Advanced Thermodynamics
Instructor: Prof. Chi-Chung Hua
Complex Fluids & Molecular Rheology Laboratory,
Department of Chemical Engineering,
National Chung Cheng University, Chia-Yi 621, Taiwan, R.O.C.
Molecular Rheology LAB
Course 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: 易函柳 (Rm414, Ext. 23482 )
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Part one: Classic Thermodynamics
Chapter 1: Basic Concepts and Principles
Chapter 2: Thermodynamic potentials and Legendre
Transformations
Chapter 3: Stabilities, Phase Transitions, and Critical
Phenomena
Chapter 4: Applications to Material Properties
Part two: Statistic Thermodynamics
Chapter 5: Statistic Ensembles and Formulism
Chapter 6: Statistical Fluctuations and Quantum Fluids
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Chapter 1 Basic Concepts and Principles
Contents
1-1
1-2
1-3
1-4
1-H
Thermodynamic Equilibrium and Postulate I
The Entropy Maximum and Postulates II & III
Intensive Parameters and Equations of State
The Euler Equation and the Gibbs-Duhem Relation
Homework
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1-1
Thermodynamic Equilibrium and
1-1
Postulate I
1-1.1 Thermodynamic coordinates
Macroscopic vs. Microscopic
Variables
Responses
Experimental variables
Experimental results
Simulation variables
Simulation data
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1-1
Q1:
How to select variables of an experimental or simulation
system?
A1:
1. Use minimum number of measurable variables.
2. Each variables must be independent.
Q2:
Select macroscopic or microscopic variables?
A2:
The basis of selection is the scale of the phenomena or substances of
primary interest.
Microscopic
1μm
Macroscopic
length scale
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1-1
Microscopic properties can spread to macroscopic properties
Microscopic
Macroscopic
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.
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Figure 1.1 Length and time scales of atoms and cats
http://www.bibalex.org/English/lectures/printable/zewail.htm
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1-1
1-1.2 Definition of equilibrium states
System has no external influences (such as flow, electrical fields).
All thermodynamic properties must be “time invariance” (time
independent).
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 equilibrium could also be different.
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1-1
1-1.3 Limitations of thermodynamic equilibrium
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.
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1-2
The Entropy Maximum Postulates:
1-2
Postulate II & III
Postulate II:
For an isolated system with constant V, N, U the equilibrium criterion requires
that the entropy S reaches a maximum value.
Postulate III:
S is a monotonically increasing function of U.
S  S (U , V , N 1 , ... , N r )
 S 
0
 U 

V , N1 , ... , N r
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1-2
The system may then be
further divided into two
sub-systems subject to
certain constraints:
S ＝ S（U , V , N）
V1
V2
U1
U2
N1
N2
thermodynamic constraints
U1  U 2  U1'  U 2'
S ＝ S（U） only
U1'  ?
U ?
'
2
By entropy
max.
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1-2
 S 
dS  
dU ; S is extreme value  dS  0

 U  N ,V
Use d2S to check S is
S  S1 (U 1 )  S 2 (U 2 )
max. or min.
 S1 
 S 2 
1
1
 dS  
 dU 1  
 dU 2  dU 1  dU 2  0
T1
T2
 U 1  N ,V
 U 2  N ,V
1 1
dS  (  )dU 1  0
T1 T2
1 1
(  )0
T1 T2
dU 1  dU 2  0
dU 1  0
T1 ＝ T2
An important equilibrium
condition
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1-2
Fixed V & U
Fixed U & N
V1
V2
V1
V2
U1
U2
U1
U2
N1
N2
N1
N2
thermodynamic constraints
thermodynamic constraints
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
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1-3
Intensive Parameters and Equations of
1-3
State
1-3.1 Extensive parameters & intensive parameters
Extensive parameters
S , U , V , N : First order extensive parameters
S (U , V ,  N )   1 S (U , V , N )
First order
S (9U , 9V , 9N )  9 S (U , V , N )
Example:
Enlarged 9 times
S
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S
S
S
S
S
S
S
S
S
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1-3
Intensive parameters
S  S (U , V , N ) ; U  U ( S , V , N )
Fundamental equations
1  S 
P  S 

 S 
 ; 

; 

 U 


T

 N , V T   V  U , N T  N  U , V
 U 
 U 
 U 
T ; 
 P ; 

 S 



N , V
 V  N , S
 N  S , V
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
Chemical potential:
The definition of chemical potential is the
energy change from a particle, molecule or
electron move into or leave the system.
 U 

 N 

S, V
Example:
Gas
Liquid
If the overall system has reached an equilibrium, the energy reduction(increase) in
the liquid phase due to the moving of a
molecule into the gas phase is the same as
the energy increase(reduction) in the gas
phase. So, the total system energy remains
unchanged.
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1-3
1-3.2 Fundamental equations & equations of state
Fundamental equations
S  S (U , V , N )
Entropy representation
U  U(S, V , N )
Energy representation
Two equations are alternative forms of the fundamental relation, and each
contains all thermodynamic information about the system.
The fundamental equations describe homogeneous first-order parameters.
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1-3
Equations of state
U  U  S ,V , N 
 U 
 U 
 U 

T
;


P
;

 S 
 V 
 N 

N , V

N , S

S, V
T  T(S, V , N )
Equations of state:
P  P( S , V , N )
  ( S , V , N )
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
Figure 1.2
U vs. S
&
U vs. T in constant V & N
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1-3
 (U )
 (U )

  0T
 ( S ) N , V  ( S ) N , V
Zero order
U & S are first order parameters
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
The equations of state describe homogeneous zero-order parameters.
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1-3
1-3.3 Unit of temperature
Thermodynamic engine
Thermodynamic engine efficiency:
Th
dQh
dWrev
Tc
dWrev
Tc
εe 
 1
dQh
Th
Used to define temperature scale.
T1  T1

T2  T2
Permissible scale (e.g. K, R)
T1 T1  k

T2 T2  k
Non-permissible scale (e.g. oC, oF)
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1-3
Question: All heat can be transfer to work?
Answer:
It is impossible unless Tc = 0K
Rotation
Tc ≠ 0K
Heat
Internal energy
Molecular
Vibration
Kinetic energy
Work
Rotation
Tc = 0K
Heat
Internal energy = 0
Molecular
Vibration
Kinetic energy
All heat transfers to
produce work
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1-3
Unit of temperature
Unit of U is energy.
 U 
T
 S 

N , V
What is the unit of
entropy?
So, unit of temperature is energy.
Unit of entropy:
In classic thermodynamics, entropy
is dimensionless parameter.
In thermostatistics, unit of entropy
correlates with states.
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
1-3.4 The general case of entropy max. in equilibrium
Stotal  S1 (U1 , V1 , N 1 )  S2 (U 2 , V2 , N 2 ) w.s.t. dU , dV , dN
1
T1
P1
T1

1
T1
 S1 
 S1 
 S1 
dS  
dU1  
dV1  
dN 1



 U1 V1 , N1
 V1 U1 , N1
 N 1 V1 ,U1
 S 2 
 S 2 
 S 2 

dU 2  
dV2  
dN 2



 U 2 V2 , N 2
 V2 U 2 , N 2
 N 2 V2 ,U 2
1
T2
P2
T2

2
T2
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1-3
 1 1
 P1 P2 
 1 2 
dS     dU1     dV1    
 dN 1  0 at equilibrium
 T1 T2 
 T1 T2 
 T1 T2 
T1  T2
This equation always holds only if
P1  P2
1  2
If dS  0 in reaching equilibrium
initial state
1. P1  P2 , 1  2 , T1  T2
dU1  0
2. P1  P2 , 1  2 , T1  T2
dV1  0
3. P1  P2 , 1  2 , T1  T2
dN 1  0
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1-4
The Euler Equation and the Gibbs-Duhem
1-4
Relation
1-4.1 The Euler relation
Homogenous first order property
U ( S , V ,  N 1 ,  N 2 , ...)  U ( S , V , N 1 , N 2 , ...)
differentiating w.r.t.  (d  )
T
S
P
V
U ( S , V ,  N 1 ,  N 2 , ...)  (  S ) U (  S , V ,  N 1 ,  N 2 , ...) ( V )

 ( S )

 (V )

U ( S , V ,  N 1 ,  N 2 , ...)  (  N 1 )

 ...  U ( S , V , N 1 , N 2 , ...)
 ( N 1 )

