1 Physical Chemistry III 01403343 Statistical Mechanics Piti Treesukol Chemistry Department Faculty of Liberal Arts and Science Kasetsart University :

1

Physical Chemistry III01403343

Statistical Mechanics

Piti TreesukolChemistry Department

Faculty of Liberal Arts and Science

Kasetsart University : Kamphaeng Saen Campus

Chem:KU-KPSPiti Treesukol

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3

Introduction

Macroscopic pictureBulk materialThermodynamic & Kinetic properties

Microscopic picture Atom, Molecule, IonPosition, Energy, Momentum

Link between micro- and macro picturesStatistical method


ประกาศสอบกลี่างภาคื 22 ม�นาคืม 2557 13:00-16:00 น.

สอบปลี่ายภาคื 22 พฤษภาคืม 2557 13:00-16:00 น.

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5

Properties

MassTemperaturePressureEnergyConductivityThermodynamic propertiesHeat capacityGibbs free energyEnthalpy

Etc.


Extensive and Intensive properties

Extensive Properties

Intensive Properties

6

n

iitotal XXAccumulative

Average

n

ii

n

iii

total

m

XmX

Xtotal

X1 X2Xtotal

X1 X2


Expectation values/Measurabl

es

Internal Propeties TemperatureT = < Ti >Ti (t)

External PropertiesTotal EnergyE = S Ei

Ei (t) 7


System & Enviroment

8

Systemn, N, T, P, V,

m, etc.

Environment T, P, m

Energy

Mass


9

Energy of a System

Energy of a macroscopic systemdepends on …

Energy of a microscopic systemdepends on …

A macroscopic system comprises of countless microscopic systems (x1023)

iii HE

iitotal EE

PVTEEE itotal ,,}{

,,,,,, iiimmlni zyxsl


10

E1, T1 E2, T2

T1 < T2 thenE1 < E2

i

itotal EE i

itotal Tn

T1

j

jjtotal TpT


State of a System

Macroscopic system!!!System composes of ???State of the system is defined by a few number of macroscopic parameters Systems with the same state may be different from each others

Properties of the system are either Acculative property orAverage property

11


12

Macroscopic description

can be derived statisticaly from microscopic descriptions of a collection of microscopic systemsDescription on average*Fluctuation of microscopic properties

Microscopic properties depends on a set of parameters of each microscopic system

Macroscopic properties depend on a small set of macroscopic parameters !!!


Distribution of Molecular StatesMolecules = Workers of a department

Energy level = Salary of each position

13

100,000

50,000

20,000

15,000

10,000

Total Energy / Expense = ?

Population of each level : Configuration = {3,2,0,2,1}

How many configuration is possible if the total energy was fixed?

* Nobody wants high salary (energy) because it has too much stress!!!


The Distribution of Molecular States

A system composed of N molecules IF Total energy (E) is constant (Equilibrium)

Posible energy state for each molecule (ei)

Molecules in different states (i) possess different energy levels

Total energy E = Ej = (ei ni)Ej is fluctuated due to molecular collision

Constraint: Ej = EThe distribution of energy is the population of a state (there are ni molecules in i energy level){0,1,5,7,1,0}

14


ExamplesTotal particle (N) = 6

15

{3,1,2,0,0,0} Etotal = 3x0 + 1x2 + 2x4 = 10

{4,0,1,1,0,0} Etotal = 4x0 + 1x4 + 1x6 = 10

{3,0,1,2,0,0} Etotal = 3x0 + 1x4 + 2x6 = 16


Configuration and WeightsConfiguration

Different configurations have different population of state

Weights

Number of ways in achieved a particular configuration

16

e6

e5

e4

e3

e2

e1Conf.1 Conf. 2 Conf.3 … w.1 w. 2 w.3 …

Conf. 1

e6

e5

e4

e3

e2

e1


17

Instantaneous ConfigurationPossible energy level (e0,

e1, e2 …)N moleculesn0 molecules in e0 staten1 molecules in e1 state …

The instantaneous configuration is {n0,n1,n2…}

Constraint: n0+n1+n2+… = N# ways to achieve instantaneous conf. (W)

!

!

!!!

