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Chem:KU-KPSPiti Treesukol
1
Physical Chemistry IV01403343
The Use of StatisticalThermodynamics
Piti TreesukolChemistry Department
Faculty of Liberal Arts and Science
Kasetsart University : Kamphaeng Saen Campus
Chem:KU-KPSPiti Treesukol
2
Thermodynamics & Partition Functions A Partition function is the bridge between
thermodynamics, spectroscopy, and quantum mechanics.Thermodynamics functions
Mean energiesHeat capacitiesEquation of stateResidue entropiesEquilibrium constants
Chem:KU-KPSPiti Treesukol
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Maxwell Relation
Chem:KU-KPSPiti Treesukol
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Thermodynamics Functions Expressions to calculate thermodynamics
functions from canonical partition function.
For independent distinguishable system
For independent indistinguishable system
Q
UUln
)0(
QkT
UUS ln
)0(
NqQ
!N
N
Chem:KU-KPSPiti Treesukol
5
Other Thermodynamics Functions The Helmholtz energy
The Pressure
The Enthalpy
QkTAA
TSUA
ln)0(
T
T
V
QkTp
V
Ap
ln
TVV
QkTV
QHH
lnln
)0(
Chem:KU-KPSPiti Treesukol
6
Other Thermodynamics Functions The Gibbs energy
For an ideal gas
For indistinguishable molecules
TV
QkTVQkTGG
pVATSHG
lnln)0(
nRTpV nRTQkTGG ln)0(
N
qnRT
nRTNkTqNkTGG
ln
!lnln)0(
Chem:KU-KPSPiti Treesukol
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The Molecular Partition Function The energy of a molecule is the sum of
contribution from its different modes of motion.
The separation of energy is only approximation.Born-Oppenheimer ApproximationRigid Rotor Approximation
The partition function is a product of each contributions.
Ei
Vi
Ri
Tii
EVRT
ii
qqqq
eeqEi
Ri
Vi
Tii
Chem:KU-KPSPiti Treesukol
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Chem:KU-KPSPiti Treesukol
9
Partition Function Contributions The Translation Contribution
The approximations are valid if many states are occupied
L is small compared to the container H2 at 25 ˚C L = 71 pm V=1L →
q=2.73x1027
O2 at 25 ˚C L = 18 pm V=1L → q=1.75x1029
3L
VqT
2/1
2
L
mh
Chem:KU-KPSPiti Treesukol
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Partition Function Contributions The Rotational Contribution
Linear Rotors:
Nonlinear Rotors:Rotational Temp.Approximation is valid when T >> qR
Symmetric Molecule
is the symmetry number (identity and rotation; E, nCx)Heter.diatomic s = 1, Homo.diatomic s = 2
J
JhcBJR eJq )1(12
Degeneracy Rotationalconstant
hcB
kTqR
2/12/3
ABChc
kTqR
Approximation is valid when many rotational levels are populated.
hcB
kTqR
khcBR / khcBR /
Chem:KU-KPSPiti Treesukol
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Rotational Partition function of HCl
B = 10.591 cm-1
hcB/kT = 0.051
J
JhcBJR eJq )1(12
J
cont
ribut
ion
Chem:KU-KPSPiti Treesukol
12
Rigid Rotor Principal Rotational Axes (A,B,C) Ic Ib Ia
Spherical rotors (Ia=Ib=Ic) CH4, CCl4 Symmetric rotors
Oblate symmetric rotors (Ia =Ib< Ic) C6H6, NH3
Prolate symmetric rotors (Ia < Ib= Ic) CHCl3 Asymmetric rotors (Ia< Ib < Ic)
i
iirmI 2,
ri
mi
a
http://www.unomaha.edu/tiskochem/Chem3360/
Oblate top Prolate top
Chem:KU-KPSPiti Treesukol
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The Rotational Energy Levels
Spherical rotors
Symmetric rotors
c
c
b
b
a
a
ccbbaaR
I
J
I
J
I
J
IIIE
222
222
2212
212
21
a
b c
Angular velocityMoment of inertia
2
3
8RmI A
24 RmI A
22
2//
22
4