1
N1
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1-4
U ( S , V , N 1 , N 2 , ...)  TS  PV  1 N 1  2 N 2  ...
 TS  PV    j N j
j=1
Let
PX
i
i
  PV    j N j
i=1
j=1
U ( S , V , N 1 , N 2 , ...)  TS   Pi X i
i=1
Another form:
 j 
1
P
S (U , V , N 1 , N 2 , ...)    U    V     N j
T 
T 
j  T 
The Euler relation is useful for deducing many important thermodynamic
relationships, including the famous Gibbs-Duhem equation.
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1-4
1-4.2 The Gibbs-Duhem equation
Euler relation: U ( S , V , N 1 , N 2 , ...)  TS   Pi X i
dU  TdS  SdT   PdX
i
i   X i dPi
i=1
Given that
i=1
i=1
dU  TdS   PdX
i
i
i=1
One obtains
SdT   X i dPi  0
the Gibbs-Duhem equation
i=1
Another form
 j 
1
P
Ud    Vd     N jd    0
T 
T  j
T 
Not all intensive properties are independent, but only n-1 of them are
independent. Another one can be deduced from the Gibbs-Duhem
equation.
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1-4
Example:
For one-component systems (having 3 intensive properties T , P &  )
SdT  VdP  Nd   0
S
V
d    dT  dP   sdT  vdP
N
N
s & v : unit number S & V
Only T & P are independent, μ
can be deduced from the GibbsDuhem 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.
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1-4
1-4.3 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
T  T(S, V , N )
P  P( S , V , N )
U  TS   Pi X i
U  U(S, V , N )
  ( S, V , N )
The totality of all equations of state can be used to reconstruct the
fundamental equation.
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1-H
1-H Homework
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)
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Thermodynamic Potentials and
Chapter 2 Legendre Transformations
Contents
2-1
2-2
2-3
2-4
Legendre Transformations
The Legendre Transformed Functions and Thermodynamic
Potentials
Minimum Principles for the Potentials
Applications of various Legendre Transformed Potentials
2-5
Maxwell Relations and Some Applications
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2-1
2-1 Legendre Transformations
2-1.1 The S max. principle & the U min. principle
An isolated system must conform with two
extremum criteria when equilibrium is reached.
Entropy max. principle:
Energy min. principle:
Entropy is max. value for
given total energy.
Energy is min. value for
given total entropy.
Equivalent
representations
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2-1
S max. principle
U min. principle
Fundamental
surface
Figure 2.1 The equilibrium state A as a point of max. S for constant
U & min. U for constant S.
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.
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2-1
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.
Can withdraw energy
dU = -dW
Energy
System
Return energy to sys.
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.
-dW = dQ = TdS
The original equilibrium state must have had min. energy consistent with
the prescribed entropy. The inverse argument is the same.
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2-1
Mathematical exercise
Entropy maximum principle:
 2S 
 S 
 X   0 and  X 2   0

U

U
X : Particular extensive parameter
 S 
 X 
 U 
 S 

U




T
 X 
 X   0

S



S

U
 U 

X
 U 
 X   0

S
By eqn. A.22 of Appendix A
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2-1
Chain Rule
  S 


  2U 
   X U
=


2 
 X  S X   S 
  U 
X

2S
 T
0
2
X
 2S 
2


S

2 

X  U   S  X U
 


2

S



 X  U  S 
 U 

 U 

X
S

X
S
at
0
X
From S max. principle:
 2S 
 S 
 X   0 ;  X 2   0

U

U
 2S 
0 & T 0

2 
 X  U
U min. principle:
  2U 
 U 
 X   0 ;  X 2   0

S

S
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Molecular Rheology LAB
2-1
2-1.2 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.
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Molecular Rheology LAB
2-1
X, Y are extensive properties & Y=Y(X).
Let slope is P 
Y
; P: intersive properties
X
If one considers the slope and the intercept
at the same time, the transformation from
Y(X) to ψ(P) is an unambiguous mapping.
Y  Y(X )
Y Y  

X X  0
  Y  PX
P
  (P)

X 
P
Y    PX
Elimination of X and Y yields
Elimination of P and  yields
  (P)
Y  Y(X )
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Molecular Rheology LAB
2-1
Generalization of the Legendre transformation
Y  Y ( X 0 , X 1 , ... , X t )
Pk 
Y
X k
   ( P0 , P1 , ... , Pn , X n1 ,
... , X t )
  Y   Pk X k
k
Elimination of Y and X 0 , X 1 ,
Xk  

Pk
Y     Pk X k
k
... , X n yields
Elimination of  and P0 , P1 ,
   ( P0 , P1 , ... , Pn , X n1 ,
... , Pn yields
... , X t )
Y  Y ( X 0 , X 1 , ... , X t )
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Molecular Rheology LAB
2-2
The Legendre Transformed Functions
2-2
and Thermodynamic Potentials
Legendre transformed functions
Y  Y ( X 0 , X 1 , ... , X t )
Pk 
Y
X k
The energy representation of the
fundamental equation: U  U ( S , V , N )
Intensive parameters
T ,  P , 1 , 2 , ...
  Y   Pk X k
k
Thermodynamic potentials
F  F (T , V , N )
H  H (S, P, N )
G  G (T , P , N )
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2-2
The Helmholtz potential, F
U(S, V , N )

F (T , V , N )
ψ P
Y X
Y  Y ( X 0 , X 1 , ... , X t )
U  U(S, V , N )
Y
Pk 
X k
  Y   Pk X k
k
   ( P0 , P1 , ... , Pn , X n1 ,
U
S
F  U  TS
F  F (T , V , N )
T
... , X t )
 F 
 F 
 F 


S
;


P
;

 T 
 V 
 N 

N , V

T , N

T , V
dF   SdT  PdV   dN
43
Molecular Rheology LAB
2-2
The Enthalpy, H
U(S, V , N )
Y

H (S, P, N )
ψ
X
P
Y  Y ( X 0 , X 1 , ... , X t )
Y
Pk 
X k
  Y   Pk X k
k
   ( P0 , P1 , ... , Pn , X n1 ,
U  U(S, V , N )
U
V
H  U  PV
H  H (S, P, N )
P 
... , X t )
 H 
 H 
 H 

T
;

V
;
 S 
 P 
 N   

P, V

S, N

P, S
dH  TdS  VdP   dN
44
Molecular Rheology LAB
2-2
The Gibbs potential, G
U(S, V , N )

G (T , P , N )
ψ P1 P2
Y X1 X2
Y  Y ( X 0 , X 1 , ... , X t )
Y
Pk 
X k
  Y   Pk X k
k
   ( P0 , P1 , ... , Pn , X n1 ,
U  U(S, V , N )
U
U
; P
S
V
G  U  ST  PV
G  G (T , P , N )
T
... , X t )
 G 
 G 
 G 


S
;

V
;

 T 
 P 
 N 

P, N

T , N

P, T
dG   SdT  VdP   dN
45
Molecular Rheology LAB
2-2
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.
The following equations are all fundamental equations:
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 fluids
H=H(S, P, N): Heat transfer
G=G(T, P, N): Chemical reactions & Phase equilibrium
46
Molecular Rheology LAB
2-3
2-3 Minimum Principles for the Potentials
The Helmholtz potential
Consider a subsystem in contact with a thermal reservoir.
Reservoir T r
Thermal reservoir (Cp → ∞, T ＝ T r )
T(1)
Subsystem
T(2)
Using U representation U=U(S, V, N) :
dU total  0
d 2U total  0
from the extremum
principle at equilibrium
47
Molecular Rheology LAB
2-3
dU r
dU
dU total  (dU r  dU )  (T r dS r  T (1)dS (1)  T (2)dS (2) )
( P (1)dV (1)  P (2)dV (2) )  (  (1) dN (1)   (2) dN (2) )  0
r
r
r







U

U

U
r
r
r
r
dU  
dS

dV

dN





r
r
r
 S V r , N r
 V  S r , N r
 N V r , S r
 T r dS r
S total  constant  dS total  dS  dS r  dS (1)  dS ( 2)  dS r  0
dS (1)  dS ( 2)   dS r
The subsystem of interest  dV (1)  dV ( 2)  0 ; dN (1)  dN ( 2)  0
dV (1)   dV ( 2) ; dN (1)   dN ( 2)
48
Molecular Rheology LAB
2-3
(T (1)  T r )dS (1)  (T (2)  T r )dS (2)  ( P (1)  P (2) )dV (1)  (  (1)   (2) )dN (1)  0
This equation always holds only if
T (1)  T (2)  T r
P (1)  P (2)
 (1)   (2)
At equilibrium
dU total  (dU  dU r )  dU  T r dS r  d (U  T r S r )  d (U  TS )  0
 dF  0
F  U  TS
d 2U total  d 2 (U  U r )  d 2U  d 2U r  d 2U  d 2U  d 2 (TS )  d 2 (U  TS )  0
 d 2F  0
By eqn. A.9 of Appendix A ;
For thermal reservoir T = T r, V r & N r = constant
49
Molecular Rheology LAB
2-3
By the same way, consider a subsystem in contact with a pressure reservoir,
or with both pressure & thermal reservoirs.
Reservoir P r
P(1)
P(2)
Reservoir T r & P r
Subsystem
T(1) P(1)
T(2) P(2)
H  H (S, P, N )
G  G (T , P , N )
dH  0
dG  0
d 2H  0
d 2G  0
50
Molecular Rheology LAB
2-3
The overall system:
Representation:
U  U ( S ,V , N )
Experimental conditions:
V total , N total , S total
Extremum criteria:
dU total  0 ; d 2U total  0
The subsystem:
Representation:
Experimental conditions:
Extremum criteria:
F  F (T ,V , N )
T, V, N
dF  0 ; d 2 F  0
H  H (S, P, N )
S, P, N
dH  0 ; d 2 H  0
G  G (T , P , N )
T , P, N
dG  0 ; d 2G  0
51
Molecular Rheology LAB
2-3
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.
52
Molecular Rheology LAB
Applications of various Legendre
2-4
Transformed Potentials
2-4
2-4.1 The Helmholtz potential
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.
53
Molecular Rheology LAB
2-4
Example:
Two simple subsystems separated by a
movable, adiabatic, impermeable wall.
V (1)  ?
V ( 2)  ?
Wrev  ?
P (1) (T r , V (1) , N (1) )  P ( 2) (T r , V ( 2) , N ( 2) )
V (1)  V ( 2)  V , a constant
Can solve V (1) & V (2)
dWrev   dU  dU r   dU  T r dS r   dU  T r dS   d (U  T r S )
  dF
The number of independent variables: V & N
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.
54
Molecular Rheology LAB
2-4
Focus our attention exclusively on the subsystem of interest, and relegate
the reservoir only to an implicit role.
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.
As shown in Chap. 16, statistical mechanical calculations are enormously
simpler for such systems, permitting calculations that would otherwise be
totally intractable.
See Example.1 in page 159, which shows that one can increases the
entropy of a subsystem without changing the associated internal energy.
55
Molecular Rheology LAB
2-4
2-4.2 The enthalpy
dH  TdS  VdP  1dN 1  2dN 2  ...
At constant P , N i  dH  TdS  dQ
As a potential for heat
H  (S, P, N )
 H 
V 
 V (S, P, N )

