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Instructor: Prof. Chi-Chung HuaInstructor: Prof. Chi-Chung Hua
Molecular Rheology LAB
Advanced ThermodynamicsAdvanced Thermodynamics
Complex Fluids & Molecular Rheology Laboratory, Department of Chemical Engineering,
National Chung Cheng University, Chia-Yi 621, Taiwan, R.O.C.
2
Course OutlineCourse Outline
Instructor: 華 繼 中 教授 ( 工二館 Rm 412 Ext. 33412)
Textbook: Herbert B. Callen, 1985, “Thermodynamics and Introduction to Thermostatistics”, 2nd ed., John Wiley & Sons, Inc. ( 歐亞書局代理 )
Grading: Homework (20 %)
Two Exams (80 %)
Teaching assistants: 李正光、林志榮 (Rm 301, Ext. 23471)
Molecular Rheology LAB
3
Part one: Classic ThermodynamicsChapter 1: Basic Concepts and PrinciplesChapter 2: Thermodynamic potentials and Legendre
TransformationsChapter 3: Stabilities, Phase Transitions, and Critical
PhenomenaChapter 4: Applications to Material Properties
Part two: Statistic ThermodynamicsChapter 5: Statistic Ensembles and Energy RepresentationsChapter 6: Some Applications of Statistic Mechanics
Molecular Rheology LAB
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Basic Concepts and PrinciplesBasic Concepts and PrinciplesChapter 1Chapter 1
Molecular Rheology LAB
Contents1-1 Thermodynamic Equilibrium and Postulate I
1-2 The Entropy Maximum and Postulates II & III
1-3 Intensive Parameters and Equations of State
1-4 The Euler Equation and the Gibbs-Duhem Relation
1-H Homework
5
Thermodynamic Equilibrium and Thermodynamic Equilibrium and Postulate IPostulate I
1-1.1 Thermodynamic coordinates1-1.1 Thermodynamic coordinates
1-1
Variables
Experimental variables
Simulation variables
Macroscopic vs. Microscopic
Responses
Experimental results
Simulation data
1-1
Molecular Rheology LAB
6
Q1: How to select variables of an experimental or simulation system?
Q2: Select macroscopic or microscopic variables?
A1: 1. Use minimum number of measurable variables.
2. Each variables must be independent.
A2: The basis of selection is the scale of the phenomena or substances of primary interest.
1-1
length scale
1μmMicroscopic Macroscopic
Molecular Rheology LAB
7
1-1
Microscopic Macroscopic
Microscopic properties can spread to macroscopic properties
But macroscopic properties cannot spread to microscopic properties
We usually used macroscopic properties to describe engineering problem, simply because it’s easier to do.
The price paid is, however, that small-scale problems cannot be studied, and some macroscopic properties themselves become difficult to understand.
Molecular Rheology LAB
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Molecular Rheology LABhttp://www.bibalex.org/English/lectures/printable/zewail.htm
Figure 1.1 Length and time scales of atoms and cats
9
1-1.2 Definition of equilibrium states1-1.2 Definition of equilibrium states
System has no external influences (such as flow, electrical fields).
All thermodynamic properties must be “time invariance” (time independent).
1-1
When all properties have reached time independence in our selected time scale, the system would seem to reach an equilibrium. Yet, since different people may have different opinions of time scale, the viewpoint of equili-brium could also be different.
Molecular Rheology LAB
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1-1.3 Limitations of thermodynamic equilibrium1-1.3 Limitations of thermodynamic equilibrium
1-1
Scale limitation: Length scale must be greater than ca.10nm.
Time invariance: All variables and properties must be time independent.
Postulate I:All macroscopic properties of an equilibrium system can be completely
characterized by extensive parameters U, V, and N1,N2,…Nr; e.g., S = S(U, V, N) for a one-component system.
Molecular Rheology LAB
11
The Entropy Maximum Postulates: The Entropy Maximum Postulates: Postulate II & IIIPostulate II & III
1-2
1-2
Postulate III:S is a monotonically increasing function of U.
Molecular Rheology LAB
Postulate II: For an isolated system with constant V, N, U the equilibrium criterion requires that the entropy S reaches a maximum value.
1
1
, , ... ,
( , , , ... , )
0r
r
V N N
S S U V N N
S
U
12
1-2
S = S ( U ) only
' '1 2 1 2
'1
'2
?
?
U U U U
U
U
By entropy max.
Molecular Rheology LAB
V1
U1
N1
V2
U2
N2
thermodynamic constraints
The system may then be further divided into two sub-systems subject to certain constraints:
S = S ( U , V , N )
13
1-2
1 1 21 2
11 2
1 1( ) 0 0
1 1( ) 0 0
dS dU dU dUT T
dUT T
T1 = T2
An important equilibriumcondition
Molecular Rheology LAB
,
1 1 2 2
1 21 2 1 2
1 2 1 2, ,
; is extreme value 0
( ) ( )
1 10
N V
N V N V
SdS dU S dS
U
S S U S U
S SdS dU dU dU dU
U U T T
Use d2S to check S is max. or min.
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Molecular Rheology LAB
Others equilibrium conditions can be derived with fixed V & U or N & U.
μ1 = μ2 for fixed V & U
P1 = P2 for fixed N & U
V1
U1
N1
V2
U2
N2
thermodynamic constraints
V1
U1
N1
V2
U2
N2
thermodynamic constraints
Fixed V & N Fixed U & N1-2
15
Intensive Parameters and Equations of Intensive Parameters and Equations of StateState
1-3
Molecular Rheology LAB
1-3
1-3.1 Extensive parameters & intensive parameters1-3.1 Extensive parameters & intensive parameters
Extensive parameters
1
, , , : First order extensive parameters
( , , ) ( , , )
S U V N
S U V N S U V N
Example:
(9 , 9 , 9 ) 9 ( , , )S U V N S U V N
S S
S
S
S
S S
S
S
S
Enlarged 9 times
First order
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1-3
Molecular Rheology LAB
Intensive parameters
( , , ) ; ( , , )S S U V N U U S V N
, , ,
, , ,
1 ; ;
; ;
N V U N U V
N V N S S V
S S P S
U T V T N T
U U UT P
S V N
Fundamental equations
At first, people didn’t know the meanings of T, P and μ. Later, they realized that T, P and μ were identical with temperature, pressure and chemical potential in our ordinary life.
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1-3
Molecular Rheology LAB
, S V
U
N
Example:
Liquid
Gas
Chemical potential:
The definition of chemical potential is the energy change from a particle, molecule or electron move into or leave the system.
If the overall system has reached an equili-brium, the energy reduction in the liquid phase due to the moving of a molecule into the gas phase is the same as the energy increase in the gas phase. So, the total system energy remains unchanged.
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1-3
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Fundamental equations
( , , )
( , , )
S S U V N
U U S V N
Entropy representation
Energy representation
Two equations are alternative forms of the fundamental relation, and each contains all thermodynamic information about the system.
1-3.2 Fundamental equations & equations of state1-3.2 Fundamental equations & equations of state
The fundamental equations describe homogeneous first-order parameters.
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1-3
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Equations of state
, , ,
( , , )
; ; N V N S S V
U U S V N
U U UT P
S V N
( , , )
( , , )
( , , )
T T S V N
P P S V N
S V N
Equations of state:
An intensive parameter is expressed in terms of independent extensive parameters.
Knowledge of a single equation of state does not constitute complete knowledge of the thermodynamic properties of a system.
Knowledge of all independent equations of state is equivalent to the knowledge of the fundamental equation.
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1-3
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Figure 1.2 U vs. S & U vs. T in constant V & N
Each line of U(S) may yield the same U(T) curve (an equation of state). But the reverse is not true.
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1-3
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0
, ,
( ) ( )
( ) ( )N V N V
U UT
S S
The equations of state describe homogeneous zero-order parameters.
U & S are first order parameters
Zero order
( , , ) ( , , )
( , , ) ( , , )
( , , ) ( , , )
T S V N T S V N
P S V N P S V N
S V N S V N
Equations of state
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1-3.3 Unit of temperature 1-3.3 Unit of temperature
1-3
Molecular Rheology LAB
Th
Tc
dQh
dWrev
Thermodynamic engine
revε 1 ce
h h
dW T
dQ T
Thermodynamic engine efficiency:
1 1
2 2
1 1
2 2
T T
T T
T T k
T T k
Used to define temperature scale.
Permissible scale (e.g. K, R)
Non-permissible scale (e.g. oC, oF)
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1-3
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Question: All heat can be transfer to work?
Heat Molecular
Rotation
Vibration
Kinetic energy
Internal energy
Work
Answer: It is impossible unless Tc = 0K
Heat Molecular
Rotation
Vibration
Kinetic energy
Internal energy = 0
All heat transfers to produce work
Tc ≠ 0K
Tc = 0K
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1-3
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Unit of temperature
,
N V
UT
S
Unit of U is energy.
What is the unit of entropy?
Unit of entropy:
In classic thermodynamics, entropy is dimensionless parameter.
In thermostatistics, unit of entropy correlates with states.
So, unit of temperature is energy.
The temperature scale, Kelvin, was used in molecular scale.
e.g. Energy of a simple molecule = 3/2 kBT = 3/2 Kelvin
Joule/Kelvin = 1.38 x 10-23 = kB
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1-3
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1-3.4 The general case of entropy max. in equilibrium1-3.4 The general case of entropy max. in equilibrium
1 1 1 1 1 1
1 1 1 1 2 2 2 2
1 1
1 1 1
1 1 11 1 1
1 1 1, , ,
2
2
( , , ) ( , , ) w.s.t. , ,
1
total
V N U N V U
S S U V N S U V N dU dV dN
P
T T T
S S SdS dU dV dN
U V N
S
U
2 2 2 2 2 2
2 22 2 2
2 2, , ,
2 2
2 2 2
1
V N U N V U
S SdU dV dN
V N
P
T T T
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1-3
Molecular Rheology LAB
at equilibrium
This equation always holds only if
1 2 1 21 1 1
1 2 1 2 1 2
1 2
1 2
1 2
1 10
P PdS dU dV dN
T T T T T T
T T
P P
If in reaching equilibrium
initial state
1.
2.
3.
1 2 1 2 1 2 1
1 2 1 2 1 2 1
1 2 1 2 1 2 1
0
, , 0
, , 0
, , 0
dS
P P T T dU
P P T T dV
P P T T dN
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The Euler Equation and the Gibbs-Duhem The Euler Equation and the Gibbs-Duhem RelationRelation
1-4
Molecular Rheology LAB
1-4.1 The Euler relation1-4.1 The Euler relation
1-4
Homogenous first order property
differentiating w.r.t. ( )1 2 1 2( , , , , ...) ( , , , , ...)U S V N N U S V N N
d
1 2 1 2
( , , , , ...) ( ) ( , , , , ...) ( )
( ) ( )
( , ,
T S P V
U S V N N S U S V N N V
S V
U S V N
1 2 1
1 21
1 1
, , ...) ( )... ( , , , , ...)
( )
N NU S V N N
N
N
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Molecular Rheology LAB
1-4
j jj = 1
i i j ji = 1 j = 1
i ii = 1
Let
Another form:
1 2 1 1 2 2
1 2
1 2
( , , , , ...) ...
( , , , , ...)
( , , , , ...)
U S V N N TS PV N N
TS PV N
P X PV N
U S V N N TS P X
S U V N N
jj
j
1 PU V N
T T T
The Euler relation is useful for deducing many important thermodynamic relationships, including the famous Gibbs-Duhem equation.