!...,,

210210

ii

n

N

nnn

NnnnW


18

Examples{2,1,1}

{1,0,3,5,10,1}

122

24

!1!1!2

!41,1,2 W

81031.9!1!10!5!3!0!1

!201,10,5,3,0,1 W


19

Principle of Equal a prioriAll possibilities for the

distribution of energy are equally probable

The populations of states depend on a single parameter, the temperature. If at temperature T, the total energy is 3

Energy levels: 0, 1, 2, 3

3

2

1

0

{0,3,0,0} {1,1,1,0} {2,0,0,1}

3

2

1

0

3

2

1

0

W=1 W=6 W=3


20

Possible configurations for 5 molecules

State 1 5 4 4 3 3 3 3 2 2 2

State 2 1 2 1 3 1 1 1

State 3 1 1 1 3 1 1 2

State 4 1 1 1 1

State 5 1 1 1

State 6 1 1 5N 5 5 5 5 5 5 5 5 5 5 5 5 5

E 5 6 7 7 812

12 8

11

11

20

17

30

W 1 5 510

20

20

20

10

10

60

120

60 1

Energy of state j = j


21

The Dominating ConfigurationSome specific

configuration have much greater weights than others

There is a configuration with so great a weight that it overwhelms all the restW is a function of all ni: W(n0, n1, n2 …)

The dominating configuration has the values of ni that lead to a maximum value of W

The number of molecule constraint :

The energy constraint :

Eni

ii Nn

ii


22

Maximum & Minimum PointF is a function of x : F(x)

Maximum point:

F ’= 0 ; F ’’ < 0Minimum point:

F ’= 0 ; F ’’ > 0

1

2

3

4 5 6

7

8

9

F(x)

x


23

Maximum & Minimum in 3D

F(x,y)


24

Configuration is defined by a set of ni, {ni}W depends on a set of ni or {ni}

At a specific condition, several configurations may be possibleThe configuration with greatest

weight (W) will dominate and that configuration can be used to represent the system

Other configurations with less weight is negligible

Weight

Configuration

Greatest weight = Dominating Configuration


Dominating ConfigurationWeight of each configuration2 energy states

Possible configurations (6 particles) :{0,6}, {1,5}, {2,4}, {3,3}, {4,2}, {5,1}, {6,0} 25


Dominating ConfigurationWeight of each configuration3 energy states

Possible configurations (10 particles) :{0,0,10}, {0,1,9}, {0,2,8}, … {1,0,9}, {2,0,8}, … {1,1,8} … 26

10 particles 30 particles20 particles


27

Maximum Value of W{ni}We are looking for the

best set of ni that yields maximum value of ln(W)Maximum W = W{ni,max} Maximum ln W = ln W{ni,max}

{ni,max} = ?


28

Maximum Value of W{ni }{ni,max} can be determined

by differentiate

ConstraintsTotal particle (N) is constant

Total energy (E) is constant

0ln

ln

ii

i

dnn

WWd

i

iinE 0i

iidndE

i

inN 0i

idndN


29

Maximum Value of W{ni }Maximum ln(W) plus

Constraints

Method of undetermined multipliers

0ln

ln

ii

i

dnn

WWd

0i

iidn

0i

idn

iii

i

iii

ii

ii

i

dnn

W

dndndnn

WWd

ln

lnln


30

Stirling’s ApproximationNatural logarithmic of the

weight

Stirling’s ApproximationThe approximation for the weight

iinN

nnnNW

nnn

NnnnW

!ln!ln

!ln!ln!ln!lnln

!!!

!...,,

210

210210

xxxx ln!ln

iii

iiii

nnNN

nnnNNNW

lnln

lnlnln

If x is large


31

xxxx ln!ln when x is a large number!

