byeong-joo lee calphad 이 병 주 포항공과대학교 신소재공학과 [email protected]

Byeong-Joo Lee www.postech.ac.kr/~calphad

이 병 주 이 병 주

포항공과대학교 신소재공학과포항공과대학교 신소재공학과[email protected]@postech.ac.kr


Warming UpWarming Up – Mathematical Skills – Mathematical Skills

xxxxxxdxxxx

ln1lnln)!(ln1

dxe x2

1. Stirling’s approximation

2. Evaluation of the Integral

dxe x2

2

10

2

dxe x

adxe ax

2

10

2

3. Lagrangian Undetermined Multiplier Method


Basic Concept of Statistical Mechanics Basic Concept of Statistical Mechanics – Macro vs. Micro– Macro vs. Micro

View Point

Macroscopic vs. Microscopic

State

Macrostate vs. Microstate


Particle in a Box Particle in a Box – Microstates of a Particle– Microstates of a Particle

EVm

22

2

02

22 E

m

nL

mE

2

2

2

222

2mL

nEn

)(8

),,( 2222

2

zyxzyx nnnmL

hnnnE

n = 1, 2, 3, …

for 66 : 8,1,1 7,4,1, 5,5,4


System with particles System with particles – Microstates of a System– Microstates of a System


Macrostate / Energy Levels / Microstates Macrostate / Energy Levels / Microstates – –


Scope and Fundamental Assumptions of Statistical Mechanics Scope and Fundamental Assumptions of Statistical Mechanics ▷(n1, n2, …, nk) 로 정의되는 하나의 macrostate 를 만들기 위해 , 있을 수 있는 수많은 경우의 수 하나하나를 microstate 라

한다 .

▷ 어떠한 시스템에 가능한 (quantum mechanically accessible 한 )

macrostate ( 하나하나가 (n1, n2, …, nk) 로 정의되는 ) 의 mental

collection 을 ensemble 이라 한다 .

▷ 같은 energy level 에서 모든 microstate 의 실현 확률은 동등하다 .

▷ Ensemble average 는 time average 와 같다 .


Number of ways of distribution : in Number of ways of distribution : in kk cells with cells with ggii and and EEii

▷ Distinguishable without Pauli exclusion principleN

kgggW )( 21

!!)()()(

!!!

!21

2121 j

njn

knn

k n

gNggg

nnn

NW

j

k

▷ Indistinguishable without Pauli exclusion principle

for gi with ni !)!1(

)!1(

ii

ii

ng

ng

!)!1(

)!1(

ii

ii

i ng

ngW

▷ Indistinguishable with Pauli exclusion principle

for gi with ni !)!(

!

iii

i

nng

g

!)!(

!

iii

i

i nng

gW


Evaluation of the Most Probable Macrostate Evaluation of the Most Probable Macrostate – Boltzman– Boltzman

)lnln(lnln iiii nngnNNW

0lnln max

i

ii n

gnW

0 in 0 in

0 ii n 0 ii n

0ln

i

i

i

g

n ieegn ii

Z

N

eg

Ne

ii

iegZ

Nn ii

knk

nn

k

gggnnn

NW )()()(

!!!

!21

2121


Evaluation of the Most Probable Macrostate Evaluation of the Most Probable Macrostate – B-E & F-D – B-E & F-D

!)!1(

)!1(

ii

ii

i ng

ngW

!)!1(

)!(

ii

ii

i ng

ngW

!)!(

!

iii

i

i nng

gW

→

1

iee

gn i

i

Bose-Einstein Distribution

Fermi-Dirac Distribution

1

iee

gn i

i


Definition of Entropy and Significance of Definition of Entropy and Significance of ββ

)( fS )()()( BABABA fffS

ln'kS

dUkdnZ

Nkdnk

Z

NdnkndnkdS iiiiiii 'ln'')ln('ln'

dNT

dVT

PdU

TdS

1

Tk

1' →

Tk '

1

▷ Thermal contact 상태에 있는 두 부분으로 이루어진 Isolated System 을 고려 . 이에 대한 평형 조건은 Classical Thermodynamics 에서는 maximum entropy (S) Statistical mechanics 에서는 maximum probability (Ω)

▷ S 와 Ω 는 monotonic relation 을 가지며


Calculation of Macroscopic Properties from the Partition FunctionCalculation of Macroscopic Properties from the Partition Function

ZkTnke

Znk

N

nnkkS ii

kTi

ii

i ln/1

ln)ln(ln /

ZNkT

US ln

ZNkTSTUF ln

VV T

ZNkTZNk

T

FS

ln

ln

VT

ZNkTU

ln2

TT V

ZNkT

V

FP

ln


Ideal Mono-Atomic GasIdeal Mono-Atomic Gas

kT

state

eZ //

)(8 2

2

2

2

2

22

c

n

b

n

a

n

m

h zyxi

)/()8/()/()8/()/()8/( 222222222 cnmkThbnmkThanmkTh zyx eeeZ

zcnmkTh

ybnmkTh

xanmkTh dnednedneZ zyx )/()8/(

0

)/()8/(

0

)/()8/(

0

222222222

adxe ax

2

10

2

2/3

2222

28

2

8

2

8

2

h

mkTV

h

mkTc

h

mkTb

h

mkTaZ

kTi

ilevelenergy

egZ //


Ideal Mono-Atomic Gas Ideal Mono-Atomic Gas – Evaluation of – Evaluation of kk

2

2ln

2

3ln

2

3lnln

h

mkTVZ

V

nRT

VTNk

V

ZTNkP

T

1

'ln

'