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1-4
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1-4.2 The Gibbs-Duhem equation1-4.2 The Gibbs-Duhem equation
i ii = 1
i i i ii = 1 i = 1
i ii = 1
i ii = 1
Euler relation:
Given that
One obtains
Another form
1 2( , , , , ...)
0
1
U S V N N TS P X
dU TdS SdT PdX X dP
dU TdS PdX
SdT X dP
PUd Vd
T T
jj
j
0N dT
the Gibbs-Duhem equation
Not all intensive properties are independent, but only n-1 of them are independent. Another one can be deduced from the Gibbs-Duhem equation.
30
Molecular Rheology LAB
1-4
Example:
For one-component systems (having 3 intensive properties & )
unit number
,
0
& : &
T P
SdT VdP Nd
S Vd dT dP sdT vdP
N Ns v S V
Only T & P are independent, μ can be deduced from the Gibbs-Duhem equation.
The Euler relation is useful for deducing many important thermodynamic relationships, including the famous Gibbs-Duhem equation.
The Gibbs-Duhem relation specifies, in a differential form, the relationship among all “intensive” properties of a thermodynamic system.
Clearly, μ is not independent of T and P in the previous example.
31
1-4
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1-4.3 Summary of formal structure1-4.3 Summary of formal structure
The fundamental equation U=U(S, V, N) contains “all” thermodynamic information.
Equations of state + the Euler equation = The full expression of U
( , , )
( , , ) ( , , )
( , , )i i
T T S V N
P P S V N U TS P X U U S V N
S V N
The totality of all equations of state can be used to reconstruct the fundamental equation.
32
HomeworkHomework1-H
1-H
Molecular Rheology LAB
Problem 1.10-1: (a), (f), (g), (i)
Problem 2.2-1
Problem 3.3-2
Read Sec. 3-5 (The ideal van der Waals fluid)
33
Thermodynamic Potentials and Thermodynamic Potentials and Legendre TransformationsLegendre Transformations
Contents2-1 Legendre Transformations
2-2 The Legendre Transformed Functions and Thermodynamic Potentials
2-3 Minimum Principles for the Potentials
2-4 Applications of various Legendre Transformed Potentials
2-5 Maxwell Relations and Some Applications
Chapter 2Chapter 2
Molecular Rheology LAB
34
Legendre TransformationsLegendre Transformations
2-1.1 The 2-1.1 The SS max. principle & the max. principle & the UU min. principle min. principle
2-1
2-1
Molecular Rheology LAB
An isolated system must conform with two extremum criteria when equilibrium is reached.
Entropy max. principle:
Entropy is max. value for given total energy.
Energy min. principle:
Energy is min. value for given total entropy.Equivalent
representations
35
2-1
Molecular Rheology LAB
Figure 2.1 The equilibrium state A as a point of max. S for constant U & min. U for constant S.
Fundamental surface
S max. principle U min. principle
S max. principle: The equilibrium value of any unconstrained internal parameter is such as to maximize the entropy for a given value of the total internal energy.
U min. principle: The equilibrium value of any unconstrained internal parameter is such as to minimize the energy for a given value of the total internal entropy.
36
2-1
Molecular Rheology LAB
Question: How to prove the two principles are equivalent?
Answer: Using either physical argument or mathematical exercise.
Physical argument
Assume: The system is in equilibrium but the energy does not have its smallest possible value consistent with the given entropy.
The system would be restored to its original energy but with an increased S. So, it is inconsistent with the principle that the initial equilibrium state is the state of maximum entropy.
System
Can withdraw energy
dU = -dW
Return energy to sys.
-dW = dQ = TdS
Energy
The original equilibrium state must have had min. energy consistent with the prescribed entropy. The inverse argument is the same.
37
2-1
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Mathematical exercise
and : Particular extensive parameter2
20 0U U
S SX
X X
Entropy maximum principle:
0U
S U
X
SXU S
P TSX XU
By eqn. A.22 of Appendix A
0S
U
X
( , )S
UP P P U X
X
2
2 at 0S X U US
U P P P PP P
X X U X X
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2-1
Molecular Rheology LAB
22
22
22
2 2
2 2
=
0 at 0 0 & 0
U U
U UU
X XU X
U
SS SXXU P S X U
S SX X X X SU U U
S S ST T
X X X
; 2
20 0U U
S S
X X
;
2
20 0S S
U U
X X
From S max. principle: U min. principle:
39
2-1
Molecular Rheology LAB
2-1.2 Legendre transformations2-1.2 Legendre transformations
Motivation:
U(S, V, N) or S(U, V, N) both are extensive parameters. Some of these parameters might be difficult to control or measure under usual experimental conditions.
Basic idea:
Using Legendre transformations is to cast the above representation into equivalent forms which, by contrast, utilize intensive parameters as independent variables.
40
2-1
Molecular Rheology LAB
X, Y are extensive properties & Y=Y(X).
Let slope is ; : intersive propertiesY
P PX
If one considers the slope and the intercept at the same time, the transformation from Y(X) to ψ(P) is an unambiguous mapping.
Elimination of and yields
( )
0
( )
Y Y X
Y YP
X XY PX
X Y
P
Elimination of and yields
( )
( )
P
XP
Y PX
P
Y Y X
41
Generalization of the Legendre transformation
2-1
Molecular Rheology LAB
Elimination of and , ,
... , yields
0 1
0 1
0 1 1
( , , ... , )
( , , ... , , ,
... , )
t
kk
k kk
n
n n
t
Y Y X X X
YP
X
Y P X
Y X X
X
P P P X
X
Elimination of and , ,
... , yields
0 1 1
0 1
0 1
( , , ... , , ,
... , )
( , , ... , )
n n
t
kk
k kk
n
t
P P P X
X
XP
Y P X
P P
P
Y Y X X X
42
The Legendre Transformed Functions The Legendre Transformed Functions and Thermodynamic Potentialsand Thermodynamic Potentials
2-2
2-2
Molecular Rheology LAB
The energy representation of the
fundamental equation: ( , , )U U S V N
Intensive parameters
, , ...1 2, ,T P
0 1( , , ... , )t
kk
k kk
Y Y X X X
YP
X
Y P X
Legendre transformed functions
Thermodynamic potentials
( , , )
( , , )
( , , )
F F T V N
H H S P N
G G T P N
43
The Helmholtz potential, F
2-2
Molecular Rheology LAB
( , , ) ( , , )U S V N F T V N
Y X ψ P
( , , )
( , , )
U U S V N
UT
SF U TS
F F T V N
, , ,
; ; N V T N T V
F F FS P
T V N
dF SdT PdV dN
0 1
0 1 1
( , , ... , )
( , , ... , , ,
... , )
t
kk
k kk
n n
t
Y Y X X X
YP
X
Y P X
P P P X
X
44
2-2
Molecular Rheology LAB
The Enthalpy, H
( , , ) ( , , )U S V N H S P N
Y X ψ P
( , , )
( , , )
U U S V N
UP
VH U PV
H H S P N
, , ,
; ; P V S N P S
H H HT V
S P N
dH TdS VdP dN
0 1
0 1 1
( , , ... , )
( , , ... , , ,
... , )
t
kk
k kk
n n
t
Y Y X X X
YP
X
Y P X
P P P X
X
45
2-2
Molecular Rheology LAB
The Gibbs potential, G
( , , ) ( , , )U S V N G T P N
Y X2ψ P2
( , , )
;
( , , )
U U S V N
U UT P
S VG U ST PV
G G T P N
, , ,
; ; P N T N P T
G G GS V
T P N
dG SdT VdP dN
X1 P1
0 1
0 1 1
( , , ... , )
( , , ... , , ,
... , )
t
kk
k kk
n n
t
Y Y X X X
YP
X
Y P X
P P P X
X
46
The following equations are all fundamental equations:
2-2
Molecular Rheology LAB
The main spirit of the Legendre transformation is that the new fundamental equation must be so derived that it does not lose any thermodynamic information as incorporated in the original one.
( , , ) ( , , ) ( , , ) ( , , ) ( , , )S U V N U S V N F T V N H S P N G T P N
Typical applications of the derived fundamental equations:
F=F(T, V, N): Incompressible fluidsH=H(S, P, N): Heat transfer G=G(T, P, N): Chemical reactions
47
Minimum Principles for the PotentialsMinimum Principles for the Potentials2-3
2-3
Molecular Rheology LAB
Reservoir T r
T(1) T(2)
Thermal reservoir (Cp → ∞, T = T r )
Subsystem
The Helmholtz potential
Consider a subsystem in contact with a thermal reservoir.
Using U representation U=U(S, V, N) :
2
0
0
total
total
dU
d U
from the extremum principle at equilibrium
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Molecular Rheology LAB
(1) (1) (2) (2)
(1) (1) (2) (2) (1) (1) (2) (2)
( ) ( )
( ) ( ) 0
total r r rdU dU dU T dS T dS T dS
P dV P dV dN dN
rdU dU
, , ,r r r r r r
r r rr r r r
r r r
V N S N V S
r r
U U UdU dS dV dN
S V N
T dS
constant
The subsystem of interest ;
;
(1) (2)
(1) (2)
(1) (2) (1) (2)
(1) (2) (1) (2)
0
0 0
total total r r
r
S dS dS dS dS dS dS
dS dS dS
dV dV dN dN
dV dV dN dN
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(1) (1) (2) (2) (1) (2) (1) (1) (2) (1)
(1) (2)
(1) (2)
(1) (2)
( ) ( ) ( ) ( ) 0
This equation always holds only if
r r
r
T T dS T T dS P P dV dN
T T T
P P
At equilibrium
2 2 2 2 2 2 2 2
2
( ) ( ) ( ) 0
0
( ) ( ) ( ) 0
0
total r r r r r
total r r
dU dU dU dU T dS d U T S d U TS
dF F U TS
d U d U U d U d U d U d U d TS d U TS
d F
By eqn. A.9 of Appendix A ;
For thermal reservoir T = T r, V
r & N r = constant
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Reservoir P r
P(1) P(2)
Reservoir T r & P
r
T(1) P(1) T(2) P(2)Subsystem
By the same way, consider a subsystem in contact with a pressure reservoir, or with both pressure & thermal reservoirs.
2
( , , )
0
0
H H S P N
dH
d H
2
( , , )
0
0
G G T P N
dG
d G
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The overall system:
The subsystem:
Representation: Experimental conditions: Extremum criteria:
, 2( , , ) , 0 ; 0total total total total totalU U S V N V N S dU d U
Representation: Experimental conditions: Extremum criteria:
2( , , ) , , 0 ; 0
( , , ) , ,
F F T V N T V N dF d F
H H S P N S P N
2
2
0 ; 0
( , , ) , , 0 ; 0
dH d H
G G T P N T P N dG d G
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Molecular Rheology LAB
The equilibrium value of any unconstrained internal parameter in contact with a heat reservoir minimizes the Helmholtz potential over the manifold of states for which T = T
r
Similarly, one can prove the Enthalpy minimum principle H(S, P, N) for systems in contact with a pressure reservoir, as well as the Gibbs potential minimum principle G(T, P, N) for systems in contact with both a heat and pressure reservoir.
53
Applications of various Legendre Applications of various Legendre Transformed PotentialsTransformed Potentials
2-4.1 The Helmholtz potential 2-4.1 The Helmholtz potential
2-4
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Molecular Rheology LAB
Question: What are the main applications?
Answer: Experiments carried out in rigid vessels with diathermal walls, so that the ambient atmosphere acts as a thermal reservoir.
Constant T Constant V, N (the subsystem)
The F(T, V, N) is admirably suited.
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Example: Two simple subsystems separated by a
movable, adiabatic, impermeable wall.(1)
(2)
rev
?