 P  S , N
H (V , P , N )
Qi  f  H (V f , P , N )  H (Vi , P , N ) At constant P , N
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.
Using a similar argument, U is a potential for heat, too.
dU  TdS  PdV   dN ; at constant V , N  dU  TdS  dQ
56
Molecular Rheology LAB
2-4
The Joule-Thomson process
u f  ui  Pi vi  Pf v f ; by conservation of energy
 u f  Pf v f  ui  Pi vi
h f  hi
A constant H process
u: initial molar energy
v: molar volume
h: molar enthalpy
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.
57
Molecular Rheology LAB
2-4
The isenthalpic curves (“isenthalps”)
The final temperature Tf can be
higher or lower then Ti in this
process, depending on the local
relationship:
 T 
dT  
dP

 P  H , N1 , N 2 , ...
By eqn. A.22 of Appendix A
& Maxwell relations
v
dT 
T  1 dP
CP
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.
58
Molecular Rheology LAB
2-4
2-4.3 The Gibbs potential
Most experiments are carried out under constant T & P.
 G  (T , P , N1 , N 2 , ...) is a particularly convenient choice.
by defition: G  U  TS  VP
the Euler relation: U  TS  VP   i N i
 G   i N i
i
As a potential for chemical potential
i
r
Consider the chemical reaction:
0   j A j
1
Aj : symbols for the chemical components
 j : stoichiometric coefficients
59
Molecular Rheology LAB
2-4
Change of G associated with changes in Nj can be described by
dG   SdT  VdP    j dN j
j
since
dN1
1

dN 2
2
 ...  dN
One has for the equilibrium condition:
 dN j   j dN
dG  dN   j j  0
j
or

j
j
0
j
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:

dH
  r
 T


 j j 
dN
T  1
 P , N1 , N 2 , ...
60
Molecular Rheology LAB
2-4
2-4.4 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.
61
Molecular Rheology LAB
2-5
2-5 Maxwell Relations and Some Applications
2-5.1 The Maxwell relations & mnemonic diagram
Maxwell relations
U  U(S, V , N )
Constant N
2nd derivatives of potential
  U
S  V
  U 


 V  S 



P
T
Maxwell relation equation
 P 
 T 





 S V , N  V  S , N
62
Molecular Rheology LAB
2-5
In a similar way, we can derive other forms as shown below:
Table 2.1. Maxwell relations for several potentials with various pairs of
independent variables. (see also pp. 182 & 183 of the textbook)
63
Molecular Rheology LAB
2-5
Mnemonic diagram
Valid
V
Facts
F
Understanding U
S
Solutions
Theoretical
T
G Generate
H
Hard
P
Problems
Pithy formula: “Valid Facts and Theoretical Understanding Generate
Solutions to Hard Problems”
64
Molecular Rheology LAB
2-5
The first derivative of potential:
Example:
 U 
T
 S 

S, N
V
("  " arrow pointed away)
F
U
 U 
 P
 V 

V , N
Maxwell relations:
Example:
 V 
 T 

 S 



 P  P  S
("  " arrow pointed toward)
("  ") S
 S 
 V 
 P     T 

T

P
S
Molecular Rheology LAB
G
S
V
T
H
P
T
P
S
P ("  ")
T
V
T ("  ")
P ("  ")
P
65
2-5
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
F  U [T ]
U
S
U [ j ]
T
X1
U [T ,  j ]
U
j
X2
U(S, Nj) mnemonic diagram
U [ P2 ]
P1
U[ P1 , P2 ]
U[ P1 ]
P2
General-form mnemonic
diagram
66
Molecular Rheology LAB
2-5
2-5.2 Procedure for the reduction of derivatives
Four customarily used derivatives
1  v 
1  V 
Thermal expansion coefficient:
 

v  T  P V  T  P
Isothermal compressibility:
T   
 

v  P  T
V  P T
1  v 
1  V 
Molar leat capacity at constant pressure:
T  S 
1  dQ 
 s 
cP  T 







 T  P N  T  P N  dT  P
Molar leat capacity at constant volume:
T  S 
1  dQ 
 s 
cV  T 







 T V N  T V N  dT V
67
Molecular Rheology LAB
2-5
For single-component systems and constant molecule numbers, the second
derivatives of Gibbs representation can be expressed by cP ,  , and  T .
2g
2g
2g
  c P /T ;
 v ;
  v T
2
2
T
T P
P
All first derivative, such as dT   T P V , N dP (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  , cP and  T .
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Molecular Rheology LAB
2-5
Reduction of derivatives
If the derivative contains any potential, bring them one by one to the
numerator and elimin2 e by the thermodynamic square.
Example:
 P 
Reduce the derivative 

 U  G , N
 P 
 U 

G , N
  U  
 
 

P
G , N 

(See also pp. 187-188 of the textbook)
1
  S 
 V  
 T 
 P

 

P

P
G , N

G , N 
 
  G 
  T 

  P  S , N
1
 G 
 G 

P
 S 
 P 

P, N

V , N
 G  
 V  

 P , N 

 S  T P  S , N  V
 S  T P V , N  V 
  T
P


S

T

S

S

T

V

P, N

 P , N 

Molecular Rheology LAB
1
1
69
2-5
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 

 V  S , N
  
 T 
 P 


s

v
 V 
 V 
 V 

S, N

S, N

S, N
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 T under S. The numerator will then be expressible as one
of the cV or cP.
Example:
 T 
Reduce the derivative 

 P  S , N
 T 
 S 
 S 
 V 



 P 




 P 

 T , N  T  P , N  T  P , N

S, N
N
c
T P
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Molecular Rheology LAB
2-5
Bring the volume to the numerator. The remaining derivative will be
expressible in terms of  and T .
Example:
 T 
Reduce the derivative 

 P V , N
T
 T 
 V 
 V 



 P 


 P 

 T , N  T  P , N 

V , N
The originally given derivative has now been expressed in terms of the
four quantities cP , cV ,  , and  T . The specific heat at constant volume is
eliminated by the follow equation.
cV  c P 
Tv 2
T
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Molecular Rheology LAB
2-5
2-5.3 Some simple applications
I. Adiabatic compression
Adiabatic wall
Ti
Adiabatic wall
Quasi-static process
Pi
Tf =?
Pf
If known: fundamental equation, Pi , Pf , and Ti
Fundamental equation
Equations of state:
ΔT=T(S, Pf , N)- T(S, Pi , N)
(see p. 190)
If known: Pi , Pf , Ti ,c P ,  , and  T
Tv
sTv
 T 
  
dT  
dP

dP
;
d


dP

(
v

)dP

 P 

P
c
c

S, N

S, N
P
P
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Molecular Rheology LAB
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II. Isothermal compression
dS  ?
Quasi-static process
Pf
Pi
Constant T
dU  ?
Q  ?
If known: fundamental equation, Pi , and Pf
Equations of state:
S  S (T , P , N )
Fundamental equation
U  U (T , P , N )
Q  T S  TS (T , Pf , N )  TS (T , Pi , N )
If known: Pi , Pf ,c P ,  , and  T
 S 
 U 
dS  
dP



VdP
;
dU

dP  ( T v  PV  T )dP



 P  T , N
 P  T , N
dQ  TVdP  Q  T  VdP
Pf
Molecular Rheology LAB
Pi
73
2-5
III. Free expansion
Adiabatic wall
Vacuum
Adiabatic wall
Gas
Vi ; Ti
Gas
Constant U
Vf ; Tf =?
If known: fundamental equation, Vi , V f , and Ti
Fundamental equation
Equations of state:
T f  Ti  T (U , V f , N )  T (U , Vi , N )
If known: Vi , V f ,Ti , c P ,  , and  T
 P
T
 T 
dT  
dV




 V U , N
 NcV NcV  T
Molecular Rheology LAB

 dV

74
2-H
2-H Homework
Problem 5.3-1
Problem 5.3-12
Problem 6.3-2
Problem 7.4-7
Problem 7.4-15
Problem 7.4-23
Molecular Rheology LAB
75
Stabilities, Phase Transitions, and
Chapter 3 Critical Phenomena
Contents
3-1
3-2
3-3
3-4
3-H
Stability Conditions for Thermodynamic Potentials
Le Chaterlier’s Principle and Effects of Fluctuations
First-order Phase Transitions in Single-Component Systems
Thermodynamic States near Critical Points
Homework
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Molecular Rheology LAB
3-1
Stability Conditions for Thermodynamic
3-1
Potentials
3-1.1 Intrinsic stability of thermodynamic systems
Basic extremum principle of thermodynamics:
dS  0
The S is an extremum value.
d 2S  0
The extremum value is a maximum, which guarantees
the stability of predicted equilibrium states.
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Molecular Rheology LAB
3-1
Remove ΔU from the
right subsystem and
transfer it to the left
subsystem.
Initial state
S(U)
S(U)
New state
S(U+ΔU)
S(U-ΔU)
S Total  S U  U   S U  U 
S Total  2S U 
Convex
Concave
S  U  U   S  U  U   2 S (U )
 Unstable
(see later dicussion
on phase separation)
S  U  U   S  U  U   2 S (U )
 Stable
(see later discussion on the
Le Chaterlier’s principle)
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Molecular Rheology LAB
3-1
The condition of stability is the concavity of the entropy.
S U  U   S U  U   2S (U )
for all 
By eqn. A.4 of Appendix A (refer to problem 8.1-1 of the textbook)
 2S 
0