RmRmI
RmI
CA
A
cIB
JhcBJI
JJI
JEJ
4
12
12
22
BJJFJF
JBJJF
2)1(
1
Rotational term
cIB
cIA
KBAJBJKJF
44
1,
//
2
J=0,1,2… K=0,±1,±2,…
Angular momentum
Chem:KU-KPSPiti Treesukol
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Metaphor ลงทะเบี�ยนเรี�ยน 3 วิ ชา
วิ ชา 1 (สอบี 4 ครี��ง ครี��งละ 25 คะแนน เกรีด A >75 B>65 C>55 D>45)
วิ ชา 2 (สอบี 5 ครี��ง ครี��งละ 20 คะแนน เกรีด S >50)
วิ ชา 3 (สอบี 3 ครี��ง ครี��งละ 33 คะแนน เกรีด A >80 B>60 C>40)
qi ค�อจำ�านวินเกรีดท��เป็�นไป็ได�ของวิ ชา iqi จำ�านวินครี��งเข�าสอบีของวิ ชา i ข��นต่ำ��าท��จำะมี�โอกาส
ผ่%านถ้�าเข�าสอบี < i qi=1 (F)
i qi เข้�าสอบ
qi
1
2
3
2
3
2
? (1-5)
1
? (1-3)qtotal =q1 x q2 x q3
3
1
2
Chem:KU-KPSPiti Treesukol
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Partition Function Contributions The Vibration Contribution
Large vibrational wavenumber
Small vibrational wavenumber
Vibrational temperatureAt high Temp (T>>qV)
i
i
hchcV
VVV
eeiq
qqq
~
~
1
1)(
...21
~
~21
hc
hcE
u = 0,1,2,…
11~ qhc
...~11
1~
hc
qkThc V
2/1
2
1~
effm
k
c
khcV /~
~1
hcqV
Chem:KU-KPSPiti Treesukol
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Partition Function Contributions The Electronic Contribution
Electronic energy separations from the ground state are usually very large. In most cases qE=1 and qE=gE
for degenerate ground-state systems. For low-lying electronically excited state (at high
temp?)
jlevelenergy
jE jegq
Chem:KU-KPSPiti Treesukol
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Energy Levels
Chem:KU-KPSPiti Treesukol
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The Overall Partition Function For a diatomic molecule with no low-lying electronically
excited states and T>>R
L
~3 1
1hcehcB
kTVgq E
3L
VqT
2/31252
3
2/112/1
2/12
10561.2
1749
2
L
L
L
molgMKTp
kT
N
q
molgMKTpm
m
h
A
m
hcBqR 1
16950.0
cmB
KT
~1
1hc
V
eq
KT
cma
e a
1~4388.1
1
1
Chem:KU-KPSPiti Treesukol
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Example: Vibrational Partion Function Three normal modes of H2O
Evaluate the vibrational partition functions at 1500 K
1594.8 cm-1 3656.7 cm-1 3755.8 cm-1
At 1500 K 16.1042 cmhckT
Mode 1 2 3
3656.7 1594.8 3755.8
3.507 1.530 3.602
1.031 1.276 1.028Vq
kThc
cm
~
~ 1
353.1028.1276.1031.1 Vq
Chem:KU-KPSPiti Treesukol
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Using Statistical Thermodynamics Any thermodynamic quantity can be calculated
from a knowledge of the energy levels of molecules.
Mean Energy of various modes of motion
The mean translation energy1-D system of length X
3-D system
EorVRTMq
qV
M
MM ,,
1
2/12/ mhXqT
kTd
dX
XV
T21
2/12/1
2
11
L
L
kTT23
Chem:KU-KPSPiti Treesukol
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The mean rotational energy (for a linear molecule)
when T< qR
when T>> qR
J
JhcBJR eJq )1(12
hcBhcBR eeq 62 531
hcBhcB
hcBhcBR
ee
eehcBE
62
62
531
306
hcB
kTqR
kTBhcd
dBhcE R
11
Chem:KU-KPSPiti Treesukol
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The mean vibrational energy (harmonic approx.)
At high temp. ex = 1+x+…
~1
1hc
V
eq
1
~1
~
1
1
~
2~
~
~
hcV
hc
hc
hc
V
e
hcE
e
ehc
ed
d
d
dq
kThc
hcEV
1
1~1
~
Relative energy related to the zero-point energy
Chem:KU-KPSPiti Treesukol
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Heat Capacities The constant-volume heat capacity
The internal energy of a perfect gas is a sum of contributions, the heat capacity is also a sum of contribution from each mode.