?
?
V
V
W
(1) (1) (1) (2) (2) (2)
(1) (2)
( , , ) ( , , )
, a constant
r rP T V N P T V N
V V V
Can solve & (1) (2)V V
The work delivered in a reversible process, by a system in contact with a thermal reservoir, is equal to the decrease in the Helmholtz potential of the system.
The number of independent variables: V & N
rev ( )
r r r r rdW dU dU dU T dS dU T dS d U T S
dF
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Focus our attention exclusively on the subsystem of interest, and relegate the reservoir only to an implicit role.
As shown in Chap. 16, statistical mechanical calculations are enormously simpler for such systems, permitting calculations that would otherwise be totally intractable.
The Helmholtz potential is often referred to as the Helmholtz free energy, though the term available work at constant temperature, would be less subject to misinterpretation.
See Example.1 in page 159, which shows that one can increases the entropy of a subsystem without changing the associated internal energy.
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2-4.2 The enthalpy2-4.2 The enthalpy
At constant 1 1 2 2 ...
, i
dH TdS VdP dN dN
P N dH TdS dQ
As a potential for heat
Heating of a system is so frequently done while the system is maintained at const. P. The enthalpy is generally useful in discussion of heat transfers, and H is as a heat content of the system.
At constant
,
( , , )
( , , )
( , , ) ( , , ) ,
S N
i f f i
H S P N
HV V S P N
P
Q H V P N H V P N P N
( , , )H V P N
Using a similar argument, U is a potential for heat, too. ; at constant ,dU TdS PdV dN V N dU TdS dQ
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The Joule-Thomson process
by conservation of energy;f i i i f f
f f f i i i
f i
u u P v P v
u P v u P v
h h
: initial molar energy
: molar volume
: molar enthalpy
u
v
hA constant H process
We do not imply anything about the enthalpy during the process; the intermediate states of the gas are non-equilibrium states for which the enthalpy is not defined.
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The isenthalpic curves (“isenthalps”)
1 2, , , ...H N N
TdT dP
P
1P
vdT T dP
C
The final temperature Tf can be higher or lower then Ti in this process, depending on the local relationship:
By eqn. A.22 of Appendix A
& Maxwell relations
If Tα >1, T will increase with a small increase in pressure, and vice versa.
For ideal gas, α = 1 / T, so that there will be no change in T.
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2-4.3 The Gibbs potential2-4.3 The Gibbs potential
Most experiments are carried out under constant T & P.
is a particularly convenient choice . . . . 1 2( , , , , )G T P N N
by defition:
the Euler relation: i ii
i ii
G U TS VP
U TS VP N
G N
As a potential for chemical potential
Consider the chemical reaction: 1
0r
j jA
symbols for the chemical components
stoichiometric coefficients
:
:
j
j
A
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Change of G associated with changes in Nj can be described by
j jj
dG SdT VdP dN
since 1 2
1 2
... j j
dN dNdN dN dN
or
0
0
j jj
j jj
dG dN
One has for the equilibrium condition:
The final state of a chemical reaction can thus be determined if the solution of the above equation is known, given that there is no depletion of any species.
While G determines the equilibrium condition, H the heat of the reaction can be obtained by the following relation:
1 21 , , , ...
r
j j
P N N
dHT
dN T
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2-4.4 Compilations of empirical data2-4.4 Compilations of empirical data
In practice, it is common to compile data on h (T, P), s (T, P), v (T, P);
v is redundant but convenient.
These relationships can, in principle, be utilized to construct the Gibbs potential via g = h –Ts, and thus all thermodynamic may be obtained using proper relationships, such as the Maxwell relationships.
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Maxwell Relations and Some ApplicationsMaxwell Relations and Some Applications
2-5.1 The Maxwell relations & mnemonic diagram2-5.1 The Maxwell relations & mnemonic diagram
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Maxwell relations
( , , )U U S V N
2nd derivatives of potential
U U
S V V S
Maxwell relation equation
, ,V N S N
P T
S V
P T
Constant N
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Table 2.1. Maxwell relations for several potentials with various pairs of independent variables. (see also pp. 182 & 183 of the textbook)
In a similar way, we can derive other forms as shown below:
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Mnemonic diagram
V F T
U G
S H P
Valid Facts Theoretical
Generate
Problems
Understanding
HardSolutions
Pithy formula: “Valid Facts and Theoretical Understanding Generate Solutions to Hard Problems”
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(" " arrow pointed away)
(" " arrow pointed toward)
,
,
S N
V N
UT
S
UP
V
Example:
The first derivative of potential:
V F T
U G
S H P
Maxwell relations:Example:
P S
T P
V T
S P
S V
P T
V
S P
T
S P
V
S P
T
P
T
(" ") (" ")
(" ")
(" ")
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If we are interested in Legendre transformations dealing with S and Nj , the mnemonic diagram takes the form shown as below in the left diagram:
Nj T
U
S
[ ]F U T
[ ]jU
[ , ]jU T
X1 P1
U
X2 P2
2[ ]U P
1[ ]U P
1 2[ , ]U P P
General-form mnemonic diagram
U(S, Nj) mnemonic diagram
j
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2-5.2 Procedure for the reduction of derivatives2-5.2 Procedure for the reduction of derivatives
Thermal expansion coefficient:
Isothermal compressibility:
Molar leat capacity at constant pressure:
1 1
1 1
P P
TT T
PP
v V
v T V T
v V
v P V P
s T Sc T
T N
Molar leat capacity at constant volume:
1
1
P P
VV V V
dQ
T N dT
s T S dQc T
T N T N dT
Four customarily used derivatives
68
For single-component systems and constant molecule numbers, the second derivatives of Gibbs representation can be expressed by
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, , and .P Tc
/ ; ; 2 2 2
2 2P T
g g gc T v v
T T P P
All first derivative, such as (involving both extensive and intensive parameters), can be written in terms of three independent second derivatives of the Gibbs potential (for the case of single-component systems), conventionally chosen as .
,V N
dT T P dP
and , P Tc
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Reduction of derivatives
If the derivative contains any potential, bring them one by one to the numerator and eliminate by the thermodynamic square.
Example:
Reduce the derivative,G N
P
U
1
, ,
1
, ,
1
, , , ,
G N G N
G N G N
S N P N V N P N
P U
U P
S VT P
P P
G G G GT P
P S P V
1
, ,
, ,
S N V N
P N P N
S T P V S T P VT P
S T S S T V
(See also pp. 187-188 of the textbook)
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If the derivative contains the chemical potential, bring it to the numerator and eliminate by means of the Gibbs-Duhem relation, .d sdT vdP
Example:
Reduce the derivative,S NV
, , ,S N S N S N
T Ps v
V V V
If the derivative contains the entropy, bring it to the numerator. If one of the four Maxwell relations of the thermodynamic square now eliminates the entropy, invoke it. If the Maxwell relations do not eliminate the entropy put a under . The numerator will then be expressible as one of the cV or cP.
T S
Example:
Reduce the derivative,S N
T
P
, , ,,P
T N P N P NS N
T S S V Nc
P T T TP
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Bring the volume to the numerator. The remaining derivative will be expressible in terms of . and T
Example:
Reduce the derivative,V N
T
P
, ,,
T
T N P NV N
T V VP TP
The originally given derivative has now been expressed in terms of the four quantities . The specific heat at constant volume is eliminated by the follow equation.
and , , ,P V Tc c
2
V PT
Tvc c
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2-5.3 Some simple applications2-5.3 Some simple applications
I. Adiabatic compression
Ti
Pi
Adiabatic wall
Tf
=?
Pf
Adiabatic wall
Quasi-static process
If known: fundamental equation, , , and i f iP P T
Fundamental equationEquations of state:
ΔT=T(S, Pf , N)- T(S, Pi , N)
TIf known: , , , , , and i f i PP P T c
; , ,
( )S N P S N P
T Tv sTvdT dP dP d d v dP
P c P c
(see p. 190)
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II. Isothermal compression
Pi Pf
Quasi-static process
Constant T
?
?
?
dS
dU
Q
If known: fundamental equation, , and i fP P
Fundamental equation
TIf known: , , , , and i f PP P c
;
, ,
( )
f
i
TT N T N
P
P
S UdS dP VdP dU dP T v PV dP
P P
dQ T VdP Q T VdP
Equations of state:
( , , )
( , , )
( , , ) ( , , )f i
S S T P N
U U T P N
Q T S TS T P N TS T P N
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III. Free expansion
VacuumGas
Vi ; Ti
Gas
Vf ; Tf =?
If known: fundamental equation, , , and i f iV V T
Fundamental equation
TIf known: , , , , , and i f i PV V T c
Equations of state:
( , , ) ( , , )f i f iT T T U V N T U V N
Constant U
Adiabatic wall Adiabatic wall
,U N V V T
T P TdT dV dV
V Nc Nc
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HomeworkHomework2-H
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Problem 5.3-1
Problem 5.3-12
Problem 6.3-2
Problem 7.4-7
Problem 7.4-15
Problem 7.4-23
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Stabilities, Phase Transitions, and Stabilities, Phase Transitions, and Critical Phenomena Critical Phenomena Chapter 3Chapter 3
Contents3-1 Stability Conditions for Thermodynamic Potentials
3-2 Le Chaterlier’s Principle and Effects of Fluctuations
3-3 First-order Phase Transitions in Single-Component Systems
3-4 Thermodynamic States near Critical Points
77
Stability Conditions for Thermodynamic Potentials
3-1.1 Intrinsic stability of thermodynamic systems3-1.1 Intrinsic stability of thermodynamic systems
3-1
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Basic extremum principle of thermodynamics:
0dS The S is an extremum value.
The extremum value is a maximum, which guarantees the stability of predicted equilibrium states.
2 0d S
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Initial state New state
S(U) S(U) S(U-ΔU)S(U+ΔU)
2TotalS S U TotalS S U U S U U
Remove ΔU from the right subsystem and transfer it to the left subsystem.
Stable
2 ( )S U U S U U S U
Unstabl e
2 ( )S U U S U U S U
Convex
Concave
(see later dicussion on phase separation)
(see later discussion on the Le Chaterlier’s principle)
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The condition of stability is the concavity of the entropy. for all 2 ( )S U U S U U S U
For 2
2
,
0 0V N
SU
U
By eqn. A.4 of Appendix A (refer to problem 8.1-1 of the textbook)
The same considerations apply to a transfer of volume.
for all
For 2
2
,
2 ( )
0 0U N
S V V S V V S V
SV
V
The differential form is less restrictive than the concavity condition, which must hold for all ΔU(ΔV) rather than for ΔU→ 0(ΔV→ 0) only.
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The portion BCDEF is unstable, because it violates the global form of concavity of entropy. Yet, only the portion CDE fails to satisfy the differential form of the stability condition.
The portions of the curve BC and EF are said to be “locally stable” but “globally unstable.” (e.g., it is susceptible to large-amplitude perturbations)
Points on the straight line BHF correspond to a phase separation in which part of the system is in state B and part in state F.
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If one simultaneously changes U and V:
22 2 2
2 2
, , 2 ( , )
0
S U U V V S U U V V S U V
S S S
U V U V
2
2
,
0V N
S
U
2
2
,
0U N
S
V
The local conditions of stability ensure the curve of the intersection of the entropy surface with the plane of constant V or U have a negative curvature.
But these two “ partial curvatures” are not sufficient to ensure concavity.