2 
 U V , N
For U  0
The same considerations apply to a transfer of volume.
S V  V   S V  V   2 S (V )
for all 
 2S 
0

2 
 V  U , N
For V  0
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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Molecular Rheology LAB
3-1
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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Molecular Rheology LAB
3-1
If one simultaneously changes U and V:
 2S 
0

2 
 U V , N
 2S 
0

2 
 V  U , N
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.
S  U  U , V  V   S  U  U , V  V   2 S (U , V )
By eqn. A.4 of Appendix A (refer to
problem 8.1-1 of the textbook)
2
 2S  2S   2S 


 0
2 
2 
 U   V   U V 
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Molecular Rheology LAB
3-1
3-1.2 Stability condition for thermodynamic potentials
Convexity of the energy surface:
U  S  S , V  V , N   U  S  S , V  V , N   2U ( S , V , N )
The local conditions of convexity:
  2U 
T
0

 2
 S V , N S
  2U 
P
0



2 
V
 V  S , N
For cooperative variations of S and V :
2
  2U    2U    2U 

 0
 2 
2 
 S   V   S V 
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Molecular Rheology LAB
3-1
Transform to other potential representations by Legendre
transformation :
Example:
F  (T , V , N )
  2F 
  2U 
1  U 
 S 
 T 
 
 1 
 1 
 1  2 




2 
  S V , N
 S V , N
 T V , N
 S V , N
 T V , N
  2F 

0
2 
 T V , N
;
  2U 
0
 2
 S V , N
  2F 
; By the same way 
0
2 
 V T , N
F T  T , V  V , N   F T  T , V  V , N   2F (T , V , N )
By eqn. A.4 of Appendix A (refer to problem 8.1-1 of the textbook)
2
  2F    2F    2F 


 0
2 
2 
 V   T   V T 
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Molecular Rheology LAB
3-1
Similarly, one has for other potentials
H  (S, P, N )
2
 2H 
 2H 
 2H  2H   2H 
0 ; 
0 ; 


 0
2 
2 
2 
2 
 S V , N
 P V , N
 S   P   S P 
G  (T , P , N )
2
  2G 
  2G 
  2G    2G    2G 
0 ;  2 
 0 ;  2  2   
 2
 0
 T V , N
 P V , N
 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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Molecular Rheology LAB
Le Chaterlier’s Principle and the Effects
3-2
of Fluctuations
3-2
3-2.1 Le Chatelier’s principle
The Le Chatelier’s principle: any inhomogeneity that somehow develops
will induce a process that eradicates the inhomogeneity.
Example:
Incident
photon
Container
Fluid
T1
In equilibrium
state
Fluid
T1
T1
Locally heating
the fluid slightly.
T1’
Heat flows away The system restores
form heated region. homogeneous state.
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Molecular Rheology LAB
3-2
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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Molecular Rheology LAB
3-2
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.
Example:
Consider a subsystem contained within a cylinder with diathermal walls and
a loosely fitting piston.
Piston 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
Pi ; Ti ; Vi
T&P
reservoirs
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87
3-2
Secondary effect: An increased Vi alters Ti ; heat flows inward if α＞0 and
outward if α＜0.
T
 T 
dT  
dV  
dV

NcV  T
 V  S
Piston is moved outward
slightly by external agent
or fluctuation.
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.
1  P 

dP  
dQ 
dQ ; dQ  TdS

2
T  S  S
NT cV  T
(note that  dQ is always positive)
Po
Pi ; Ti ; Vi
Q
T&P
reservoirs
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Molecular Rheology LAB
3-3
First-order Phase Transitions in Single3-3
Component Systems
3-3.1 First-order & second-order phase transitions
Fluctuation
Fluctuations (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.
• Global min. is path independent state, yet the metastable one is path
dependent.
The system spends almost all the time in the more stable minimum.
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Molecular Rheology LAB
3-3
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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3-3
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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Molecular Rheology LAB
3-3
3-3.2 Latent heat
At first-order phase transition:
 ( I )   ( II )
Discontinued in u, f , h, v , s. But P ( I )  P ( II )
T ( I )  T ( II )
Example:
Latent heat of fusion:
dQ  Tds  Q  l LS  T ( s ( L )  s ( S ) )  T s
since h  Ts   ; at phase transition point   ( L )   ( S )
 h  T s  l LS
l LS  80 cal
Ice
cP  2.1kJ
kg-K
g
lGL  540 cal
Water
cP  4.2 kJ
g
Steam
Saturated steam table
kg-K
Superheated steam table
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3-3
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3-3
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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Molecular Rheology LAB
3-3
The Clapron equation
Clapron equation is the slope of coexistence curves.
dP
Slope:
dT
At phase equilibrium   A   A ;  B   B
  B   A   B   A ; d    sdT  vdP
  B   A   sdT  vdP   B   A   sdT  v dP
(v   v )dP  ( s  s )dT 
and l  T s  s 
dP
l

dT T v
dP ( s'  s ) s


dT (v'  v ) v
l
T
The Clapeyron equation
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Molecular Rheology LAB
3-3
The Clapeyron equation embodies the Le Chatelier principle. Consider a
solid-liquid transition with a positive latent heat (sl > ss):
dP
If (vl  v s )  Slope 
0
dT
Increase P at constant T  System tends towards the more dense (solid) phase
Increase T  System tends towards the more entropic (liquid) phase
dP
If (vl  v s )  Slope 
0
dT
Increase P at constant T  System tends towards the more liquid phase
The Clapeyron-Clausius approximation:
v  v g  vl  v g ; assume ideal gas  v g 
From Clapeyron equation 
RT
P
dP
P

l
2
dT RT
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Molecular Rheology LAB
3-3
3-3.4 1st 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.
RT
a
P
 2
(v  b) v
van der Waals E.O.S.
 P 
Stability criteria:  T  0 or 
0

 V T
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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Molecular Rheology LAB
3-3
The given P-v isotherms may be used to compute the molar Gibbs free
energy to better perceive the stability criterion.
d    sdT  vdP or    v ( P )dP   (T )
 (T ) is an undetermined function of the T
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).

O
D
v( P )dP  0
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3-3
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.
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.
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Molecular Rheology LAB
3-3
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.
x g  xl  1 ; x g v g  xl v l  v z
xl 
v g  vz
v g  vl
v z  vl
xg 
v g  vl
Molecular Rheology LAB
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3-3
The entropy change during the phase transition can be computed using the
following relation along hypothetical isotherm OMKFD.
 s 
ds    dv
 v  T
 P 
s   
dv

T  v
OMKFD 
 s 
 P 

 v 
 T  the Maxwell relation
 T 
v
The total change in the molar energy:
u  T s  Pv
Limitation: Can’t apply near critical point.
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3-3
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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Molecular Rheology LAB
3-3
3-3.6 1st phase transitions in multi-component system
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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Molecular Rheology LAB
3-3
Gibbs phase rule
Gibbs phase rule is the thermodynamic degrees of freedom (f).
f   2  M ( r  1)  r ( M  1)  r  M  2
variations
equations
r : No. of components
M : No. of phases
f : No. of intensive parameters capable of independent variations
 2  M ( r  1): P , T  each phase have ( r - 1) variations  No. of phases ( M )
 1I  1II  ...  1M 


r ( M  1): 

 rI  rII  ...   rM 


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Molecular Rheology LAB
3-3
Application of the Gibbs phase rule
Example:
I. Single-component system (r＝1)
a: M＝2 → f＝1 → at a given T, another variable 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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Molecular Rheology LAB
3-3
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 twophase state is composed
of A and B, and only A
and B are physical point.
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Molecular Rheology LAB
3-3
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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Molecular Rheology LAB
3-4
Thermodynamic States near Critical
3-4
Points
3-4.1 Features in the vicinity of the critical point
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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Molecular Rheology LAB
3-4
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 shortrange 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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Molecular Rheology LAB
3-4
The generalized susceptibilities, such as cp, κT and α diverge.
Example:
1  v 
T  
v  P T
 v 
 P    at first phase transition or near critical point

T
  T diverges;  and c p diverge in a similar way.
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Molecular Rheology LAB
3-4
3-4.2 The failure of the classical theory in predicting
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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Molecular Rheology LAB
3-4
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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3-H
3-H Homework
Problem 9.1-1
Problem 9.3-3
Problem 9.4-11
Problem 9.7-1
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Chapter 4 Applications to Material Properties
Contents
4-1
4-2
4-3
4-4
The general ideal gas
Small Deviations from Ideality—The Virial Expansion
Law of Corresponding States for Gases
Applications to Dilute Solutions—The van’t Hoff Relation
and the Raoult’s law
4-H Homework
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Molecular Rheology LAB
4-1
4-1 The general ideal gas
4-1.1 The essence of ideal gas behavior
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.  T 
1
1
,   , c p  cv  R have the same value for all ideal gas
P
T
115
Molecular Rheology LAB
4-1
The F , U and S of general multicomponent ideal gas:
F (T , V )   U j (T )  T  S j (T , V )  U  TS
j
j
U j  N j u j (T )  N j u j 0  N j  cvj (T  )dT 
T
T0
T0 : arbitrarily chosen standard temperature
 v N 
1
S   N j s j0   N j 
c (T  )dT    N j R ln 