1.67%


32

0ln

0ln

ln

ii

iii

i

n

W

dnn

WWd

N

nNn

n

W ii

i

ln1ln1lnln

0ln ii

N

n ieN

ni

jj

jiji eNenNNen

j

jee 1

j i

jj

ii n

nn

n

NN

n

W lnlnln

1lnln

ln

N

n

NN

n

N

n

NN

iii

1ln

lnln

ln

i

j i

jjj

i

j

j i

jj

n

n

nnn

n

n

n

nn

Eq. 1 is possible if (and only if) …

Eq. 1

ii n

N

NN

n

NN

1ln

ei is relative

energy


33

The Boltzmann DistributionThe populations in the

configuration of the greatest weight depend on the energy of the state

The fraction of molecules in the state i (pi) is

i

i

i

i

e

e

N

n

Z

e

N

np

ii

i

jj

i

j

i

eg

eZ

The Molecular Partition Function

(Z,q,Q)Sum over energy level (j)

degeneracy

kT

1

Sum over all states (i)

Boltzmann constant = 1.38x10-23 J/K

***


34

The Molecular Partition Function

An interpretation of the partition functionat very low T ( T0) b ∞

at very high T ( T∞) b 0

The molecular partition function gives an indication of the average number of states that are thermally accessible to a molecule at the temperature of the system

j

jjegZ

kT

1

000

lim0lim gZeTT

i

Ze

TT

i

00lim1lim


Uniform Energy Levels Equally spaced non-degenerate

energy levels

Finite number

Infinite number

35

x

SS

SxxxxS

xxS

1

1

132

2

e0= 0 e1= e e2= 2e e3= 3e …

e3

e2

e1

e0

e n

i

ieZ

e

eee

eee

eee

eZi

i

1

1

1

132

32

210

Infinite # of energy levels

Finite # of energy levels


36

What are the possible states of particles at high temperature?High-energy states?Low-energy states?All states?


37

The Possibility *The possibility of molecules in the state with energy ei (pi)

The possibilities of molecules in the 2-level system

i

i

eeZ

epi

1

Z of infinite # of energy levels*

e

ep

11

ep

1

10

As T the populations of all states (pi’s) are equal.


38

The possibilities of molecules in the infinite-level system* ep 10

eep 11

22 1 eep

As T the populations of all states are equal.


39

Temperature


40

ExamplesVibration of I2 in the ground,

first- and second excited states (Vibrational wavenumber is 214.6 cm-1)

036.1226.207

6.2141

1

cm

cm

kT

hc

1226.207

15.2982,1,0

cmhc

kT

KTandvfor

081.0

229.0

645.0

)1(

2

1

0

p

p

p

eep vv

Relative energy


41

Approximations and Factorizations

In general, exact analytical expression for partition functions cannot be obtained. Closed approximation expressions to estimate the value of the partition functions are required for each systems

Energy levels of a molecule in a box of length X

,2,18 2

22

nmX

hnEn

08 12

2

1 mX

hE

2

22

81

mX

hnn Relative

energy


42

Translational Partition Function

Partition function of a molecule in a box of length X

1

12

n

nX eq ,2,1

81

2

22 n

mX

hnn

The translation energy levels are very close together, therefore the sum can be approximated by an integral.

01

1 22

dnedneq nnX

Make substitution: x2=n2be and dn = dx/(be)1/2

Xh

mdxeq x

X

2/1

2

2/12/11

0

2/11 2

2

2

Transitional partition function


43

When the energy of a molecule arises from several different independent sourcesE = Ex+Ey+Ezq = qxqyqz

A molecule in 3-d box

)()()(,,

Zn

Yn

Xnnnn zyxzyx

zyx

nnn

qqq

eeeqz

Zzn

y

Yyn

x

Xxn

)()()(

XYZh

mq

2/3

2

2


44

is called the thermal wavelength

The partition function increases with The mass of particle (m3/2)The volume of the container (V)

The temperature (T3/2)

XYZh

mq

2/3

2

2

3

Vq

2/1

2/1

22 mkT

h

mh


45

ExampleCalculate the translational

partition function of an H2 molecule in 100 cm3 vessel at 25C

About 1026 quantum states are thermally accessible at room temperature

26

311

34

31077.2

1012.7

1000.1

m

mVq

m

KJKkg

Js

mkT

h

11

2/112327

34

2/1

1012.7

2981038.1106605.1016.22

10626.6

2


46

The Internal Energy and

EntropyThe molecular partition function contains all information needed to calculate the thermodynamic properties of a system of independent particlesq Thermal wave function