By eqn. A.4 of Appendix A (refer to
problem 8.1-1 of the textbook)
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3-1.2 Stability condition for thermodynamic potentials3-1.2 Stability condition for thermodynamic potentials
Convexity of the energy surface:
, , , , 2 ( , , )U S S V V N U S S V V N U S V N
The local conditions of convexity:
For cooperative variations of and :
2
2
,
2
2
,
22 2 2
2 2
0
0
0
V N
S N
U T
S S
U P
V V
S V
U U U
S V S V
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22 2 2
2 2 0F F F
V T V T
Transform to other potential representations by Legendre transformation :
( , , )F T V N
; ; By the same way
2 2
22, ,, ,,
2 2 2
2 2 2
, , ,
11 1 1
0 0 0
V N V NV N V NV N
V N V N T N
F S T U US S ST T
F U F
T S V
, , , , 2 ( , , )F T T V V N F T T V V N F T V N
Example:
By eqn. A.4 of Appendix A (refer to problem 8.1-1 of the textbook)
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Similarly, one has for other potentials
; ; 22 2 2 2 2
2 2 2 2
, ,
( , , )
0 0 0V N V N
H S P N
H H H H H
S P S P S P
; ; 22 2 2 2 2
2 2 2 2
, ,
( , , )
0 0 0V N V N
G T P N
G G G G G
T P T P T P
In summary, for constant N the thermodynamic potentials are convex functions of their extensive variables and concave function of their intensive variables.
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Le Chaterlier’s Principle and the Effects of Fluctuations
3-2.1 Le Chatelier’s principle3-2.1 Le Chatelier’s principle
3-2
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Molecular Rheology LAB
Example:
The Le Chatelier’s principle: any inhomogeneity that somehow develops will induce a process that eradicates the inhomogeneity.
Fluid
T1
In equilibrium state
Incident photonContainer
T1 T1
Locally heating the fluid slightly.
Heat flows away form heated
region.
The system restores homogeneous state.
Fluid
T1’
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3-2.2 Effect of fluctuations 3-2.2 Effect of fluctuations
From the perspective of statistical mechanics all physical systems undergo continual local fluctuations even in the absence of externally induced vibrations, only to attenuate and dissipate in accordance with the Le Chatelier principle.
The curvature of the potential well then plays a crucial and continual role, restoring the system toward the “expected state” after each Brownian impact (fluctuation).
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3-2.3 The Le Chatelier-Braun principle 3-2.3 The Le Chatelier-Braun principle
The Le Chatelier-Braun principle: Various other secondary processes that are also induced indirectly act to attenuate the initial perturbation.
T & P reservoirs
Pi ; Ti ; Vi
Example:
Poston is moved outward slightly by external agent or fluctuation.
Primary effect: The Pi decreased-the pressure difference across the piston then acts to push it inward; this is the Le Chatelier principle.
Po
Consider a subsystem contained within a cylinder with diathermal walls and a loosely fitting piston.
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Secondary effect: An increased Vi alters Ti ; heat flows inward if α > 0 and outward if α < 0.
S V T
T TdT dV dV
V Nc
; 2
1
S V T
PdP dQ dQ TdS
T S NT cdQ
Then, Pi is increased for either sing of α due to the heat flow. Thus a secondary induced process also acts to diminish the initial perturbation. This is the Le Chatelier-Braun principle.
T & P reservoirs
Pi ; Ti ; Vi
Poston is moved outward slightly by external agent or fluctuation.
Po
Q
(note that is always positive)
dQ
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First-order Phase Transitions in Single-Component Systems
3-3.1 First-order & second-order phase transitions3-3.1 First-order & second-order phase transitions
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FluctuationFluctuations (e.g. Brownian motions) play an important role bringing the system form one local min. to the other. The possibility associated with a successful jump is exponentially inverse to the energy gap. It is therefore possible that the system trapped in a metastable min. is effectively in stable equilibrium.
The system spends almost all the time in the more stable minimum.
Global min. is path independent state, yet the metastable one is path dependent.
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First-order phase transitions
Definition: Shift of the equilibrium state from one local minimum to the other, induced by a change in certain thermodynamic parameters.
Equilibrium state at constant T & P correspond to the min. in G(T, P, N).
The slope of the curve in the left figure is simply difference in pressure (?).
At T4, the molar Gibbs potential of two minima are equal, but other properties (e.g. u, f, h, v ,s) are discontinuous across the transition.
High-density phase is more stable below the transition temperature, T4
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Second-order phase transitions
Definition: At the so-called critical point (point D), the difference between two distinct states disappears and there is only one equilibrium state.
Moving along the liquid-gas coexistence curve toward higher temperatures, the discontinuities of two phase in molar volume and molar energy become progressively smaller.
The existence of the critical point (D) precludes the possibility of a sharp distinction between the generic term liquid and the generic term gas.
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3-3.2 Latent heat 3-3.2 Latent heat
At first-order phase transition:I II
I II
I II
Discontinued in , , , , . But
( ) ( )
( ) ( )
( ) ( )
u f h v s P P
T T
Example:Latent heat of fusion:
since ; at phase transition point
( ) ( )
( ) ( )
( )L SLS
L S
LS
dQ Tds Q l T s s T s
h Ts
h T s l
Ice Water SteamkJ2.1 kg-KPc kJ4.2 kg-KPc
cal80 gLSl cal540 gGLl
Saturated steam table
Superheated steam table
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3-3.3 The Clapyron equation 3-3.3 The Clapyron equation
Some interesting natural observations:
Ice skating: The pressure applied to ice on the blade of the skate shifts the ice, providing a lubricating film of liquid on which the skate slides.
Mountain climbing: The pressure decreased and the water boiling point decreased with the higher altitude. Therefore, it is difficult to cook on the high altitude mountain.
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The Clapron equation
Clapron equation is the slope of coexistence curves.Slope:
At phase equilibrium ;
;
'
'
and
the Cl
( )( ) ( )
( )
A A B B
B A B A
B A B A
dP
dT
d sdT vdP
sdT vdP s dT v dP
dP s s sv v dP s s dT
dT v v v
ll T s s
TdP l
dT T v
apeyron equation
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The Clapeyron equation embodies the Le Chatelier principle. Consider a solid-liquid transition with a positive latent heat (sl > ss):
If Slope
at constant System tends towIncrease more liquid phards the ase
( ) 0l s
dP
P
v vdT
T
Increase dense (solid) phase
Increase entropic (liquid)
If Slope
at constant System tends towards the more
System tends towards the mo asre ph e
( ) 0l s
dPv v
dP
T
TT
The Clapeyron-Clausius approximation:
; assume ideal gas
From Clapeyron equation 2
g l g g
RTv v v v v
PdP P
ldT RT
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3-3.4 13-3.4 1stst phase transition based on van der Waals eq. phase transition based on van der Waals eq.
The van der Waals equation of state describes rather well the general features of many gases and liquids.
van der Waals E.O.S.2( )
RT aP
v b v
Stability criteria: or 0 0TT
P
V
Not all curves conform to the stability criteria. Clearly, portion FKM of T1 curve violates the stability criterion.
Setting constant P will get one solution T in the T9 curve, and three solutions O, K and D in the T1 curve. But only O and D have physical meaning because K violates the stability criterion.
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The given P-v isotherms may be used to compute the molar Gibbs free energy to better perceive the stability criterion.
It should be obvious that B, C, D are more stable than M, L, K and R, Q, O are more stable than F, J, K etc.
or
is an undetermined function of the
( ) ( )
( )
d sdT vdP v P dP T
T T
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In general, for P < PD, the right-hand branch (ABCD) of the isotherm in right figure is physically significant, whereas for P > PD the left-hand branch (SRQD) is physically significant. The intermedium region is physically unstable.
The area of region I is the same as that of region II, and for the region in between liquid and gas (OD) coexist. This is the way to determine the phase transition points (O, D).
( ) 0O
Dv P dP
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The entropy change during the phase transition can be computed using the following relation along hypothetical isotherm OMKFD.
The total change in the molar energy:
Limitation: Can’t apply near critical point.
the Maxwell relation
T
v T vOMKFD
sds dv
v
P s Ps dv
T v T
u T s P v
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The O, D, O’, D’, O’’, D’’ are first-order phase transitions, and O’’’ (D’’’) corresponds to the critical point, termination of the gas-liquid coexistence.
The component in each of the phases is governed by the lever rule.
; 1g l g g l l z
g zl
g l
z lg
g l
x x x v x v v
v vx
v v
v vx
v v
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3-3.5 Summary of stability criteria 3-3.5 Summary of stability criteria
U=U(x1, x2, …) or U=U(P1, P2, …) must be a “convex function” of its “extensive parameters” and a “concave function” of its “intensive parameter.”
Geometrically, the function must lie above its tangent hyper-planes in the x1, … , xs-1 subspace and below its tangent hyper-planes in the Ps, … , Pt subspace.
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3-3.6 First-order phase transitions in multi-component 3-3.6 First-order phase transitions in multi-component systems systems
For a multi-component system the fundamental relation:
U = U(S, V, N1 ,N2, …,Nr)
or the molar form:
u = u(s, v, x1 ,x2, …, xr - 1)
At the equilibrium stable the “energy” , i.e., the enthalpy and the Helmholtz and Gibbs potentials are “convex functions” of the mole fraction x1, x2, … , xr - 1.
The mole fractions, like the molar entropies and the molar volumes, differ in each phase during a phase transition. Can be utilized for the purpose of purification by distillation.
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Gibbs phase rule is the thermodynamic degrees of freedom (f).
Gibbs phase rule
: No. of components
: No. of phases
: No. of parameters capable of independent v
: , each phase have variations No. of phase
intensi ariati
:
v
s
o se n
2 ( 1) ( -1)
( 1)
2 ( 1) ( 1)
( )
2f M r r M r M
r
M
f
M r M
r M
r P T
I II M
I II M
...
...
1 1 1
r r r
variations equations
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Application of the Gibbs phase rule
Example:
I. Single-component system (r= 1)
a: M= 2 → f= 1 → at a given T, P must be unique
b: M= 3 → f= 0 → the triple point
II. Two-component system (r= 2)
a: M= 1 → f= 3 → T, P, x1 (or x2) can be arbitrarily assigned
b: M= 2 → f= 2 → T, P can be arbitrarily assigned
c: M= 3 → f= 1 → T, for example, can be arbitrarily assigned
d: M= 4 → f= 0 → four phases coexist at unique point
※There cannot be five phases coexist in this case
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3-3.7 Phase diagrams for binary systems 3-3.7 Phase diagrams for binary systems
Example:
The three-dimensional phase diagram of a typical gas-liquid binary system. The two-dimensional sections are constant P planes, with P1 < P2 < P3 < P4.
The C state in the two-phase state is composed of A and B, and only A and B are physical point.
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Example:
Typical phase diagram for a binary system.
Phase diagram for a binary system in equilibrium with its vapor phase. (P is not constant)
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Thermodynamic States near Critical Thermodynamic States near Critical Points Points
3-4.1 Features in the vicinity of the critical point3-4.1 Features in the vicinity of the critical point
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As the point approaches the critical point the two minima of the underlying Gibbs potential coalesce.
The distinction in phases utilized in the first-order transition breaks.
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As the critical point is reached the single minimum is a flat bottom, which implies the absence of a “restoring force” for fluctuations away from the critical state.
In the critical region, long-range correlated fluctuation become the dominant factor affecting the thermodynamic properties, and the short-range atomistic feature becomes unimportant.
Universal scaling behavior is predicted by the “ renormalization group theory” and verified in experiments for the behavior of the susceptibilities “ near the critical point.”
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The generalized susceptibilities, such as cp, κT and α diverge.Example:
at first phase transition or near critical point
diverges; and diverge in a similar way.