T0 T  vj

j
j
j
 v0 N j 
T 1
v
 N
c v (T  )dT   NR ln  NR  x j ln x j
T0 T 
v0
j
T
(see also Eq. (3.40)
entropy of mixing
Molecular Rheology LAB
(at the same temperature and particle density)
116
4-1
Step-like
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.
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
117
Molecular Rheology LAB
4-1
4-1.2 Chemical reactions in ideal gases
Partial molar Gibbs poential:  j  RT  j (T )  ln P  ln x j 
Since  v j  j  0 at chemical equilibrium
j
  v j ln x j    v j ln P   v j ln  j (T )
j
j
j
The only temperaturedependent term
Let K (T ), equilibrium constant, be defined by ln K (T )    v j ln  j (T )
j
 j (T ) can in principle be known, so that K (T ) can be considered as known.
For given T & P together with
x
j
 1 each of x j can be determined.
j
 The mass action law:  x j j  P  j j K (T )
v
v
j
118
Molecular Rheology LAB
4-1
Logarithmic additivity of K(T)
(1)
(2)




reaction1: 0 
v
A
;
reaction2:
0
v
 j j

 j Aj

 v (3)
If 0 

j A j can be obtained by summing the previous reaction,
(1)
(2)

 v (3)
that is, 0 
A

(
B
v

B
v
; B1 & B2 are constant


j
j
1 j
2 j )A j
Then it can be shown that:
ln K 3 (T )  B1 ln K 1 (T )  B2 ln K 2 (T )
Can help tabulations of K(T)
for various chemical reactions.
The reaction heat can also be computed from K (T ) as
dH
d
 RT 2
ln K (T )
(the van't Hoff relation)
dN
dT
(see an example in page 295 of the textbook.)
119
Molecular Rheology LAB
4-2
Small Deviations from Ideality—The Virial
4-2
Expansion
4-2.1 Virial expansion
P R
B(T ) C (T )

 1
 2  
T v
v
v

Correction
term for v
B(T ): "Second virial coefficient" which accounts for the contribution from
Ideal behavior
two-body interactions.
C (T ): "Third virial coefficient" which accounts for the contribution from
three-body interactions etc.
For simple gases, use of the second virial coefficient may suffice to
capture their physical behaviors; for complex fluids, more correction terms
120
must be introduced.
Molecular Rheology LAB
4-2
4-2.2 Other thermodynamic quantities may be
expanded in a self-consistent fashion
 B(T ) C (T ) D(T )
 (f is the molar Helmholtz potential)
f  f ideal  RT 







2v 2
3v 2
 v

f
(from P   )
v
 1 d 2 ( BT )

1 d 2 (CT )
1 d 2 ( DT )
cv  cv ,ideal  RT 
 2
 2
 
2
2
2
2v dT
3v
dT
 v dT

f
 s 
(from cv  T 
; s
)

T
 T  v
1 dC
1 dD
 1 dB

u  uideal  RT 2 
 2
 2
 
 v dT 2v dT 3v dT

(from u  f  Ts )
121
Molecular Rheology LAB
4-2
The second virial coefficient vanishes at some special temperatures for
several simple gases.
122
Molecular Rheology LAB
4-3
4-3 Law of Corresponding States for Gases
4-3.1 Important observations for gases
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)
123
Molecular Rheology LAB
4-3
4-3.2 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
Z
Pv
RT
or
 P  v   T 
Z  0.27 

 

P
v
T
 cr   cr   cr 
as functions of P/Pcr, and T/Tcr.
124
Molecular Rheology LAB
4-3
Tr curve
125
Molecular Rheology LAB
4-3
The Lennard-Jones potential
1
r6
1
Repulsive force  12
r
Attractive force 
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.
126
Molecular Rheology LAB
4-4
Applications to Dilute Solutions—the van’t
4-4
Hoff Relation and the Raoult’s law
4-4.1 the van’t Hoff relation for osmotic pressure
Consider a single-component fluid system, the Gibbs potential of the dilute solution
can be written in the general form:
G (T , P , N 1 , N 2 )  N 1 10 ( P , T )  N 2 ( P , T )  N 1 RT ln
 N 2 RT ln
N1
N1  N 2
N2
N1  N 2
 represents the effect of the interaction energy between the two types of molecules,
while the last two terms accounts for the entropy of mixing
In dilute region (N 2  N 1 ) 
ln
N1
N
N1
 2 , N 1 ln
  N2
N1  N 2 N1
N1  N 2
 G (T , P , N 1 , N 2 )  N 1 10 ( P , T )  N 2 ( P , T )  N 2 RT  N 2 RT ln
Molecular Rheology LAB
N2
N 1 127
4-4
Solvent and solute partial molar Gibbs potentials:
1 ( P , T , x ) 
G
 10 ( P , T )  xRT
N 1
2 ( P , T , x ) 
N
G
  ( P , T )  RT ln x ; ( x  2 )
N 2
N1
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:
1 ( P , T , 0)  1 ( P , T , x )
P : The pressure on the solute side of the membrane
128
Molecular Rheology LAB
4-4
1 ( P , T , 0)  1 ( P , T , 0)  xRT
Expanding 1 ( P , T , 0) around the pressure P
1 ( P , T , 0)
 ( P  P )
P
 1 ( P , T , 0)  ( P   P )v
1 ( P , T , 0)  1 ( P , T , 0) 
( P   P )v  xRT
Multiplying by N 1
 V P  N 2 RT
the van’t Hoff relation for osmotic
pressure in dilute solutions
129
Molecular Rheology LAB
4-4
4-4.2 Application to vapor pressure
The vapor pressure reduced by addition of low concentration of nonvolatile solute:
 liq ( P , T )   gas ( P , T )
  liq ( P , T )  xRT   gas ( P , T )
Expanding  liq ( P , T ) and  gas ( P , T ) around the pressure P
 liq ( P , T )   liq ( P , T )  v liq  ( P   P )
 gas ( P , T )   gas ( P , T )  v gas  ( P   P )
P  P  
xRT
v g  vl
v g  vl
;

(the ideal gas equation: v g 
P
 x
P
RT
)
P
Raoult’s law
130
Molecular Rheology LAB
4-H
4-H Homework
Problem 13.2-2
Problem 13.5-2
Problem 13.5-3
131
Molecular Rheology LAB
Chapter 5 Statistic Ensembles and Formulism
Contents
5-1
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)
132
Molecular Rheology LAB
5-1
The Entropy Representation—the
5-1
Boltzmann’s law
5-1.1 Physical significance of entropy for closed system
Classical Thermodynamics
Erected on a basis of few hypotheses of entropy.
e.g. S=S(U, V, N) ; extremum principle
Statistical Thermodynamics
Statistical mechanics provides the physical interpretation of the
entropy and a heuristic justification for the extremum principle.
133
Molecular Rheology LAB
5-1
Atomic systems
Quantum mechanics
If the system is macroscopic, there
may exist many discrete quantum
states consistent with the specified
values of U, V, and N.
Macroscopic systems
Closed system
given V & N
U, V, and N are the
only constraints on
the system.
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.
134
Molecular Rheology LAB
5-1
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.
135
Molecular Rheology LAB
5-1
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.
136
Molecular Rheology LAB
5-1
Entropy (S )
additive (extensive)
No. of microstates ( )
Multiplicative
6
6
＋
？
＝
6
×
6
S  kB ln 
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.
137
Molecular Rheology LAB
5-1
5-1.2 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 N atoms in the crystal are free to vibrate
around its equilibrium position in any of three coordinate
directions, and can be replaced by 3N harmonic oscillators,
all with the same natural frequency 0 .
Einstein introduced the energy of harmonic oscillators discretely:
 =n 0
;
n  0, 1, 2, ...
 h / 2
;
h: Planck's constant
Discrete values of energy: n 0
138
Molecular Rheology LAB
5-1
If the energy of the system is U it
can be considered as constituting
U  0 quanta. Theres quanta are
to be distributed among 3 N
vibrational modes.
Number of permutations of
(3 N  1  U
0 )
3 N  U   1 !

H
C

 3 N  1 !  U   !
3 N  U   1 !  3 N  U   !

 

 3 N  1 !  U   !  3 N  !  U   !
3N
U 0
3 N U
U 0
0 1
0
0
0
0
0
(
N  )
0
139
Molecular Rheology LAB
5-1

u
u 
u
s  3 R ln  1    3 R ln  1   ; u0  3 N A 0
u0 
u0 
u0 

From S  k B ln  & ln( M !)  M ln M  M     (if M  1)
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.
k
1 S
3N



 B ln  1 
N A 0 
T U
0 
U

 There are 3 NN A oscillator in the system
Thermal equation of state:
 mean energy per oscillator 
U

3 NN A e
0
0 k BT
1
The quantity  0 k B is called the "Einstein temperature" of the crystal,
and it generally is of the same order of magnitude as the melt point of the solid.
140
Molecular Rheology LAB
5-1
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.
141
Molecular Rheology LAB
5-1
5-1.2 The two-state system ; a polymer model
Two-state system
Total atoms N
U : system energy
(N - U ) atoms are
Ground state
Excited state
in the ground state
(with energy 0)
(with energy ε)

   CUN  
N!
U   ! N  U
U
 atoms are in
the excited state
 !
U  U
U
U


 S    N  k B ln  1 
 ln

N  
N



From S  k B ln  & ln( M !)  M ln M  M     (if M  1)
142
Molecular Rheology LAB
5-1
Thermal equation of state:
N
U 
1  e  k BT

1 S k B  N 


ln 
 1
T U
  U

( U  N )
;
du
2
e  k BT
c
 NA
dT
k BT 2 1  e  2 k B T


2
 NA
2
k BT
2
e
 2 k BT
e
  2 k BT

2
The energy approaches N  2 as the temperature approaches infinity in
this model.
At infinite temperature half the atoms are excited and half are in their
ground state.
143
Molecular Rheology LAB
5-1
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.
144
Molecular Rheology LAB
5-1
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.
145
Molecular Rheology LAB
5-1
N x  N x  N y  N y  N
Lx
 Lx
a
Ly


