβ) = 10

PHY 341/641 Thermodynamics and Statistical Physics Lecture...

Documents

Transcript of PHY 341/641 Thermodynamics and Statistical Physics Lecture...