1T
T
T
T p
v
v P
v
P
c
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3-4.2 The failure of the classical theory in predicting 3-4.2 The failure of the classical theory in predicting the scaling behavior near the critical point.the scaling behavior near the critical point.
Over-simplification of the extremum principle in classical thermodynamics. It must be replaced by the associated “probability distribution” and the consequent average properties.
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The basic postulate of thermodynamics incorrectly identifies the most probable value of the energy as the equilibrium or average value.
The deviation becomes very pronounced near critical points when the thermodynamic potentials become very shallow.
Renormalization-group theory demonstrates that the numerical values of the scaling exponents for a large class of materials are identical, dependent primarily on the dimensionality of the system and on the order parameter.
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HomeworkHomework3-H
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Problem 9.1-1
Problem 9.3-3
Problem 9.4-11
Problem 9.7-1
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Applications to Material Properties Applications to Material Properties
Contents4-1 The general ideal gas
4-2 Small Deviations from Ideality—The Virial Expansion
4-3 Law of Corresponding States for Gases
4-4 Applications to Dilute Solutions—The van’t Hoff Relation and the Raoult’s law
4-H Homework
Chapter 4Chapter 4
Molecular Rheology LAB
115
The general ideal gasThe general ideal gas
4-1.1 The essence of ideal gas behavior4-1.1 The essence of ideal gas behavior
4-1
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Molecular Rheology LAB
Ideal gas behavior (which is good at sufficiently low density and fails evidently in the vicinity of their critical points) means that the gas do not interact, implying the following properties:
1. PV = NRT
2. For single-component ideal gas, T = T(u) or u = u(T)
3. The Helmholtz potential F(T, V, N1, …) is additive (Gibbs’s theorem)
4. have the same value for all ideal gas , , 1 1
T p vc c RP T
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The , and of general multicomponent ideal gas:
: arbitrarily chosen standard temperature0
0
0
0
0
( , ) ( ) ( , )
( ) ( )
1( ) ln
j jj j
T
j j j j j j vjT
T
j j j vj jTj j j
F U S
F T V U T T S T V U TS
U N u T N u N c T dT
T
vS N s N c T dT N R
T v
0
0
0
1( ) ln ln
j
T
v j jTj
N
N
vN c T dT NR NR x x
T v
entropy of mixing
(see also Eq. (3.40)
(at the same temperature and particle density)
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In general, however, the heat capacity, for instance, depends on the internal degrees of freedom, as well as on the system temperature which has to do with different modes of excitation.
e.g. stretching of interatomic bond, kinetic energy of vibration, rotation, translational motions of the molecules, other potential and kinetic energy
The heat capacity of real gases approaches a constant value at high temperatures. At intermediate temperatures, each such mode contributes additively and independently to the heat capacity. This accounts for a “step-like” feature in the cv versus T curves.Step-like
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4-1.2 Chemical reactions in ideal gases 4-1.2 Chemical reactions in ideal gases
Partial molar Gibbs poential:
Since at chemical equilibrium
Let , equilibrium constant, be defined by
can in p
( ) ln ln
0
ln ln ln ( )
( ) ln ( ) ln ( )
( )
j j j
j jj
j j j j jj j j
j jj
j
RT T P x
v
v x v P v T
K T K T v T
T
rinciple be known, so that can be considered as known.
For given & together with each of can be determined.
The mass action law:
( )
1
( )j jj
j jj
vv
jj
K T
T P x x
x P K T
The only temperature-dependent term
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reaction1: ; reaction2:
If can be obtained by summing the previous reaction,
that is, ; & are constant
Then it
(1) (2)
(3)
(3) (1) (2)1 2 1 2
0 0
0
0 ( )
j j j j
j j
j j j j j
v A v A
v A
v A B v B v A B B
can be shown that:
The reaction heat can also be compute
(see an example in
d from as
(the van'
page 295 of the textb
t Hoff relat
ook.
ion)
)
3 1 1 2 2
2
ln ( ) ln ( ) ln ( )
( )
ln ( )
K T B K T B K T
K T
dH dRT K T
dN dT
Can help tabulations of K(T) for various chemical reactions.
Logarithmic additivity of K(T)
120
Small Deviations from Ideality—The Virial Small Deviations from Ideality—The Virial Expansion Expansion
4-2
4-2
Molecular Rheology LAB
: "Second virial coefficient" which accounts for the contribution from
.
: "Third virial coefficient" which accounts for the con
two-body interactions
tribution
h
fr
t
om
2
( ) ( )1
( )
( )
P R B T C T
T v v v
B T
C T
ree-body interactions etc.
Ideal behavior Correction term for v
For simple gases, use of the second virial coefficient may suffice to capture their physical behaviors; for complex fluids, more correction terms must be introduced.
4-2.1 Virial expansion4-2.1 Virial expansion
121
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ideal
ideal
ideal
(from )
(from ; )
(fr
2 2
2 2 2
, 2 2 2 2 2
22 2
( ) ( ) ( )
2 3
1 ( ) 1 ( ) 1 ( )
2 3
1 1 1
2 3
v v
vv
B T C T D Tf f RT
v v v
fP
v
d BT d CT d DTc c RT
v dT v dT v dT
s fc T s
T T
dB dC dDu u RT
v dT v dT v dT
om )u f Ts
(f is the molar Helmholtz potential)
4-2.2 Other thermodynamic quantities may be 4-2.2 Other thermodynamic quantities may be expanded in a self-consistent fashionexpanded in a self-consistent fashion
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The second virial coefficient vanishes at some special temperatures for several simple gases.
123
Law of Corresponding States for GasesLaw of Corresponding States for Gases
4-3.1 Important observations for gases4-3.1 Important observations for gases
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Molecular Rheology LAB
In the equation of state of a given fluid, there is one unique point - the critical point—characterized by Tcr, Pcr, and vcr.
The dimensionless constant Pcr vcr/R Tcr has value on the order of 0.27 for all “normal” fluids of the three parameters that characterizes the critical point. It thus appears that “only two are independent”.
(see more data in Table13.1 of the textbook)
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4-3.2 The Law of corresponding states4-3.2 The Law of corresponding states
There exists, at least semi-quantitatively, a universal equation of state containing no arbitrary constants if expressed in the reduced variables v /vcr, P/Pcr, and T/Tcr. This empirical fact is known as the “Law of corresponding state”.
e.g. It has been customary to plot the compressibility factor in a way
as functions of P/Pcr, and T/Tcr.
or 0.27cr cr cr
Pv P v TZ Z
P v TRT
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Tr curve
126
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A possible origin of the universal, two-parameter equations of state - the “van der Waals force” of two spherical-like, non-polar molecules.
The force between two molecules can be parameterized by the radius of the molecule, which describes the short-range repulsion, and the strength of the long-range attraction.
The long-range attractive force falls with distance as 1/r6 and has arisen from the instaneous dipole moment fluctuations.
Attractive force
Repulsive force
6
12
1
1r
r
The Lennard-Jones potential
127
Applications to Dilute Solutions—the van’t Applications to Dilute Solutions—the van’t Hoff Relation and the Raoult’s lawHoff Relation and the Raoult’s law
4-4.1 the van’t Hoff relation for osmotic pressure4-4.1 the van’t Hoff relation for osmotic pressure
4-4
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Molecular Rheology LAB
Consider a single-component fluid system, the Gibbs potential of the dilute solution can be written in the general form:
represents the effect of the interaction energy between the two types of molecules,
while the last two te
0 11 2 1 1 2 1
1 2
22
1 2
( , , , ) ( , ) ( , ) ln
ln
NG T P N N N P T N P T N RT
N N
NN RT
N N
rms accounts for the entropy of mixing
In dilute region ( )
1 2 12 1 1 2
1 2 1 1 2
0 21 2 1 1 2 2 2
1
ln , ln
( , , , ) ( , ) ( , ) ln
N N NN N N N
N N N N N
NG T P N N N P T N P T N RT N RT
N
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Molecular Rheology LAB
Solvent and solute partial molar Gibbs potentials:
, , ,
, , , ;
01 1
1
22
2 1
( ) ( )
( ) ( ) ln ( )
GP T x P T xRT
N
NGP T x P T RT x x
N N
The pressure on the pure solvent side is maintained constant (=P), but the mixture side can alter by change in height of the liquid level in a vertical tube.The equilibrium condition:
, , , ,
: The pressure on the solute side of the membrane1 1( 0) ( )P T P T x
P
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, , , ,
Expanding , , around the pressure
, , , , , ,
, ,
Multiplying by
1 1
1
11 1
1
1
2
( 0) ( 0)
( 0)
( 0)( 0) ( 0) ( )
( 0) ( )
( )
P T P T xRT
P T P
P TP T P T P P
PP T P P v
P P v xRT
N
V P N RT
the van’t Hoff relation for osmotic pressure in dilute solutions
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4-4.2 Application to vapor pressure4-4.2 Application to vapor pressure
liq gas
liq gas
liq gas
liq
The vapor pressure reduced by addition of low concentration of nonvolatile solute:
, ,
, ,
Expanding , and , around the pressure
,
( ) ( )
( ) ( )
( ) ( )
(
P T P T
P T xRT P T
P T P T P
P
liq liq
gas gas gas
,
, ,
; (the ideal gas equation: )
) ( ) ( )
( ) ( ) ( )
g l
g l g
T P T v P P
P T P T v P P
xRTP P
v v
RTv v v
PP
xP
Raoult’s law
131
HomeworkHomework4-H
4-H
Molecular Rheology LAB
Problem 13.2-2
Problem 13.5-2
Problem 13.5-3
132
Statistic Ensembles and FormulismStatistic Ensembles and FormulismChapter 5Chapter 5
Molecular Rheology LAB
Contents5-1 The entropy representation—the Boltzmann’s law 5-1.1 Physical significance of entropy for closed system (15.1)
5-1.2 The Einstein model of a crystalline solid (15.2) 5-1.3 The two-state system (15.3); a polymer model (15.4)
5-2 The Canonical Formalism 5-2.1 The probability distribution (16.1)5-2.2 The partition function (16.2)
5-2.3 The classical ideal gas (16.10)
5-3 Generalized Canonical Formulations 5-3.1 Entropy as a measure of disorder (17.1)5-3.2 The grand canonical formalism (17.3)
133
The The EEntropy ntropy RRepresentation—the epresentation—the Boltzmann’s law Boltzmann’s law
5-1.1 Physical significance of entropy for closed system 5-1.1 Physical significance of entropy for closed system
5-1
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Molecular Rheology LAB
Classical Thermodynamics
Statistical Thermodynamics
Erected on a basis of few hypotheses of entropy.
e.g. S=S(U, V, N) ; extremum principle
Statistical mechanics provides the physical interpretation of the entropy and a heuristic justification for the extremum principle.
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Macroscopic systemsAtomic systems
Closed system
given V & NU, V, and N are the only constraints on the system.
Quantum mechanics
If the system is macroscopic, there may exist many discrete quantum states consistent with the specified values of U, V, and N.
Naively, we might expect the system in a particular quantum state would remain forever in that state.
The apparent paradox is seated in the assumption of isolation of physical system. No physical system is, or ever can be, truly isolated.
A realistic view of a macroscopic system is one in which the system makes enormously rapid random transitions among its quantum states. A macroscopic measurement senses only an average of the properties of myriads of quantum states.
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Equal probability assumption
Consider a “isolated” system and “constraint N, V, & U.” Not all of the system microstates are determined, and hence one is concerned with the probability associated with each of these microstates.
Equal probability assumption brought out by Boltzmann:
A macroscopic system samples every permissible quantum state with equal probability.
The assumption of equal probability of all permissible microstates is the fundamental postulate of statistical mechanics.