Ny  Ny 
 Ly
a
U
N y  N y   U 
N x  N x 

1
2
1

Nx 
2
1
N y 
2
1
N y 
2
N x 
 N  U   L 
x
 N  U   L 
x
 U   L 
y
 U   L 
y
The number of configurations of the polymer
consistent with given corrdinates Lx and Ly
of its terminus and given energy U :
(U , Lx , Ly , N ) 
N!
N x ! N x ! N y ! N y !
146
Molecular Rheology LAB
5-1
S  k B ln   Nk B ln N  N x k B ln N x  N x k B ln N x
 N y k B ln N y  N y k B ln N y
or
1
1

N  U   Lx k B ln  N  U   Lx 
2
2

1
1



 N  U  Lx k B ln  N  U   Lx 
2
2

S  Nk B ln N 








1
1

U   Ly k B ln  U   Ly 
2
2

1
1

 U   Ly k B ln  U   Ly 
2
2

Ty
S k B U   Ly
 

ln
 0 ; (  y  0)
T Ly 2a U   Ly









 Ly  Ly  N y  N y 
Molecular Rheology LAB
Ly
a
 Ly & a  0  Ly  Ly  0  N y  N y
147
5-1
S k B N  U   Lx
 

ln
T Lx 2a N  U   Lx
Tx
N  U   Lx
e

N  U   L
kB
kB
1 S k B






ln N  Lx  U 
ln N  Lx  U 
ln U 
T U 2
2

 2 Tx a k B T

 e 2
k BT

N  U 


2


 Lx 2
U2
sinh  Tx a k BT 
Lx

N cosh  Tx a k BT   e 
k BT
Tx Na
2
1
For small Tx a (relative to k BT )  Lx 
k BT 1  e  
k BT
 
 The modulus of elasticity of the rubber band:
1  Lx 
a2

 

1

e

N  Tx T k BT
Molecular Rheology LAB

k BT

1
148
5-2
5-2 The Canonical Formalism
5-2.1 The probability distribution
Limitation of microcanonical formalism:
The microcanonical formalism is simple in principle, but it is computationally
feasible only for a few highly idealized models.
Isolated
system
Must calculate the no. of combinatorial ways that a given total
energy can be distributed in arbitrarily sized boxes.
Generally beyond our mathematical capabilities!
Canonical formalism (Helmholtz representation)
T reservoir
system
Remove the limitation on the amount of energy available
－to consider a system in contact with T reservoir rather
than an isolated system.
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Canonical formalism
Closed system
T reservoir
Closed system ＝ Subsystem ＋ T reservoir
subsystem
(note that the energy of the subsystem fluctuates)
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.
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A Simple analogy
thrown
Sum 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 ?
Sum is12 corresponds to
given total energy
2/25
3/25
Reservoir
subsystem
The numbers shown
correspond to the
energy of the system
4/25 Each state does
not have the same
5/25 probability.
6/25
5/25
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Probability of subsystem
The probability f j of the subsystem in state j:
fj 

 res Etot  E j
 tot  Etot 
 fj 


; E j : the energy in state j

exp k B1 S res Etot  E j


exp k Stot  Etot 
1
B

;
 from S  k B ln  
If U is the average value of the subsystem
 Stot  Etot   S  U   S res  Etot  U 


Expanding S res Etot  E j around the equilibrium point Etot  U :




 S res Etot  E j  S res Etot  U  U  E j  S res  Etot  U  
 f j  e
U  Ej
T
1 k BT U TS ( U )   1 k RT  E j
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f j  e
1 k BT U TS ( U )   1 k RT  E j
e
1
k BT
define  

f j  e F e
 Ej
e  F is state-independent normalization factor
  f j  e  F e
j
 Ej
j
define Z   e
 Ej
 e F  e
 Ej
1
j
 e F  Z
j
Fundamental relation in the canonical formalism:
  F  ln  e
 Ej
 ln Z
j
The probability of occupation of the j -th state:
fj  e
 Ej
e
  Ei
i
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The average energy:
U   E j f j   E je
j
j
 Ej
e
  Ei
i
or
 d
U  
 d

 ln Z

;
 F 
U  F T 
&   F  ln Z

 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 The partition function
Consider a system composed of Ñ “distinguishable elements,” an element
being an “independent (noninteracting) excitation mode” of the system.
System
Element
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:
   ε  ε  ε  
ε i j : the energy of the i -th element
Z   e 1 j 2 j 3 j
j , j  , j  , ...
in its j -th orbitial state
ε
ε
ε
  e 1 j e 2 j e 3 j   
j , j  , j  , ...
 e
  ε1 j
  ε 2 j
j
j
zi   e
e
  εi
j
;
e
  ε 3 j
j 
   z1z 2 z 3   
z i : the "partition sum of the i -th element"
j
The Helmholtz potential :
  F  ln Z  ln z1  ln z 2    
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:

Z z  e e
N
0
ε
  F  ln  e

 Ej
N

 1 e
j

 F   Nk BT ln 1  e


N
Total atoms Ñ
1
& 
k BT
 ln Z
ε
ε
N
Ground state
Excited state
(with energy 0)
(with energy ε)
If the number of orbitals had been three rather than two, the partition sum
per particle z would merely have contained three terms.
z  e
ε j
 e   ε1  e   ε2  e   ε3
j
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Molecular Rheology LAB
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Applied to the Einstein model:
Einstein introduced the energy of harmonic oscillators discretely:
1



 h / 2
;
 = 0  n  
2
n  0, 1, 2, ...
;
h: Planck's constant
The "elements" are the vibrational modes.
ze
e
1
  0
2
1
  0
2
e
3
  0
2
1  e
1
  0
2
e
z
1  e
0
F    1 ln z 3 N
e
  0
5
  0
2
 e 2 
0
e
7
  0
2
 e 3 
0
 
 


1
  xn  1  x  x2      xn    
1  x n 0
3
  0 Nk BT  3 Nk BT ln 1  e   0
2

(1  x  1)

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5-2.3 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).
Consider monatomic classical ideal gas:
Neglect interactions
between gas molecules
T reservoir
No. of atoms
Ñ(＝NNA)
The energy is the sum of one-particle
“kinetic energies,” and the partition
sum factors.
Container
volume V
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The one-particle translational partition sum ztransl:
kp
;
p: generalized momentum
H( x , y , z , p x , p y , pz ): Hamiltonian function
 dq j dp j 
Z   e   1 2 1 2   z transl  z internal
j
h h 



1
   p 2x  p 2y  pz2  2 m
 z transl  3  dxdydz    dp xdp y dpz e
  
h

   p2 2 m
   p2
1
  p 2x 2 m
y
 3  dxdydz  e
dp x  e
dp y  e z



h
H
V
32
V
 3  2 mk BT 
h
(


-
e  ax =
2

a
2m
dpz
)
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Molecular Rheology LAB
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32
V
2

mk
T


B 
h3 
If one identifies Z as z N and thereby
z transl 
calculates F .
Finding F is not extensive!
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!.
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.
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The partition sum for a classical monatomic ideal gas:
 N!
Z= 1
N
z transl
 V  2 mk BT  3 2 
 F   k BT ln Z   Nk BT ln  
  Nk BT

2
h
 
 N 
 U 3 2V 
F
5 3

2
S
 Nk B   ln 3
m   Nk B ln  5 2 
T
2 2

 N



Thermodynamic context for monatomic ideal gases:
 U  3 2  V   N  5 2 
S  Ns0  NR ln 
  
 
 U 0   V0   N 0  
The constant s0, undetermined in classic thermodynamic context, has now
been evaluated in terms of fundamental constants.
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Orbital model
Reflection on the problem of counting states reveals that division by Ñ! is a
rather crude classical attempt to account for indistinguishability.
Classical counting
Fermi particles
Bose particles
1
2
1
2
1
2
2
1
“Corrected” number
of states = 4．1/2!
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Molecular Rheology LAB
Classical counting
Fermi particles
5-2
Bose particles
1
2
1
2
1
2
2
1
“Corrected” number
of states = 4．1/2!
“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”.
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The Equipartition Theorem
Suppose the energy (Hamiltonian) to be of the form
E  Aq  Bp
2
2
Then, the classical partition function yields
z~

dq dp
  ( Aq  Bp )
2
e
2
  k BT 
~
1/ 2
  k BT 
 
1/ 2

h
 hA   hB 
The significance is that, at sufficiently high temperature when the classical density of
1
states is applicable, every quadratic term in the energy contributes a term Nk
B
2
to the heat capacity. This is the "equipartition theorem" of classical statistical mechanics.
Exp. A heteronuclear diatomic molecule has three translational modes,
one vibrational mode (with both kinetic and potential energy), two 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
3
2
2
7
7
k B  k B  k B  k B ( or R per mole).
2
2
2
2
2
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5-3 Generalized Canonical Formulations
5-3.1 Entropy as a measure of disorder
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 observing 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:
 1 
I f j  log b     log b f j
 fj 
 
b: the positive constant b was used in different units
e.g.
log 2 units are bits (from "binary")
 
 
log 3 units are trits (from "trinary")
log e units are nats (from "natural logarithm")
log10 units are Hartleys
Define the disorder (entropy) of the porbability distribution
n
H 
j 1
f ,f ,
1
2

