byeong-joo lee calphad 이 병 주 포항공과대학교 신소재공학과 calphad@postech.ac.kr
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Byeong-Joo Lee www.postech.ac.kr/~calphad
이 병 주 이 병 주
포항공과대학교 신소재공학과포항공과대학교 신소재공학과calphad@postech.ac.krcalphad@postech.ac.kr
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
System with particles System with particles – Microstates of a System– Microstates of a System
Byeong-Joo Lee www.postech.ac.kr/~calphad
Macrostate / Energy Levels / Microstates Macrostate / Energy Levels / Microstates – –
Byeong-Joo Lee www.postech.ac.kr/~calphad
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 .
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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 //
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
Entropy Entropy – S = – S = kk ln W ln W
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
Einstein and Debye Model for Heat Capacity Einstein and Debye Model for Heat Capacity – Comparison– Comparison
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
Einstein and Debye Model for Heat Capacity Einstein and Debye Model for Heat Capacity – More about Cp– More about Cp
Tce '
for T << TF
Byeong-Joo Lee www.postech.ac.kr/~calphad
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.
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