Equal probability assumption applies only to isolated systems.
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Correlation of Ω & S
The No. of microstates (Ω) among which the system undergoes transitions, and which thereby shares uniform probability of occupation, increases to the max. permitted by the imposed constraints.
According to the postulate of the entropy increases to the max. permitted by the imposed constraints.
It suggests that the entropy can be identified with the number of microstates consistent with the imposed macroscopic constraints.
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Entropy ( )
additive (extensive)
S No. of microstates ( )
Multiplicative
6
6
+6
6
×=?
lnBS k
The constant prefactor merely determines the scale of S, and agrees with Boltzmann’s constant kB = R/NA=1.3807 × 10-23 J/K.
Nevertheless, the microcanonical formulation establishes the clear and basic logical foundation of statistical mechanics.
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5-1.2 5-1.2 The Einstein model of a crystalline solid The Einstein model of a crystalline solid
Consider a nonmetallic crystalline solid, and just focus on the vibrational modes. Electronic excitations, nuclear modes, and various other types of excitations were ignored.
Assumed the atoms in the crystal are free to vibrate
around its equilibrium position in any of thr
ee coordinate
directions, and can be replaced b harmonic oscillatoy
all with the
,
rs
same natural
3N
N
0 frequen y .c
0
0
Einstein introduced the energy of harmonic oscillators discretely:
= ;
; : Planck's constant
Discrete values of energy:
10, 1, 2, ...
2
/ 2
n n
h h
n
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Number of permutations of
( )03 1N U
If the energy of the system is it
can be considered as constituting
quanta. Theres quanta are
to be distributed among
vibrational modes.
0
3
U
U
N
H C
( )
0
0 0
03 13
0
0 0
0 0
3 1 !
3 1 ! !
3 1 ! 3 !
3 1 ! ! 3 ! !
N UNU U
N U
N U
N U N UN
N U N U
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;
From & if
0 00 0 0
3 ln 1 3 ln 1 3
ln ln( !) ln ( 1)
A
B
u u us R R u N
u u u
S k M M M M M
According to this equation, which is independent of the volume, the pressure is identically zero. Such a nonphysical result is a reflection of the naive omission of volume dependent effect from the model.
Thermal equation of state:
There are oscillator in the system
mean energy per oscillator
The quantity is ca Einstein tempelled rthe atur"
0
00
0
0
1 3ln 1
3
3 1B
BA
A
T
B
kA
kS NN
T U U
NN
U
N
k
N e
" of the crystal,
and it generally is of the same order of magnitude as the melt point of the
e
solid.
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cv approaches a constant value (3R) at high temperature.
cv is zero at zero temperature.
Einstein model is in qualitative agreement with experiment, but the rate of increase of the heat capacity is not quantitatively correct.
At low temperature cv rises exponentially, whereas experimentally the heat capacity rises approximately as T 3.
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5-1.2 5-1.2 The two-state system ; a polymer model The two-state system ; a polymer model
Two-state system
Ground state
(with energy 0)
Excited state
(with energy ε)
Total atoms N: system energyU
( ) atoms are
in the ground state
-UN atoms are in
the excited state
U
C
From & if
!
! !
ln 1 ln
ln ln( !) ln ( 1)
NU
B
B
N
U N U
U U U US N k
N N
S k M M M M M
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Thermal equation of state:
;
2 222 2
22 22
1ln 1
( )1
1
B
B
B B
B
B
k T
k Tk T k T
A Ak T
B B
kS N
T U U
NU U N
e
du ec N N e e
dT k T k Te
The energy approaches as the temperature approaches infinity in this model.
At infinite temperature half the atoms are excited and half are in their ground state.
2N
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Schottky hump
The max. is taken as an indication of a pair of low lying energy states, with all other energy states lying at considerably higher energies.
It is an example of the way in which thermal properties can reveal information about the atomic structure of materials.
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A polymer model
Each monomer unit of the chain is permitted to lie +x, -x, +y, and –y.
The length between two monomer unit is a.
Represent interference by assigning a positive energy ε to such a perpendicular monomer.
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x x y y
xx x x
yy y y
y y
N N N N N
LN N L
aL
N N La
UN N U
1
21
21
21
2
x x
x x
y y
y y
N N U L
N N U L
N U L
N U L
The number of configurations of the polymer
consistent with given corrdinates and
of its terminus and given energy :
!
( , , , )! ! ! !
x y
x yx x y y
L L
U
NU L L N
N N N N
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or
ln ln ln ln
ln ln
1 1ln ln
2 2
1 1ln
2 2
1 1ln
2 2
1 1ln
2 2
B B x B x x B x
y B y y B y
B x B x
x B x
y B y
y B y
S k Nk N N k N N k N
N k N N k N
S Nk N N U L k N U L
N U L k N U L
U L k U L
U L k U L
; ( )
&
ln 0 02
0 0
y yBy
y y
yy y y y y y y y y
U LkS
T L a U L
LL L N N L a L L N N
a
T
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2
2 2
22
ln2
1ln ln ln
2 2
x B
B
x xB
x x
a k T x
B B Bx x
xk T
N U LkS
T L a N U L
N U Le
N U Lk k kS
N L U N L U UT U
N U Le
U
T
T
For small (relative to )
The modulus of elasticity of the rubber band:
2
21
sinh
cosh
1
1
11
B
B
B
x Bxk T
x B
xx B x k T
B
k Tx
x BT
a k TL
N a k T e
Naa k T L
k T e
L ae
N k T
T
T
TT
T
149
The Canonical Formalism The Canonical Formalism
5-2.1 The probability distribution 5-2.1 The probability distribution
5-2
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Molecular Rheology LAB
Limitation of microcanonical formalism:
The microcanonical formalism is simple in principle, but it is computationally feasible only for a few highly idealized models.
Isolatedsystem
Must calculate the no. of combinatorial ways that a given total energy can be distributed in arbitrarily sized boxes.
Generally beyond our mathematical capabilities!
system
T reservoir Remove the limitation on the amount of energy available- to consider a system in contact with T reservoir rather than an isolated system.
Canonical formalism (Helmholtz representation)
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Canonical formalism
subsystem
T reservoir
Closed system
In “subsystem,” each “state” does not have the same probability, as for an closed system” to which the principle of equal probability of “microstates” applies.
i.e., the subsystem does not spend the same fraction of time in each state.
The key to the canonical formalism is the determination of the probability distribution of the subsystem among its microstates.
Closed system = Subsystem + T reservoir(note that the energy of the subsystem fluctuates)
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A Simple analogy
thrownSum of no. =12
Sum of no. 12
Record red dice no.
What fraction of these recorded throws has the red die shown a one, a two,…, a six ?
2/25
3/25
4/25
5/25
6/25
5/25
Reservoir
subsystem
Sum is12 corresponds to given total energy
The numbers shown correspond to the energy of the system
Each state does not have the same probability.
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Probability of subsystem
The probability of the subsystem in state :
; : the energy in state
; from
If is the average value of the su
res tot
tot tot
1 res tot
1tot tot
expln
exp
j
j
j j
B j
j B
B
f j
E Ef E j
E
k S E Ef S k
k S E
U
bsystem
Expanding around the equilibrium point :
tot tot res tot
res tot tot
res tot res tot res tot
11 ( ) R jB
j
jj j
k T Ek T U TS Uj
S E S U S E U
S E E E U
U ES E E S E U U E S E U
T
f e e
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1define
is state-independent normalization factor
define
Fundamental relation in the canonical formali
11 ( )
1
Z Z
R jB
j
j j
j
k T Ek T U TS Uj
EFj
B
F
E EF Fj
j j j
E F
j
f e e
f e ek T
e
f e e e e
e e
sm:
The probability of occupation of the -th state:
ln ln Zj
j i
E
j
E Ej
i
F e
j
f e e
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The average energy:
or
; & ln Z ln Z
j iE E
j j jj j i
U E f E e e
d FU U F T F
d T
Two equations are very useful in statistical mechanics, but it must be stressed that these equations do not constitute a fundamental relation.
The quantity β is merely the reciprocal temperature in “natural units.”
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5-2.2 5-2.2 The partition function The partition function
Consider a system composed of Ñ “distinguishable elements,” an element being an “independent (noninteracting) excitation mode” of the system.
SystemElement
Each element can exist in a set of orbital states (use the term orbital state to distinguish the states of an element from the state of the collective system).
The identifying characteristic of an “element” is that the energy of the system is a sum over the energies of the element, which are independent and noninteracting.
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Each element is permitted to occupy any one of its orbitial states independently
of the orbital states of the other elements.
Then the partition sum:
1 2 3
1
ε ε ε
, , , ...
ε
Z j j j
j
j j j
e
e
2 3
1 2 3
ε ε
, , , ...
ε ε ε1 2 3z z z
j j
j j j
j j j
j j j
e e
e e e
: the energy of the -th element
in its -th orbitial state
ε i j i
j
; : the "partition sum of the -th element"
The Helmholtz potential :
ε
1 2
z z
ln Z ln z ln z
i j
i ij
e i
F
The energy is additive over elements, and each element is permitted to occupy and its orbital states independently.
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Applied to two- state model:
Ground state
(with energy 0)
Excited state
(with energy ε)
Total atoms Ñ
1 &
0 ε ε
ε
Z z 1
ln ln Z
ln 1
j
N NN
E
j B
N
B
e e e
F ek T
F Nk T e
If the number of orbitals had been three rather than two, the partition sum per particle z would merely have contained three terms.
31 2ε εε εz j
j
e e e e
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Applied to the Einstein model:
0
Einstein introduced the energy of harmonic oscillators discretely:
= ;
; : Planck's constant
The "elements" are the vibrational modes.
10, 1, 2, ...
2
/ 2
n n
h h
0 0 0 0
00 0 0
0
0
0
1 3 5 7
2 2 2 2
12 32
1
22
0
1 30
z
1
1z 1 ( 1 1)
11
3ln z 3 ln 1
2
n n
n
NB B
e e e e
e e e e
ex x x x x
xe
F Nk T Nk T e
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5-2.3 5-2.3 The classical ideal gas The classical ideal gas
Definition of the classical ideal gas:
Can neglect the interactions between gas particles (unless such interactions make no contribution to the energy - as, for instance, the instantaneous collisions of hard mass points).
No. of atoms Ñ( = NNA)
T reservoir
Container volume V
The energy is the sum of one-particle “kinetic energies,” and the partition sum factors.
Consider monatomic classical ideal gas:Neglect interactions between gas molecules
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; : generalized momentum
: Hamiltonian function
d d
2 2 2
transl internal1 2 1 2
2
transl 3
3
p p
( , , , , , )
Z z z
1z
1
x y z
x y z
j j
j
p p p m
x y z
k
x y z p p p
dq dpe
h h
dxdydz dp p p eh
dh
H
H
d d
( = )a
2 22 2 22
3 2
3 -2
y zxp m p mp m
x y z
axB
xdydz e dp e p e p
Vmk T e
h
The one-particle translational partition sum ztransl:
V
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Definition of indistinguishability in quantum mechanics:
Indistinguishability does not imply merely that the particles are “identical” - it requites that the identical particles behave under interchange in ways that have no classical analogue.
If one identifi as es and thereby
calculates
3 2
transl 3
Z z
z 2
.
B
N
Vmk T
h
F
Finding is not extensive!F
To identify Z as zÑ is to assume the particles to be distinguishable. To correct this partition sum by taking indistinguishability into account, the result must be further divided by Ñ!.
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The partition sum for a classical monatomic ideal gas:
transl
3 2
2
3 22
5 2
1Z = z!