, f j is
 1 
f j log b  
 fj 
 
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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 S and it's possibility distribution:
S   s1 , s2 , s3 , s4  ; possibility distribution  f1 , f 2 , f 3 , f 4 
Situation I:
f1  f 2  f 3  f 4  1 / 4
1
1
1
1
log 2 4  log 2 4  log 2 4  log 2 4  2 bits
4
4
4
4
Situation II:
H P 
Have max.
disorder
f1  1 / 2, f 2  1 / 4, f 3  f 4  1 / 8
1
1
1
1
log 2 2  log 2 4  log 2 8  log 2 8  1.75 bits
2
4
8
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H P 
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At max.
f  1- f  1/ 2
H
1
1
log 2 2  log 2 2  1 bits
2
2
Entropy in the case of two possibilities
with probabilities f and (1－f).
C. E. Shannon, J. Bell Syst. Tech. 27, 379-423 (1948).
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.
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From the definition of disorder (entropy):
n
H 
j 1
n
 1 
f j log b     k  f j ln f j
 fj 
j 1
 
If the system is isolated, it spends equal time in each of the permissible states.
1
fj 
; j  1, 2,

1 1
1

H   k  f j ln f j   k  ln    k    ln    k ln 



j 1
j 1 
 H  k ln 
Microcanonical formalism: S  k B ln 
 k  kB
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
fj 
e
 Ej
Z
; j  1, 2,

H   k B  f j ln f j   k B  f j ln e
j 1
j 1
 H  k B   f j E j  k B ln Z
j 1
 Ej


Z  k B  f j  E j  ln Z
j 1

S  k B   f j E j  k B ln Z
j 1
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.
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5-3.2 The grand canonical formalism
Closed system
Closed system ＝ Subsystem ＋ T reservoir
T & μ reservoir
Consider a system of fixed V in contact with both energy and
particle reservoirs, and given energy Ej and mole number Nj.
subsystem
Probability of a state of the system:
fj 

 res Etot  E j , N tot  N j
 fj 
 tot  Etot , N tot 



exp k B1 S res Etot  E j , N tot  N j


Expanding S res Etot  E j , N tot  N j
 fj  e

 U TS   N    E j   N j
Molecular Rheology LAB
e
;

 & S E
exp k Stot  Etot , N tot 
1
B

tot
tot
 from S  kB ln  
, N tot 

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5-3
fj  e

 U TS   N    E j   N j
e

defined  =U  TS   N  U T ,   ;  is the "grand canonical potential"
 f j  e  e

 E j N j

e  is state-independent normalizing factor
  E j N j 
  E j N j 
  f j   e  e
 e   e
1
j
j
j
define Z   e

 E j N j

;
Z : the grand canonical partition sum
j
 e    Z
     ln Z
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
F  U T 
Useful relationships :
Multiplied
1
kB
   
 lnZ 
U
 





 
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Molecular Rheology LAB
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To corroborate the grand canonical formalism from maximizing the disorder
(entropy):
S   k B  f j ln f j
j 1
Auxiliary conditions:
f
j 1
j
1
j 1
f
j 1
j
 S   kB   ln f j  1  f j  0
j 1
Auxiliary conditions:
1   f j  0
j 1
fE
j
Taking differentials
j
E
Nj  N
2  E j f j  0
Taking differentials &
multiplying by Lagrange 3
multiplier λ1, λ2, λ3.
j 1
N  f
j 1
j
0
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Molecular Rheology LAB
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 S   kB   ln f j  1  f j  0
j 1
Adding 4
equations
Auxiliary conditions:
1   f j  0
j 1
  ln f
j 1
j

 1  1  2 E j  3 N j  0
2  E j f j  0
If this equation is always to hold true,
3  N j f  0
each term must be equal to zero.
 ln f j  1  1  2 E j  3 N j  0
j 1
j 1
 fj  e
fj  e

 1 1  2 E j  3 N j
f j  e  e

 1 1  2 E j  3 N j


1  1   

  E j  N j

Compare the two
equations to identify λ1, λ2,
& λ3.
2  
 3  
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Molecular Rheology LAB
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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  0 


Each site has an energy: singly occupied   1 
 doubly occupied   
2

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.
177
Molecular Rheology LAB
5-3
The surface sites do not interact, the grand partition sum:
Z  zN
z = e

 E j N j

 e0  e
  1   
e
   2  2  
 1 e
  1   
e
   2  2  
j
The mean number of molecules adsorbed per site is
  E j N j 
e
Nj
    E j   N j 
n   fjN j  e e
Nj  
z
j
j
j

0e
   0  0
z

1 e
z

2e
   2  2  
z
 2e  2 
n
z
The mean energy per site is
e
  1   
  1   
e    z
   2
   fjEj  e e

   E j   N j
j
j

Ej
 
 1e    
1 

  2e
z
   2  2  
178
Molecular Rheology LAB
5-3
An alternative route to these two results, and to the general thermodynamics
of the system, is via calculation of the grand canonical potential:
   k BT ln Z .

   k BT ln Z =  k BT ln z N   k BT ln 1  e
  1   
e
   2  2  

The number Nˆ of adsorbed atoms on the N sites:
   

ˆ
N  Nn  

   

;

T is constant 
   
ˆ
U  N  

179
Molecular Rheology LAB
Density of States
" sum "    ...  e
 Ej
j
The quantity in the parenthesis is unity
 " sum "    1 e
 Ej
Z
j
The quantity in the parenthesis is the energy ,and the "sum" divided by Z
" sum " 
 Ej 

  E j e
 Z U
Z
 j

 
Under the circumstance that   E j 1  E j   1 ,the sum can be replaced
by integral.
" sum "
≈


Emin
 ...  e   E
j
D  E  dE
Density of state
180
Molecular Rheology LAB
Example: Problem 16.10-4
Consider a particle of mass m in a cubic container of volume V.
2
2
Show that  E ≈
2mV 2 3
h
k k
2
Planck constant : h  2
Momentum : P 
Just consider the kinetic energy
p2
 Ej 
2m
Substitute
k
P for

Wave factor : k   1 3   n1 , n2 , n3 
V 
2 2
k2

2
2
2


n

n

n
2
3
2m
2mV 2 3 1
2
 E  E j  1  E j
Molecular Rheology LAB

≈

2
2

n1  0,1, 2, ...
n2  0,1, 2, ...
n3  0,1, 2, ...
2mV 2 3
181
Example: Problem 16.10-4
Show that the partition sum can be approximated well by an
integral under certain temperature criterion.
Under the circumstance that   E j 1  E j   1 ,the sum can be replaced
by integral.
 ztransl   e
 Ej
≈
j

  E j 1  E j
 T 



Emin
e
 Ej
D  E  dE
2 2
1

  E 

 1
2 3
kbT 2mV
2
2
2kb mV 2 3
Molecular Rheology LAB
182
5-H
5-H Homework
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
Molecular Rheology LAB
183
Statistical Fluctuations and Solution
Chapter 6 Strategies
Contents
6-1
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)
184
Molecular Rheology LAB
6-1
6-1 Fluctuations
6-1.1 The probability distribution of fluctuations
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.
185
Molecular Rheology LAB
6-1
The statistical mechanical form of the probability distribution for a fluctuating
extensive parameter is familiar.
Case I:
T reservoir
The probability that the system occupies a particular
microstate of energy Eˆ :
subsystem
f  e F  E
ˆ
Case II:
T, P reservoir
subsystem
The probability that the system occupies a particular
microstate of energy Eˆ and volume Vˆ :
f e

 G   Eˆ  PVˆ

186
Molecular Rheology LAB
6-1
Generally, for a system in contact with reservoirs corresponding to the
extensive parameters X 0 , X 1 , , X s , the probability that the system
occupies a particular microstate with parameters Xˆ , Xˆ , , Xˆ :
0
f Xˆ
0,
Xˆ 1 , , Xˆ s
S  F0 ,
F0 ,
e.g.
 k
=e
1
B S
 F0 ,

, Fs  k B1 F0 Xˆ 0  F1 Xˆ 1 , , Fs Xˆ s
1
s

, Fs : the Massueu function
, Fs : the entropic intensive parameters

1 
  S , 
T
T T 
1

F0 
; F1 
T
T
Xˆ 0  Eˆ ; Xˆ 1   Nˆ
F
1
  S 
T
T 
1
F0 
T
Xˆ 0  Eˆ
ˆ
 f  e F  E
 f e

   Eˆ   Nˆ

G
1 P
  S , 
T
T T 
1
P
F0 
; F1 
T
T
Xˆ 0  Eˆ ; Xˆ 1  Vˆ
 f e

 G   Eˆ  PVˆ

187
Molecular Rheology LAB
6-1
6-1.2 Moments and the energy fluctuations
Suppose that the Ê is the only fluctuating variable, all other extensive
parameters being constrained by restrictive walls.
 Eˆ  U  : the deviations
Eˆ  U ;

Eˆ  U


Eˆ  U




2
: the mean-square deviation (the second central moment)
2
Eˆ  U


Eˆ  U  0


  Ej U

2

fj   Ej U
j

2

2
e

 F Ej

j
 

j

Ej U e
j
Molecular Rheology LAB

  F Ej 
Ej U
e

 F Ej



U






 F   U 


188
6-1

Eˆ  U

 E
2







j U e


Eˆ  U

 F Ej
j
 F Ej
j

Ej U e




U

  Ej U fj 
E U 
0
j

2

U
U

  k BT 2 Ncv

  1 k BT 
A very useful expression to estimate
the heat capability of a system in
computer simulations.
The mean square energy fluctuations are proportional to the size of the
system.
U , which measures
The relative root-mean-square dispersion  Ê  U 
the amplitude of the fluctuations relative to the mean energy, is
proportional to N-1/2 .
2 12
For large systems (N→∞) the fluctuation amplitudes become negligible
relative to the mean values, and thermodynamics becomes precise.
189
Molecular Rheology LAB
6-1
Higher-order moments of the energy fluctuations:

