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
4/20/2012 PHY 341/641 Spring 2012 -- Lecture 33 2
-- student presentations 4/30, 5/2 (need to pick topics)
4/20/2012 PHY 341/641 Spring 2012 -- Lecture 33 3
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”
4/20/2012 PHY 341/641 Spring 2012 -- Lecture 33 4
∫ −=
∂∂
+=
∂∂
−=
∂∂
−=
+=−−=−=
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:
4/20/2012 PHY 341/641 Spring 2012 -- Lecture 33 5
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
4/20/2012 PHY 341/641 Spring 2012 -- Lecture 33 6
( )∫
∫−−−−−
−
=
=
)()()()(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
4/20/2012 PHY 341/641 Spring 2012 -- Lecture 33 7
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
4/20/2012 PHY 341/641 Spring 2012 -- Lecture 33 8
( )( )
( )( )( )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
4/20/2012 PHY 341/641 Spring 2012 -- Lecture 33 9
−
−−
1612114
2 xxexεβ
1/(4εβ) = 1
1/(4εβ) = 10
4/20/2012 PHY 341/641 Spring 2012 -- Lecture 33 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)
4/20/2012 PHY 341/641 Spring 2012 -- Lecture 33 11
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
4/20/2012 PHY 341/641 Spring 2012 -- Lecture 33 12
∫ ∫
= − 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
4/20/2012 PHY 341/641 Spring 2012 -- Lecture 33 13
( )
∫
∫
≈−=
−=
−= −
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:
4/20/2012 PHY 341/641 Spring 2012 -- Lecture 33 14
4/20/2012 PHY 341/641 Spring 2012 -- Lecture 33 15
Experiment measures neutron scattering signal as a function of the change in neutron wave number Q:
( ) ( )∫∞
−+=0
sin1)(41)( drQrrrgQ
QS πρ
4/20/2012 PHY 341/641 Spring 2012 -- Lecture 33 16
Estimate of g(r) from theory:
For systems described by pair potentials in the dilute limit:
)()( ruerg β−≈
1/(4εβ) = 1
1/(4εβ) = 10