2ln Z ln
5 3ln 3 ln
2 2
N
BB B B
B B
N
V mk TF k T Nk T Nk T
N h
F U VS Nk m Nk
T N
Thermodynamic context for monatomic ideal gases:
3 2 5 2
00 0 0
lnU V N
S Ns NRU V N
The constant s0, undetermined in classic thermodynamic context, has now been evaluated in terms of fundamental constants.
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Reflection on the problem of counting states reveals that division by Ñ! is a rather crude classical attempt to account for indistinguishability.
12
12
1 2
12
Classical counting Fermi particles Bose particles
“Corrected” number of states = 4 . 1/2!
Orbital model
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“Corrected classical counting” is incorrect for either type of real particle!
At sufficiently high temperature the particles of a gas are distributed over many orbital states, form very low to very high energies.
The probability of two particles being in the same orbital state becomes very small at high temperature. The error of classical counting then becomes insignificant, as that error is associated with the occurrence of more than one particle in a one-particle state.
All gases approach “ideal gas” behavior at “sufficiently high temperature”.
12
12
1 2
12
Classical counting Fermi particles Bose particles
“Corrected” number of states = 4 . 1/2!
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B B
Suppose the energy (Hamiltonian) to be of the form
Then, the classical partition function yields
2 2
2 2
1 / 2
( )~ ~Aq Bp
E Aq Bp
dq dp k T k Tz e
h hA hB
1 / 2
The Equipartition Theorem
B
The significance is that, at sufficiently high temperature when the classical density of
states is ap every quadratic term in the energy contributes a term
to the heat c
plicable,
This iapacity.
1
2Nk
s the " " of classical statistical mechanics.
A heteronuclear diatomic molecule has three translational modes,
one vibrational mode (with both kinetic
equiparti
and poten
tion
tial
theo
ener
Exp.
gy)
rem
, tw
B B B B
o rotational modes
(i.e., it requires two angles to specify its orientation and has only kinetic energy).
Therefore, the heat capacity per molecule at high temperature is
p
or 3 2 2 7 7
2 2 2 2 2(k k k k R er mole).
166
Generalized Canonical Formulations Generalized Canonical Formulations
5-3.1 Entropy as a measure of disorder 5-3.1 Entropy as a measure of disorder
5-3
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Information Theory
Define the information I in terms of the probability fj, subject to the following requirements:
Information is a non-negative quantity, I(fj) 0≧ .
If an event has probability 1, it’s no information from the occurrence for event, I(1) = 0 .
If two independent events occur, then the information we get from observ-ing the events is the sum of the two of them: I(f1 × f2) = I(f1) + I(f2).
I(fj) is monotonic and continuous in fj.
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Four properties can thus be derived:
: the positive constant was used in different units
e.g.
units are bits (from "binary")
units are trits (from "trinary"2
3
1log log
log
log
j b b jj
I f ff
b b
)
units are nats (from "natural logarithm")
units are Hartleys
Define the disorder (entropy) of the porbability distribution , , , is
10
1 2
1
log
log
1log
e
j
n
j bj j
f f f
H ff
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Several requirements of the measure of disorder reflect our qualitative concepts:
The measure of disorder should be defined entirely in terms of the set of numbers ﹛fj﹜.
The maximum disorder corresponds to all event have equal possibility.Example
:
Given sample space and it's possibility distribution:
, , , ; possibility distribution , , ,
Situation I:
Situation II:
1 2 3 4 1 2 3 4
1 2 3 4
2 2 2 2
1
1/ 4
1 1 1 1log 4 log 4 log 4 log 4 2
4 4 4 4
1
S
S s s s s f f f f
f f f f
H P bits
f
, ,
2 3 4
2 2 2 2
/ 2 1/ 4 1/ 8
1 1 1 1log 2 log 4 log 8 log 8 1.75
2 4 8 8
f f f
H P bits
Have max. disorder
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The conceptual framework of “information theory,” erected by Claude Shannon in 1948, provides a basis for interpretation of the entropy in terms of Shannon’s measure of disorder.
Entropy in the case of two possibilities with probabilities f and (1- f).
C. E. Shannon, J. Bell Syst. Tech. 27, 379-423 (1948).
At max.
2 2
1- 1/ 2
1 1log 2 log 2 1
2 2
f f
H bits
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From the definition of disorder (entropy):
1 1
1log ln
n n
j b j jj jj
H f k f ff
If the system is isolated, it spends equal time in each of the permissible states.
; , ,
Microcanonical formalism:
1 1
11 2
1 1 1ln ln ln ln
ln
ln
j
j jj j
B
B
f j
H k f f k k k
H k
S k
k k
For a closed system the “entropy” corresponds to “Shannon’s quantitative measure of the maximum possible disorder” in the distribution of the system over its permissible microstates.
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If the system is in contact with a thermal reservoir, the probability of j-th sate is
; , ,
1 1 1
1
1 2
ln ln ln Z
ln Z
j
j
E
j
E
B j j B j B j jj j j
B j j Bj
ef j
Z
H k f f k f e Z k f E
H k f E k
This agreement between entropy and disorder is preserved for all other boundary conditions - that is for systems in contact with pressure reservoirs, with particle reservoirs, and so forth.
Thus we recognize that the physical interpretation of the entropy is that the entropy is a quantitative measure of the disorder in the relevant distribution of the system over its permissible microstates.
In thermodynamics the entropy enters as a quantity that is maximum in equilibrium. Identification of the entropy as the disorder simply brings theses two viewpoints into concurrence for closed systems.
1
ln ZB j j Bj
S k f E k
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5-3.2 5-3.2 The grand canonical formalism The grand canonical formalism
T & μ reservoir
subsystemConsider a system of fixed V in contact with both energy and particle reservoirs, and given energy Ej and mole number Nj.
Closed system = Subsystem + T reservoir
Closed system
Probability of a state of the system:
; from
Expanding &
res tot tot
tot tot tot
1 res tot tot
1tot tot tot
res tot tot tot tot
,
,
exp ,ln
exp ,
, ,
j j
j
B j j
j B
B
j j
E E N Nf
E N
k S E E N Nf S k
k S E N
S E E N N S E
tot
j jE NU TS Nj
N
f e e
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defined = ; is the "grand canonical potential"
is state-independent normalizing factor
define
,
1
Z
j j
j j
j j j j
j j
E NU TS Nj
E N
j
E N E N
jj j j
E N
f e e
U TS N U T
f e e
e
f e e e e
e
: the grand canonical partition sum ; Z
Z
ln Z
j
e
The conventional view is that Ψ(T, V, μ) is the Legendre transform of U, or Ψ(T, V, μ) = U[T, μ].
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Mnemonic figure
Useful relationships :
lnZU
F U T
Multiplied 1
Bk
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To corroborate the grand canonical formalism from maximizing the disorder (entropy):
1
lnB j jj
S k f f
1
ln 1 0B j jj
S k f f
Taking differentials & multiplying by Lagrange multiplier λ1, λ2, λ3.
Auxiliary conditions:
11
21
31
0
0
0
jj
j jj
jj
f
E f
N f
Auxiliary conditions:
1
1
1
1jj
j jj
j jj
f
f E E
f N N
Taking differentials
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1 2 31 j jE N
jf e
1
2
3
1
j jE N
jf e e
1
ln 1 0B j jj
S k f f
If this equation is always to hold true,
each term must be equal to zero.
1 2 3
1 2 31
1 2 3
1
ln 1 0
ln 1 0
j j
j j jj
j j j
E N
j
f E N
f E N
f e
Auxiliary conditions:
11
21
31
0
0
0
jj
j jj
jj
f
E f
N f
Adding 4 equations
Compare the two equations to identify λ1, λ2, & λ3.
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Example: Molecular Adsorption on a Surface
Consider a gas in contact with a solid surface. The molecules of the gas can adsorb on specific sites on the surface.
T & μ reservoir
Solid surface(system)
There are Ñ such sites, and each site can adsorb zero, one, or two molecules.
the site is empty
Each site has an energy: singly occupied
doubly occupied 1
2
0
We seek the “fractional coverage” of the surface.The gaseous phase which bathes the surface establishes the values of T
and μ, being both a thermal and a particle reservoir.
In such a case μ, the Gibbs potential per particle of the gas, must first be evaluated from the fundamental equation of the gas, if known, or from integration of the Gibbs-Duhem relation if the equation of state are known.
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The surface sites do not interact, the grand partition sum:
The mean number of molecules adsorbed per site is
1 2 1 2
1 2
1 2
2 20
0 0 2
Z z
z = 1
z
0 1 2
z z z
2
j j
j j
j j
N
E N
j
E NE N j
j j jj j j
e e e e e e
e Nn f N e e N
e e e
e en
The mean energy per site is
1 2
2
21 2
z
zj jE N
j j jj j
e ef E e e E
ze
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An alternative route to these two results, and to the general thermodynamics
of the system, is via calculation of the grand canonical potential:
.ln ZBk T
The number of adsorbed atoms on the sites:
; is constant
1 2 2ln Z = ln z ln 1
ˆ
ˆ
ˆ
NB B Bk T k T k T e e
N N
N Nn T
U N
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HomeworkHomework5-H
5-H
Molecular Rheology LAB
Problem 15.1-1
Problem 15.2-1
Problem 15.4-4
Problem 16.1-1
Problem 16.1-4
Problem 16.2-1
Problem 16.10-4
181
Statistical Fluctuations and Solution Statistical Fluctuations and Solution StrategiesStrategiesChapter 6Chapter 6
Molecular Rheology LAB
Contents6-1 Fluctuations
6-1.1 The probability distribution of fluctuations (19.1)
6-1.2 Moments and the energy fluctuations (19.2)
6-2 Quantum Fluids 6-2.1 Quantum particle - fermions and bosons (18.1)
6-2.2 The ideal Fermi fluid (18.2)6-2.3 The ideal Bose fluid (18.5) 6-2.4 The classical limit and the quantum criterion (18.3)6-2.5 The strong quantum regime: electrons in a metal (18.4)6-2.6 Bose condensation (18.7)
182
FluctuationsFluctuations
6-1.1 The probability distribution of fluctuations 6-1.1 The probability distribution of fluctuations
6-1
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Molecular Rheology LAB
T reservoir
subsystem
The subsystem and the reservoir together undergo incessant and rapid transitions among their joint microstates.
The subsystem energy thereby fluctuates around its equilibrium value
reservoir
subsystem
The “subsystem” may, in fact, be a small portion of a larger system, the remainder of the system then constituting the “reservoir.”
In that case the fluctuations are local fluctuations within a nominally homogeneous system.
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The statistical mechanical form of the probability distribution for a fluctuating extensive parameter is familiar.
T reservoir
subsystem
Case I:
The probability that the system occupies a particular
microstate of energy :ˆ
ˆ
F E
E
f e
T, P reservoir
subsystem
Case II:
The probability that the system occupies a particular
microstate of energy and volume :ˆ ˆ
ˆ ˆ
G E PV
E V
f e
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Generally, for a system in contact with reservoirs corresponding to the
extensive parameters , , , , the probability that the system
occupies a particular microstate with parameters , , ,
0 1
0 1ˆ ˆ
sX X X
X X
:ˆsX
, ,
, , , =e
: the Massueu function
: the entropic intensive parameters
1 10 0 0 1 1
0 1
ˆ ˆ ˆ, ,
ˆ ˆ ˆ
0
0
, ,
, ,
B s B s s
s
k S F F k F X F X F X
X X X
s
s
f
S F F
F F
0
0
ˆ
1
1
ˆ ˆ
F E
FS
T T
FT
X E
f e
e.g.