Eˆ  U

n1

  E j  U

  F Ej 
e

n
j



 E
j
U
e

 F Ej

 e
j

 F Ej
j



E U



E U
e.g. the third moment:

n
 n e
n

 F Ej


Ej U
j

n
Eˆ  U
n

3
E U 
n 1
 
Ej U



n 1

n
U

U

c 

 Nk B2 T 2  2Tcv  T 2 v 
T 

The higher-order moments of the energy fluctuations can be generated
from the lower-order moments by the recursion relation.
190
Molecular Rheology LAB
6-1
For a system in which both E and V fluctuates:
 
Eˆ
2
 U 
  kB 


(1
/
T
)

 P / T , N1 ,...
 kBT 2 Nc p  kBT 2 PV   k BTP 2V  T

Eˆ Vˆ

2
 V 
  kB 


(1
/
T
)

 P / T , N1 ,...
 kBT 2V   k BTPV  T
 
Vˆ
2
 V 
  kB 


(
P
/
T
)

1/ T , N1 ,...
 kBT V  T
The energy and the volume fluctuations are correlated, as expected.
191
Molecular Rheology LAB
6-2
6-2 Quantum Fluids
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
Fermions
Bosons
Fermions are the quantum
analogues of the material
particles of classical physics.
Bosons are the quantum
analogues of the “waves” of
classical physics.
e.g. electrons, protons, and
neutrons
e.g. photons (quanta of light)
Molecular Rheology LAB
192
6-2
(From Astro-group, department of physics of Hong Kong university)
193
Molecular Rheology LAB
6-2
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:
2
n ,
n  1, 3, 5,
(odd multiples)
Spin quantum number m s :
1
1
(  ),  (  )
2
2
194
Molecular Rheology LAB
6-2
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.
There are therefore six orbital states:
1
1
( n, ms ) ; n  1, 2, 3 ; m s   , 
2
2
T & μ reservoir
subsystem
The Grand canonical partition sum factors:
Z  z1, 1 2 z1, 1 2 z 2, 1 2 z 2, 1 2 z 3, 1 2 z 3, 1 2
z n , ms 
   n   N n 
   n   
   n   
0
e

e

e

1

e

n ,ms
f n , ms
  
   
e n 
e n 
1


     
   n   
z n , ms
1 e
e n 1
195
Molecular Rheology LAB
6-2
Pair the two orbital states with the same n but with ms  1:
   
   
2    
 1  e  n    1  2e  n   e  n 


2
z n, 1 2 z n, 1 2
   
2e  n 
1
empty
2 singly occupied
e
2    n   
double occupied
The fundamental equation:
e
 
 Z  1  e

  1   
2
 1  e
 
   2   
2
 1  e
 
   3   


2
196
Molecular Rheology LAB
6-2
N
N


f
n, m
or N 

2
e
 1   
   
U 




 

n, m
n, m
n, m
n, m
U
f
f n, m 
n, m
or
1

U
2
e
  2   

n, m
1

2
  
e  3  1
f n, m
n, m
2 1
e
 1   
1

2 2
e
  2   
1

2 3
  
e  3  1
 

By the same way,  S 
determines the entropy.

T 

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.
197
Molecular Rheology LAB
6-2
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 k , m s :
z k , ms 
  k   Nk 
   k   
   k   
0
e

e

e

1

e

k , ms
 The partition sum factors: Z   z k , ms
k , ms
The probability of occupation: f k , ms
   
e k 
1

     
z k , ms
e k 1
198
Molecular Rheology LAB
6-2
The grand canonical potential:
   k BT  lnZ   k BT  ln 1  e

k
k
  k   


2
The energy of an occuppied orbital state k , m s :
2 2
p2
k
 k ,ms 

, k: wave vector (can refer to eq. 16.43 in textbook)
2m
2m
The density of orbital states: (see Chap. 16)
V 2 dk
V  2m 
D   d 
k
d 
2
2
d
4 2  2 
32
 1 2d 

  
   2k BT  ln 1  e    D    d 
0
V  2m 
  k BT
2 2  2 
32


0
 
ln 1  e 

 
 1 2 d 


   
 
which can be utilized to determine U and N .  U  

 ;N 







 


199
Molecular Rheology LAB
6-2
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:
2
n ,
n  0, 2, 4,
(even multiples)
200
Molecular Rheology LAB
6-2
Ideal Bose fluids
The partition sum of a single orbital state is independent of m s , and is, for
each value of m s ,
  
  2  2 
  3  3 
z k  z k , ms  e 0  e  k   e  k   e  k  


1
n

;

x



  k   
1

x
1 e
n 0


The probability of occupation with various occupated no. i in orbital state k, m s is
1
f k , ms , i
  i  i 
e  k 

z k , ms
The average number of bonsons in the orbital state k , m s :
  
  2  2 
  3  3 
nk , ms   i  f k , ms ,i   e  k   e  k   e  k  

i

ln z k , ms

Molecular Rheology LAB
 k BT
 zk , m
s

201
6-2


 e  k 
  k   
nk , ms  k BT
ln z k , ms   k BT
ln 1  e
  k BT
  


1 e  k 
1
     
e k 1
1
occupation probability (or
 nk , ms  f k , ms       
mean occupation number)
e k 1


   

  
     ln 1  e    D    d 
0
2 V  2m 

3  2  2  2 
32

32

0
e
 k   
d   which determines U and N .
1
The quantity fk , ms is not necessarily less than unity, and therefore is more
properly identified as a “mean occupation number”
.nk , ms
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.
202
Molecular Rheology LAB
6-2
6-2.2 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.
1
The probability of occupancy: f k , ms       
1
e
If fugacity e   1 ;  classical regime 
 The occupation probability reduces to
f k , ms  e  e    e
     
The Boltzmann distribution
 The number of particles N reduces to
g V  2m 

 0 2 2 
N 
  2  

λT 
h
2 mk BT
Molecular Rheology LAB
;
32
e



0
e    1 2 d  
g0V 
e
3
λT
λ T : the thermal wave length
203
6-2
 The energy reduces to
32
     

g0V 
2V  2m 
3

  3 2
U
e  e  d   k BT 3 e
 
2 
2 
0
2
λT

     2  
U 
3
Nk BT
2
N
The equation of state of
classical ideal gas
The criterion that divides the quantum and classical regimes:
e   1
or
 gV 
λ T3  0   1  classical-quantum boundary 
 N 
The classical limit the fugacity is the ratio of the “thermal
volume” λ3T to the volume per particleV/(Ñ/g0).
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).
204
Molecular Rheology LAB
6-2
6-2.3 Example of Fermi fluid: electrons in a metal
At T＝0 K, to calculate the number of particles:
2m 

12
 d 
T  0K
for   0
1
N

f T ,    
 2 3 0
3 2 3
for   0
0
The number of conduction electrons per unit volume in metals is of the order
2m 3 2V
0
32
V
32
0
of the order of 1022 to 1023 electrons cm 3 , the Fermi temperature  TF  is
TF 
0
kB
 104 K to 105 K
Metal
TF (K)
Metal
TF (K)
Li
5.5×104
Cs
1.8×104
Na
3.7×104
Cu
8.2×104
K
2.4×104
Ag
6.4×104
Rb
2.1×104
Au
6.4×104
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Molecular Rheology LAB
6-2
All reasonable temperatures μ>> kBT, the electrons in a metal are an
example of an ideal Fermi gas in the strong quantum regime.
The heat capacity:
  2 k BT
3
C  Nk B 
2
 3 0

3
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.
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Molecular Rheology LAB
6-2
6-2.4 Bose condensation
The number of particles N e :
g V  2m 
Ne  0 2  2 
 2   
32

1 2

0
1 
 e 1
d
Fugacity   e  , and 0    1
Expand the integral of equation of N e in powers of the fugacity:
 g V  2m  3 2  
g0V
32
0


Ne 
 k BT   λ 3 F3 2   
2 
2 
2
 
  2  
T
h
λT 
; λ T : the thermal wave length
2 mk BT

F3 2     
r 1
r
r
32
 
2
2 2

3
3 3

207
Molecular Rheology LAB
6-2
The energy:
g0V  2m 
U
2 
2 


 2 
32

32

0
1 
 e 1
d
Expand the integral of equation of U in powers of the fugacity:
 2V  2m  3 2  3 
g0V
3
52

U
k
T
F


k
T
F5 2   



B 
52
B
2 
2 
3
2
λT
  4
  2  

F5 2     
r
52
r 1 r
Ne 
 
2
4 2

3
9 3

g0V
g0V
Ne
F





λ T3 3 2
λ T3
F3 2   
F5 2   
3
 U  N e k BT
2
F3 2   
The ratio F5 2   F3 2   measures the deviation from the classical
equation of state.
208
Molecular Rheology LAB
6-2
At   1
F5 2  1.34
At   0
slope of F5 2  1
F3 2  2.612
slope of F3 2  1
The two functions F5 2    and F3 2    satisfy the relation:
d
1
F5 2     F3 2   
d

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Molecular Rheology LAB
6-2
If
g0V
 2.612, to first evaluate the temperature at which the failure of the
3
N e T
"integral analysis" occurs.
gV
 Set 0 3  2.612
N e T
λT 
h
2 mk BT
2 2  1
N 
 k BTc 


m  2.612 g0V 
23
Tc : the Bose condensation temperature
Example of Bose condensation of 4He
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.
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Molecular Rheology LAB
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211
Molecular Rheology LAB
6-2
ρs(T): superfluid density of 4He
ρn(T): normal fluid density of 4He
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.
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Molecular Rheology LAB