Classical partition function
q
e
j
j
Molecular partition functions – sum over all possible states
Energy levels εj – in classical limit (high temperature) – they become a continuous function
(root temp translational quantum numbers ~ 108)
qclass
... e
Monoatomic gas:
H ( p ,q )
dpdq
H
H ... Hamiltonian function (p, q)
1
px2
2m
p y2
pz2
( px2
qclass
Rigid rotor:
H
pz2 )
2m
... e
1 2
p
2I
p 2y
dpdq
V
e
p2
2m
p2
sin 2
2
qrot
dp dp
d
0
Missing h-s in classical function !
d e
0
H
8 2 IkT
3
dp
2 mkT
3/ 2
V
Classical limit:
Q
1
N !h sN
H
Monoatomic gas:
Qclass
1 2 mkT
N!
h2
... e
1
2m
dpdq
Hamiltonian function for the system of
interacting molecules.
p y2
U x1 , y1 ,..., z N
H ( p ,q )
N
px2
pz2
j 1
3N / 2
ZN
Classical configuration integral
Description of real gases and liquids
ZN
e
U ( x1 ,..., z N )
dx1...dz N
V
U
0
ZN
VN
Classical limit – suitabel for translation and rotation degrees of freedom
Splitting Hamiltonian into classical and quantum parts:
H H class H quant
q qclass qquant
Q
Qclass Qquant
Qquant
N !h
sN
e
H class ( p , q )
dpclass dqclass
General – for systems of interacting particles
Imperfect gases
Respecting inter-molecular interactions – starting with monoatomic gas for the case of simplicity
Q( N , V , T )
1
N ! h3 N
... e
H
dp1...dpN dr1...drN
Integration over the momenta
Q( N , V , T )
1 2 mkT
N!
h2
ZN
H
1
2m
2
pxn
2
p yn
pzn2
U x1 , y1 ,...z N
n 1
3 N /2
... e
ZN
U N / kT
dr1dr2 ...drN
Non-ideal behavior
p
kT
N
B2 (T )
2
B3 (T )
3
Looking for the expressions for
virial coeficients in terms of the
partition function - from
configuration integral:
Bn depends on the interactions
1
1– –dependence
0.00064 + 0.00000
Anaysis of virial coefficients
on U + … (+0.00000)
10 1 – 0.00648 + 0.00020 + … (-0.00007)
100 1 – 0.06754 + 0.02127 + … (-0.00036)
1000 1 – 0.38404 + 0.68788 + … (0.37232)
Virial coeficinents
V ,T ,
Q( N , V , T )
N
N 0
Q( N , V , T )e N
/ kT
N 0
V ,T ,
1
QN (V , T )
N
N 1
GCE:
pV
N
kT ln
kT
V ,T ,
ln
For virial equation of state – the
depnedence on Ξ should be eliminated.
Activity z
ln
V ,T
V ,T
0
ln
N
Q1
V
Q1
V ,T
V ,T ,
1
N 1
QNV N N
z
Q1N
ZN
1
V
N!
Q1
N 1
Z N (V , T ) N
z
N!
z
This holds only if the interactions are
moderate (r-3). For strongly
interacting systems it cannot be used
(plazma).
N
QN
It becomes configuration integral in classical limit.
V ,T ,
Z N (V , T ) N
z
N!
1
N 1
p
Pressure can be expressed as:
bj z j
kT
j 1
b1
1
Z1
1!V
b2
1
2!V
Z2
Searching for bj as
function ofZN
1
Z12
jb j z j
j 1
N
V
ln
V
pV
kT
z
kT
p
z
V ,T
z
V
ln
z
V ,T
ln
p and ρ both depend on zj
V ,T
z
Assuming that:
a1
2
a2
jb j z
a3
a1
3
j
j 1
z
p
1
a2
2b2
a3
3b3
8b22
n
f
kT
bj z
p
kT
j
B2 (T )
2
B3 (T )
3
j 1
ZN
V
N!
Q1
N
QN
B2 (T )
b2
B3 (T )
4b2 2
Z 2 Z12
2!V
1
2b3
V Z3
2
3V
Replacing N-particle problem to much simpler one.
3Z 2 Z1
2Z13
3 Z2
Z z2
2
Virial coefficients – classical limit (monoatomic gas)
Q1 (V , T )
QN V , T
2 mkT
h2
3/ 2
V
dr1
Z2
e
Z3
3
3N / 2
1 2 mkT
N!
h2
Z1
V
ZN
N ! 3N
ZN
Konfigurační
integrál
ZN
V
N!
Q1
QN
1 Q1
N! V
N
QN
N
ZN
V
U 2 / kT
e
U2 – for calculations of the second virial coef.
dr1dr2
U3 / kT
U3 – for third virial coef.
dr1dr2 dr3
It is assumed for monoatomic gas that U2 = u(r12)
Assuming that u(r12) is non-zero only if atoms are close enough (depending on the type of inter.)
B2 (T )
B2 (T )
Z2
Z12
2V
1
2V
dr1
1
2V
e
u ( r12 )
e
u ( r12 )
1 dr12
1 dr1dr2
2
e
0
u ( r12 )
1 r 2 dr
Can be solved for particular
forms of u – either
analytically of numerically
For particular inter-molecular potential – second virial coef. as a function of temperature Bx(T)
Hard spheres:
(B2 independent of T)
B2 (T )
1
2
2
4 r dr
0
3
2
3
„Square-well“ potential
u (r )
B2 (T )
,
b0 1
r
3
1 e
1
B3 coefficient for argon
B3 – depends on three-body interaction
Is it enough to take is as a pair-representative?