,
;
;
0 1
0 1
ˆ ˆ
1
1
ˆ ˆ ˆ ˆ
E N
ST T T
F FT T
X E X N
f e
,
;
;
0 1
0 1
ˆ ˆ
1
1
ˆ ˆ ˆ ˆ
G E PV
G PS
T T T
PF F
T T
X E X V
f e
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6-1.2 6-1.2 Moments and the energy fluctuations Moments and the energy fluctuations
Suppose that the Ê is the only fluctuating variable, all other extensive parameters being constrained by restrictive walls.
: the deviations
;
: the mean-square deviation (the second central moment)
2
2 2 2
2
ˆ
ˆ ˆ 0
ˆ
ˆ
ˆ
j
j
j
F E
j j jj j
F E
jj
F E
jj
E U
E U E U
E U
E U E U f E U e
E U E U e F U
UE U e
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2
22
ˆ
0
ˆ1
j
j
F E
jj
F E
j j jj j
B vB
UE U E U e
E U e E U f E U
U UE U k T Nc
k T
The mean square energy fluctuations are proportional to the size of the system.
The relative root-mean-square dispersion , which measures the amplitude of the fluctuations relative to the mean energy, is proportional to N-1/2 .
For large systems (N→∞) the fluctuation amplitudes become negligible relative to the mean values, and thermodynamics becomes precise.
1 22
E U U
A very useful expression to estimatethe heat capability of a system in computer simulations.
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Higher-order moments of the energy fluctuations:
e.g. the third moment:
1
1
1
ˆ
ˆ
j
j j
j
n n F E
jj
n nF E F E
j jj j
nn F E
jj
n n
E U E U e
E U e e E U
UE U n e E U
UE U n E U
E
32 2 22 vB v
cU Nk T Tc T
T
The higher-order moments of the energy fluctuations can be generated from the lower-order moments by the recursion relation.
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For a system in which both E and V fluctuates:
B
B B B
B
B B
B
1
1
2
/ , ,...
2 2 2
2
/ , ,...
2
2
ˆ(1 / )
ˆ ˆ(1 / )
ˆ( / )
P T N
p T
P T N
T
UE k
T
k T Nc k T PV k TP V
VE V k
T
k T V k TPV
VV k
P T
B
11/ , ,...T N
Tk T V
The energy and the volume fluctuations are correlated, as expected.
189
Quantum Fluids Quantum Fluids 6-2
6-2
Molecular Rheology LAB
As so often happens in physics, the formalism points the way to reality. The awkwardness of the formalism is a signal that the model is unphysical - there are no classical particles in nature!
Quantum Mechanical Particles
BosonsFermionsFermions are the quantum analogues of the material particles of classical physics.
e.g. electrons, protons, and neutrons
Bosons are the quantum analogues of the “waves” of classical physics.
e.g. photons (quanta of light)
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(From Astro-group, department of physics of Hong Kong university)
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6-2.1 Fermions and bosons & ideal quantum fluids 6-2.1 Fermions and bosons & ideal quantum fluids
Fermions
Fermions obey “law of impenetrability of matter”, only a single fermion can occupy a given orbital state.
The intrinsic angular momentum for fermions:
, (odd multiples)2
Spin quantum number : ( ), ( )
1,3,5,
1 1
2 2s
n n
m
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Consider a pre-gas model system in which only three spatial orbits are permitted; particles in these spatial orbits have energies ε1, ε2, and ε3. The model system is in contact with a T reservoir and a μ reservoir.
T & μ reservoir
subsystem
There are therefore six orbital states:
; ; , 1 1
( , ) 1, 2, 32 2s sn m n m
The Grand canonical partition sum factors:
1, 1 2 1, 1 2 2, 1 2 2, 1 2 3, 1 2 3, 1 2
0,
,
,,
Z z z z z z z
z 1
1
z 1 1
n n n n
s
s
n n
s n n
s
Nn m
n m
n mn m
e e e e
e ef
e e
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Pair the two orbital states with the same but with :2
2, 1 2 , 1 2
1
z z 1 1 2n n n
s
n n
n m
e e e
empty 2 singly occupied double occupied
The fundamental equation:
31 222 2
Z 1 1 1e e e e
21 2 n ne e
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or
or
By the same way,
1 2 3
1 2 3
,,
,,
, ,,
31 2, ,
,
2 2 2
1 1 1
22 2
1 1 1
n mn m
n mn m
n m n mn m
n m n mn m
N N f
N fe e e
U U f
U fe e e
ST
determines the entropy.
The fundamental equation is an attribute of the thermodynamic system, independent of boundary conditions. In general, the Fermi level μ as a function of T and Ñ must be determined.
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Ideal Fermi fluids
The ideal Fermi fluid is a quantum analogue of the classical ideal gas; it is a system of fermion particles between which there are no (or negligibly small) interaction forces.
Composite “particles,” such as atoms, behave as fermion particles if they contain an odd number of fermion constituents e.g. 3He.
The partition sum of orbital state , :
The partition sum factors:
The probability of occupation:
k k k k
k
k
0k ,
k ,
k , k ,
k , k ,
k
z 1
Z z
1
z 1
s
s
s
s
s
s
s
Nm
m
mm
mm
m
e e e e
ef
e
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The grand canonical potential:
The energy of an occuppied orbital state , :
, : wave vector (can refer to eq. 16.43 in textbook)
The density of
k2
k k
2 2 2
,
lnZ ln 1
k
2 2s
B B
s
k m
k T k T e
m
p kk
m m
orbital states: (see Chap. 16)
which can be utilized to determine and .
3 2
2 1 22 2 2
0
3 2
1 22 2 0
2
2 4
2 ln 1
2ln 1
2
B
B
V dk V mD d k d d
d
k T e D d
V mk T e d
U N U
; N
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Bosons
Fermions as waves can be freely superposed, so an arbitrary number of bosons can occupy a single orbital state.
The intrinsic angular momentum for fermions:
, (even multiples)2
0,2,4,n n
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Ideal Bose fluids
The partition sum of a single orbital state is independent of , and is, for
each value of
;
k k k
k
2 2 3 30k k ,
,
z z
1 1
11
s
s
s
m
n
n
m
m
e e e e
xxe
,
The probability of occupation with various occupated no. in orbital state is
The average number of bonsons in the orbital state , :
k
0
k , ,k ,
k k , ,
k,
z
k
s
s
s s
s
i i
m im
s
m m ii
i m
ef
m
n i f e
k k k2 2 3 3k ,
k ,
z
ln z
s
s
m
B m
e e
k T
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,
, ,
which determines and .
k
k
k
k
k
k
k k,
k k
0
3 2 3 2
2 2 0
ln z ln 11
1
11
1
ln 1
2 2
3 12
s s
s s
m B m B B
m m
en k T k T e k T
e
e
n fe
e D d
V md U N
e
The quantity is not necessarily less than unity, and therefore is more properly identified as a “mean occupation number” .
If μ were positive the orbital state with εk equal to μ would have an infinite occupation number! So, the molar Gibbs potential μ is always negative.
, sk mn, k smf
occupation probability (or mean occupation number)
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6-2.2 6-2.2 The classical limit and the quantum criterion The classical limit and the quantum criterion
At low densities or high temperatures, the probability of occupation of each orbital state is small, thereby minimizing the effect of the fermion prohibition against multiple occupancy.
,
,
The probability of occupancy:
If fugacity ; classical regime
The occupation probability reduces to
The number of particles reduces to
k
k
02 2
1
1
1
2
2
s
s
m
m
fe
e
f e e e
N
g V mN
; : the thermal wave length
3 2
1 2 030 λ
λ λ2
T
T T
B
g Ve e d e
h
mk T
The Boltzmann distribution
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The energy reduces to3 2
3 2 02 2 30
2 2 3
2 λ2
3
2
BT
B
g VV mU e e d k T e
U Nk T
The equation of state of classical ideal gas
The criterion that divides the quantum and classical regimes:
or classical-quantum boundary3 01 λ 1T
g Ve
N
The system is in the quantum regime if the thermal volume is larger than the actual volume per particle (of a single spin orientation) either by virtue of large Ñ or by virtue of low T (and consequently of large λT).
The classical limit the fugacity is the ratio of the “thermal volume” λ3
T to the volume per particleV/(Ñ/g0).
N
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6-2.3 Example of Fermi fluid6-2.3 Example of Fermi fluid: electrons in a metal : electrons in a metal
At T = 0 K, to calculate the number of particles:
The number of conduction electrons per unit volume in metals is of the order
of the order of 10 to 10 electrons cm , the Fermi temperature is
to
0
3 23 21 2 3 2
02 3 2 30
22 23 3
40
22
3
10 K 10
F
FB
m Vm VN d
T
Tk
5 K
for
for 0
0
0
1,
0
T K
f T
Metal TF (K)
Li 5.5×104
Na 3.7×104
K 2.4×104
Rb 2.1×104
Metal TF (K)
Cs 1.8×104
Cu 8.2×104
Ag 6.4×104
Au 6.4×104
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The heat capacity:
23
0
3
2 3B
B
k TC Nk O T
The prefactor 3ÑkBT/2 is the classical result, and the factor in parentheses is the “quantum correction factor” due to the quantum properties of the fermions, which is of the order of 1/10 at room temperature.
This drastic reduction of the heat capacity from its classically expected value is in excellent agreement with experiment for essentially all metals.
All reasonable temperatures μ>> kBT, the electrons in a metal are an example of an ideal Fermi gas in the strong quantum regime.
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6-2.4 6-2.4 Bose condensation Bose condensation
The number of particles :
Fugacity , and
Expand the integral of equation of in powers of the fugacity:
3 2 1 20
2 2 10
3 23 20 0
2 2
2
12
0 1
2
2 λ2
e
e
e
e B
N
g V mN d
e
e
N
g V g VmN k T
; : the thermal wave length
3 23
2 3
3 2 3 21
λ λ2
2 2 3 3
T
T T
B
r
r
F
h
mk T
Fr
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The energy:
Expand the integral of equation of in powers of the fugacity:
3 2 3 20
2 2 10
3 25 2 0
5 2 5 22 2 3
2 3
5 2 5 21
2
12
2 2 3 3
4 2 λ2
4 2 9 3
B BT
r
r
e
g V mU d
e
U
g VV mU k T F k T F
Fr
N
0 03 23 3
3 2
5 2
3 2
λ λ
3
2
e
T T
e B
g V g V NF
F
FU N k T
F
The ratio measures the deviation from the classical equation of state.
5 2 3 2F F
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The two functions and satisfy the relation:5 2 3 2
5 2 3 2
1
F F
dF F
d
At
5 2
3 2
1
1.34
2.612
F
F
At
slope of
slope of
5 2
3 2
0
1
1
F
F
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If , to first evaluate the temperature at which the failure of the
"integral analysis" occurs.
Set
: the Bose condensation tempe
03
03
2 32
0
2.612
2.612
λ2
2 1
2.612
e T
e T
T
B
B c
c
g V
N
g V
N
h
mk T
Nk T
m g V
T
rature
For temperature greater than Tc the “integral analysis” is valid. At and below Tc a “Bose condensation” occurs, associated with anomalous population of the orbital ground state.
Example of Bose condensation of 4He
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Below Tc the fluid flows freely through the finest capillary tubes, as its name denotes, “superfluid.”
This component cannot easily dissipate energy through friction, as it is already in the ground state.
The condensed phase has a quantum coherence with no classical analogue.
These electron pairs then act as bosons. The Bose condensation of the pairs leads to superconductivity, the analogue of the superfluidity of 4He.
ρs(T): superfluid density of 4He
ρn(T): normal fluid density of 4He