VV TUC
V
V
UkC
d
dk
d
d
kTd
d
dT
d
dT
d
2
22
1
V
MM
MV Nk
TNC
2
dT (+)
dU (+)
Chem:KU-KPSPiti Treesukol
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Individual Contributions The molar constant-volume heat capacity
Translation is the only mode of motion for a monoatomic gas
Rotation contributes to the heat capacity only at high T At high temp,
Diatomic molecules
Vibration contributes to the heat capacity only at high T
The maximum contribution is R and the lowest is 0.
R
dT
kTdNC A
TmV 2
323
, RCRC
molJKRC
mVmp
mV
25
,,
1123
, 47.12
RC RmV 2
3,
RC RmV ,
For non linear molecules
For linear molecules
ThighatR
TlowatRC mV
25
23
,
T
TVV
mV V
V
e
e
TfRfC 2
22
, 1
Chem:KU-KPSPiti Treesukol
25
The Overall Heat Capacity The total heat capacity of a molecular
substance is the sum of each contribution.If T >> qm, equipartition is valid. The heat
capacity can be estimated by counting the number of modes that are active.
For gases; Each of three translational modes: ½ R Each of active nR rotational modes: ½ R
Each of active nV vibrational modes: R
RC VRmV 2321
,
--------
CV
,m/R
qR qV
| |
Temp.Dissociation
4
3
2
1
In thermal equilibrium, energy is shared equally among all of its various forms; for example, the average kinetic energy per degree of freedom in the translational motion of a molecule should equal that of its rotational motions
Chem:KU-KPSPiti Treesukol
26
Example Estimating the CV,m of H2O(g) at 373 K
Vibration modes: 3656.7 1594.8 3755.8 cm-
1 Rotational constants: 27.9 14.5 9.3 cm-
1
M of each modes:
khcBR /
khcV /~
K cm 439.1/ khc
Mode cm-1 K
Rotation
27.9 40.15
14.5 20.87
9.3 13.38
Vibration
3656.7 5261.99
1594.8 2294.92
3755.8 5404.60
1
11
23
23
,
JK 25
JK 5.12JK 5.12
0
RRRC mV
Expt. Value = 26.1 JK-1
Chem:KU-KPSPiti Treesukol
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Equation of State An equation of state is
a relation between state variables.a thermodynamic equation describing the
state of matter under a given set of physical conditions.Ideal gasVDW VerialEtc.
1RT
pVm
2
1mm
m
V
C
V
B
RT
pV
Chem:KU-KPSPiti Treesukol
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Equations of State Statistical Mechanics
Configuration partition function (Z)
Canonical Partition Function (Q)
pair potentials
ideal gas (Ep=0)
TV
QkTp
ln
!! 3 N
q
N
VQ
N
N
N
L
NN
NN ZV
NN
33!
1
! L
L
NE drdrdre
NZ p ...
!
121
Ep = intermolecular potential
!...
!
121 N
Vdrdrdr
NZ
N
N
21),( 21
2
1drdreZ rrE p
Chem:KU-KPSPiti Treesukol
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Residual Entropy The entropy at any temperature may be determined from
The experimental entropy could be less than the calculated value*. Some disorder is present in the solid even at T=0. The entropy at T=0 is sometime greater than 0 and is called
residual entropy.
phase
phasep
T
H
T
dTCSTS )0()(
Temperature
Ent
ropy
Sfus
Svap
Chem:KU-KPSPiti Treesukol
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Origin of Residual Entropy A crystal composed of AB molecules, wher A and
B are similar atoms (such as C and O in CO).AB AB AB AB … AB BA BA AB … Random orientations in solid with small energy different The entropy arises from residual disorder by
The total number of ways of achieving the same energy
For solids composed of molecules that can adopt s orientations
WkS ln
2ln2ln2ln
2
nRNKkS
WN
N
5.8 JK-1mol-12 possible orientations
sRSm ln
Chem:KU-KPSPiti Treesukol
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Residual entropies of moleculesCO ( 5 JK-1)FClO3 ( 10.1 JK-1)
H2O(s) ( 3.4 JK-1)
1JK 8.52ln RSm
1JK 5.114ln RSm
NNNW
2
3
16
62
1JK 4.32
3ln RSm
Chem:KU-KPSPiti Treesukol
32
Equilibrium Constant The equilibrium constant K of a reaction is
related to standard Gibbs energy of reaction:
The relation between K and the partition function Standard Molar Gibbs energy of species J
Standard molar partition function: Standard pressure; only qT depends on p
KRTGr ln
A
mJmm N
qRTJGJG
,ln0,
p
RTVm
mJq ,
react
reactfprod
prodfr GGG ,,
Chem:KU-KPSPiti Treesukol
33
Equilibrium constant
The equilibrium constant for the reaction aA + bB cC + dD is
is the difference in molar energies of the ground states of the products and reactants.
can be calculated from the bond dissociation energies of the species.