NkTT

NkTT

ZNkTU

V 2

3

2

3ln 22

Nkh

mkNkVNkTNk

T

ZNkTZNkS

V 2

32ln

2

3lnln

2

3lnln

2

ov sVRTcs lnlnfor 1 mol of gas

2/3

2

2

h

mkTVZ


Entropy Entropy – S = – S = kk ln W ln W


Equipartition TheoremEquipartition Theorem

kT2

1

'''''' ''''''/)''''''(/ ZZZeeZ kTkT

VT

ZNkTU

ln2

Z

N

U ln

translational kinetic energy :rotational kinetic energy : vibrational energy :kinetic energy for each independent component of motion has a form of

221 mv

2

2

1 I2

212

21 kxmv

2ii pb

2222

211 ff pbpbpb f

pbpbpb dpedpedpeP ff22

22211

02010

ii py 2/1 iiyb

ipb Kdyedpe iiii 2/1

0

2/1

0

22

ff

f KKKKKKZ 212/2/1

22/1

12/1

The average energy of a particle per independent component of motion is


Equipartition TheoremEquipartition Theorem

kTffP

22

ln

RTu2

3

RTu2

5

RTqu )3(

The average energy of a particle per independent component of motion is kT2

1

ff KKKZ 21

2/

※ for a monoatomic ideal gas :

for diatomic gases :

for polyatomic molecules which are soft and vibrate easily with many frequencies, say, q:

※ for liquids and solids, the equipartition principle does not work


Einstein and Debye Model for Heat Capacity Einstein and Debye Model for Heat Capacity – – Background & Concept Background & Concept

3N independent (weakly interacting) but distinguishable simple harmonic oscillators.

hii )( 21

kTh

kTh

e

eP

/

2/

1

)1ln(

2

1ln / kThe

kT

hP

for N simple harmonic vibrators

12

1ln/

2kTh

V e

hhN

T

PNkTU

average energy per vibrator 12 /

kThe

hh


Einstein and Debye Model for Heat Capacity Einstein and Debye Model for Heat Capacity – number density– number density

Let dNv be the number of oscillators whose frequency lies

between v and v + dv

dgdN )(

where g(v), the number of vibrators per unit frequency band, satisfy the condition

NdgdN 3)(

The energy of N particles of the crystal

dge

hhdNU

kTh)(

12 /

dge

ekThk

T

UC

kTh

kTh

VV )(

)1(

)/(2/

/2


Einstein and Debye Model for Heat Capacity Einstein and Debye Model for Heat Capacity – Einstein– Einstein

2/

/2

)1(

)/(3

kTh

kThE

V E

E

e

ekThkNC

k

h EE

2/

/2

)1(3

T

TEV

E

E

e

e

TR

c

All the 3N equivalent harmonic oscillators have the same frequency vE

Defining Einstein characteristic temperature

dge

ekThk

T

UC

kTh

kTh

VV )(

)1(

)/(2/

/2


Einstein and Debye Model for Heat Capacity Einstein and Debye Model for Heat Capacity – Debye– Debye

A crystal is a continuous medium supporting standing longitudinal and transverse waves

23

9)(

m

Ng

de

ekTh

Nk

C m

kTh

kThmV

0 2/

/232

)1(

)/)(/3(

3

kT

hx

TkT

hx m

m

dxe

ex

TR

c T

x

xV

/

0 2

4

3 )1()/(

3

3

dge

ekThk

T

UC

kTh

kTh

VV )(

)1(

)/(2/

/2

set


Einstein and Debye Model for Heat Capacity Einstein and Debye Model for Heat Capacity – Comparison– Comparison


Einstein and Debye Model for Heat Capacity Einstein and Debye Model for Heat Capacity – More about Debye– More about Debye

Behavior of dxe

ex

TR

c T

x

xV

/

0 2

4

3 )1()/(

3

3

at T → ∞ 2

4

)1( x

x

e

ex → x2

RcV 3

34

5

4

3

T

R

cV : Debye’s T3 law

at T → ∞ and T → 0

at T → 0


Einstein and Debye Model for Heat Capacity Einstein and Debye Model for Heat Capacity – More about Cp– More about Cp

Tce '

for T << TF


Statistical Interpretation of Entropy Statistical Interpretation of Entropy – Numerical Example– Numerical Example

A rigid container is divided into two compartments of equal volume by a partition. One compartment contains 1 mole of ideal gas A at 1 atm, and the other compartment contains 1 mole of ideal gas B at 1 atm.

(a) Calculate the entropy increase in the container if the partition between the two compartments is removed.(b) If the first compartment had contained 2 moles of ideal gas A, what would have been the entropy increase due to gas mixing when the partition was removed?(c) Calculate the corresponding entropy changes in each of the above two situations if both compartments had contained ideal gas A.

byeong-joo lee calphad 이 병 주 포항공과대학교 신소재공학과 [email protected]

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