The Internal Energy **

i

iinE

i

iie

q

NE

d

dee

i

ii

d

dq

q

NE

ed

d

q

Ne

d

d

q

NE

ii

ii

Boltzmann distribution q

eNn

i

i


47

Total energy

ei is relative energy (e0=0)E is internal energy relative

to its value at T=0The conventional Internal

Energy (U)

e3

e2

e1

e0

e

Relative energy

3e

2e

e

0

d

dq

q

NnE

iii

A system with N independent molecules • q=q(T,X,Y,Z,…)

EUU )0(

V

q

q

NUU

)0(

V

qNUU

ln

)0(

Only the partition function is required to determine the internal energy relative to its value at T=0.

***


48

ExampleThe two-level partition function

e

d

d

e

N

d

dq

q

NE 1

1

e

N

e

eNE

11

0

0.1

0.2

0.3

0.4

0.5

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

kT/

E/N

At T = 0 : E 0all are in lower state (e=0)

As T : E ½ Netwo levels become equally populated


49

The value of bThe internal energy of monatomic ideal gas

For the translational partition function

nRTUq

q

NUU

V

23)0()0(

3

Vq

d

dVV

Vq

VV

4333

1

222

1

2 2/12/12/1

2/1

m

h

m

h

d

d

d

d

32

3

Vq

V

2

3)0(

2

3)0(

3

3 NU

V

VNUU

V

A

A

N

Rk

kTnRT

nN

nRT

N

NnRT

1

2

3

2

3

This result is also true for general cases.


1 amu = ? g12C 1 mol = 12 g12C 1 atom = 12 amu12C 1 mol = 6.02x1023 atom

1 amu = 1g/6.02x1023 =1.66x10-27 kg

50


51

Temperature and PopulationsWhen a system is heated,The energy levels are

unchangedThe populations are changed

0 0.2 0.4 0.6 0.8

e10

e9

e8

e7

e6

e5

e4

e3

e2

e1

e0 0 0.2 0.4 0.6 0.8

e10

e9

e8

e7

e6

e5

e4

e3

e2

e1

e0

Increase T

HEAT

2

22)(

81

mX

hnX

n


52

Volume and PopulationsTranslational energy levels

When work is done on a system, The energy levels are changed

The populations are changed

2

22)(

81

mX

hnX

n

decrease V

e10

e9

e8

e7

e6

e5

e4

e3

e2

e1

e0 0 0.2 0.4 0.6 0.8

e5

e4

e3

e2

e1

e00 0.2 0.4 0.6 0.8

WORK


53

The Statistical EntropyThe partition function

contains all thermodynamic information.Entropy is related to the disposal of energy

Partition function is a measure of the number of thermally accessible states

Boltzmann formula for the entropy

As T 0, W 1 and S 0

WkS ln ***


54

A change in internal energy

When the system is heated at constant V, the energy levels do not change.From thermodynamics,

Entropy and Weight

i

iii

iii

ii dndndUdUnUU )0()0(

i

iidndU

iii

iiirev

dnkT

dUdS

dnTdSdqdU

iii

i i

ii

dnkdnn

WkdS

n

W

ln

0ln

Wdkdnn

WkdS i

i i

lnln

qNk

T

UUS

WkS

ln0

ln


55

Calculating the EntropyCalculate the entropy of N

independent harmonic oscillators for I2 vapor at 25ºCMolecular partition function:

The internal energy:

The entropy:

eq1

1

11)0(

e

N

e

eNq

q

NUU

V

e

eNkS

qNkT

UUS

1ln1

ln)0(

Entropy

0

5

10

15

20

25

30

35

0 1000 2000 3000 4000 5000

T(K)

S(J

K-1

mo

l-1)


56

Entropy and Temperature

What do we know from the graph?T increases, S increasesWhat else?

Entropy

0

5

10

15

20

25

30

35

0 1000 2000 3000 4000 5000

T(K)

S(J

K-1

mo

l-1)

Documents

1 Physical Chemistry III 01403343 Statistical Mechanics Piti Treesukol Chemistry Department Faculty of Liberal Arts and Science Kasetsart University :