Chem:KU-KPS Piti Treesukol 1 Physical Chemistry IV 01403343 The Use of Statistical Thermodynamics...

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


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


3

Maxwell Relation


4

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

qQ

N


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(


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(


7

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


8


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


10

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 /


11

Rotational Partition function of HCl

B = 10.591 cm-1

hcB/kT = 0.051

J

JhcBJR eJq )1(12

J

cont

ribut

ion


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


13

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


14

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


15

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


16

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


17

Energy Levels


18

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


19

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


20

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


21

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


22

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


23

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 (+)


24

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


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


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


27

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


28

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

qQ

33!

1

! L

L

NE drdrdre

NZ p ...

!

121

Ep = intermolecular potential

!...

!

121 N

Vdrdrdr

NZ

N

N

21),( 21

2

1drdreZ rrE p


29

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


30

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


31

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


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 ,,


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


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


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


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


37

q

Nen

i

i

q(1000 K,1) = 3.5

q(2000 K,1) = 6.4

q(1000 K,2) = 2.2


38

Contributions to the equilibrium constant Reaction at equilibrium

(reactants products)

DE0

Reactants

Products


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*


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


41

)(2)(2 gNagNa

Chem:KU-KPS Piti Treesukol 1 Physical Chemistry IV 01403343 The Use of Statistical Thermodynamics...

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Transcript of Chem:KU-KPS Piti Treesukol 1 Physical Chemistry IV 01403343 The Use of Statistical Thermodynamics...