RTEb
AmB
a
AmA
d
AmD
c
AmC reNqNq
NqNqK /
,,
,, 0
0Er
0Er
D0(reactants) D0(products)
0Er
RTE
J A
mJ r
J
eN
qK /, 0
product
reactant
atoms
Chem:KU-KPSPiti Treesukol
34
A Dissociation Equilibrium Equilibrium of the dissociation of diatomic molecule
X2(g) 2X(g)
Dissociation energy of X2 moleculePartition function of X atom
Partition function of X2 molecule
pp
p
p
p
p
p
a
aK
X
X
X
X
X
X
222
222
RTE
AmX
mXRTE
AmX
AmX rr eNq
qe
Nq
NqK /
,
2
,/1
,
2
, 0
2
0
2
XXDXUXUE mmr 020 0,0,2
33,X
X
X
mXmX p
RTgVgq
L
L
33,22
22
2
2
X
VX
RXXV
XRX
X
mXmX p
qqRTgqq
Vgq
L
L
Chem:KU-KPSPiti Treesukol
35
The equilibrium constant of the dissociation of diatomic molecule is
RTD
XVX
RXX
XX eqqgp
kTgK /
6
32
0
222
2
L
L
Boltzmann constant = R/NA
Degeneracy of the electronic ground state
Thermal wavelength depends on T and m
Dissociation energy of X2 molecule
Chem:KU-KPSPiti Treesukol
36
Example: K of Na2 dissociation Na2(g) 2 Na(g) at 1000 K
B = 0.1547 cm-1 = 159.2 cm-1 D0 = 70.4 kJ/molThe Na atom has doublet ground state of homonuclear diatomic molecule is 2
Calculate each terms
~
42.2
m 1015.1885.42246Pa 10
m 1014.84JK 1038.1
1 2
885.4 2246
pm 5.11)( pm 14.8)(
47.86115
312123
2
22
2
LL
eK
NagNag
NaqNaq
NaNaVR
][
][
2
2
Na
NaK
Chem:KU-KPSPiti Treesukol
37
q
Nen
i
i
q(1000 K,1) = 3.5
q(2000 K,1) = 6.4
q(1000 K,2) = 2.2
Chem:KU-KPSPiti Treesukol
38
Contributions to the equilibrium constant Reaction at equilibrium
(reactants products)
DE0
Reactants
Products
Chem:KU-KPSPiti Treesukol
39
The ratio of reactants and products at equilibrium At equilibrium both reactants
and products are present
The equilibrium constant is the ratio of reactants and products at equilibrium
q
Nen
i
i
q
Nqe
q
NnN R
rrrR
r
q
Nqe
q
NnN p
pppP
P '
*PR qqq
kTE
R
P
R
P req
q
N
NK /0
RTEp
p
rP eq
Nqe
q
N /00
DE0
Reactants
Products
P
EP
PP
RR
PP
R
eqeq
eq
0*
Chem:KU-KPSPiti Treesukol
40
Calculations
2/12/1
2/12
3
)/()/(
1749/
2
1molgK
pm
L
L
L
MWT
m
hVqT
2/1
2/32/1
3/2 )/(
/0270.1
)(
1)(
)/(
/6950.01)(
3
1
cm
K
cm
K
ABC
T
ABChcNonlinearq
B
T
hcBLinearq
R
R
K
cm
/
)/(4388.1
1
1
1
1 1
Ta
eeq
ahcV
33,0
,0,
X
VX
RXV
XRX
X
m
p
qqRTgqq
Vgq X
XmXL
L
3,0,
X
VX
RXX
A
mX
p
qqkTg
N
q
L
0gqE )/exp(
)/()/(
)/()/(0
,,
,, RTENqNq
NqNqK
dDcCbBaA
rbAmB
aAmA
dAmD
cAmC
productsreactants
)()( 000 DDEr
Chem:KU-KPSPiti Treesukol
41
)(2)(2 gNagNa
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