Transcript Other Partition Fnct..

Other Partition
Functions
Motivation
So far, we considered the statistics of systems of constant
volume and mole numbers (or initial mole numbers in the
case of reacting systems) kept at constant temperature.
By considering this setup, we derived the canonical
partition function, Q, for a several ideal gas situations
(monoatomic, diatomic, non-reacting and reacting
mixtures):
Q  Q T , V , N 
We also established connections to several properties
of macroscopic systems, e.g.:
A T ,V , N   kT ln Q T ,V , N 
2
Motivation
The structure of classical thermodynamics is such
that if one knows:
A T , V , N 
it is possible to derive all thermodynamic properties
of a macroscopic system.
3
Motivation
But classical thermodynamics is also rich in the
possibility of adopting independent variables other
than temperature, volume, and mole numbers. For
example, if one knows the Gibbs energy as function
of temperature, pressure, and mole numbers:
G T , P, N 
it is possible to derive all thermodynamic properties
of a macroscopic system.
4
Motivation
Can we develop the statistical thermodynamics of
systems subject to other specifications?
5
Microcanonical Ensemble for a Pure Fluid
In the microcanonical ensemble, the system is subject
to the following specifications:
• constant number of molecules (in non-reactive
systems) or constant number of initial molecules
(in reactive systems);
• constant volume;
• constant energy.
According to the first postulate of statistical
thermodynamics:
All microstates of the system of volume V that have
the same energy and number of particles are equally
probable.
6
Microcanonical Ensemble for a Pure Fluid
For simplicity, let us consider the case of a pure
substance that does not undergo any chemical reaction.
The system has N molecules, energy E, and volume V.
The number of microstates accessible to the system is its
degeneracy at this energy level:
    N ,V , E 
Each of these microstates is equally probable according
to the first postulate of statistical thermodynamics. The
probability of observing one of them is:
1
p  N ,V , E  
  N ,V , E 
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Microcanonical Ensemble for a Pure Fluid
Therefore:
1
p  N ,V , E   

microstates
microstates   N , V , E 
In Chapter 3, the following expression was derived:
S  k
 p  N ,V , E  ln p  N ,V , E 
j
j
states j
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Microcanonical Ensemble for a Pure Fluid
Applying it to the microcanonical ensemble:
1
1
S  k 
ln
  N ,V , E j 
states j   N , V , E j 
But all the states have the same energy and the sum
of the probabilities equals 1:
1
1
S  k ln
 k ln   N ,V , E 

  N ,V , E  states j   N ,V , E j 
1
9
Microcanonical
Ensemble
for a Pure Fluid
S  k ln 
10
Microcanonical Ensemble for a Pure Fluid
Let us now recall some results of classical
thermodynamics:
 S 
 S 
 S 
dS  
 dU  
 dV  
 dN 
 U V , N
 V U , N
 N U ,V
1
P

dU 
dV 
dN
T
T
T
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Microcanonical Ensemble for a Pure Fluid
Therefore:
  ln  U , V , N  
1  S 


 k
T  U V , N
U

V , N
  ln  U , V , N  
P  S 


 k
T  V U , N
V

U , N
  ln  U ,V , N  
 S 
 

 dN  k 
T
N
 N U ,V

U ,V

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Grand canonical Ensemble for a Pure Fluid
In the grand canonical ensemble, the system is
subject to the following specifications:
• constant chemical potential (by being in contact
with an infinite reservoir of constant chemical
potential of the species of interest);
• constant volume;
• constant temperature.
This ensemble is often used in studies of adsorption,
assuming the chemical potential in the bulk (nonadsorbed) phase is constant.
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Grand canonical Ensemble for a Pure Fluid
The grand canonical partition function is:
 N
Ei  N ,V  



kT
 V , T ,     e kT
e



N 
energy states i
for N molecules


 NkT

N  e Q  N ,V , T  


This ensemble is often used in studies of adsorption,
assuming the chemical potential in the bulk (nonadsorbed) phase is constant
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Grand canonical Ensemble for a Pure Fluid
The average number of molecules in the system is:
 NkT

 Ne Q  N ,V , T  

N 

N
 V , T ,  
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Grand canonical Ensemble for a Pure Fluid
Also:
  


  T ,V

 N
Ei  N ,V   

   e kT

kT
e

 

 
N 
energy states i


for N molecules













T ,V
 N
Ei  N ,V  

1
 Ne kT

kT
e



kT N 
energy states i
for N molecules


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Grand canonical Ensemble for a Pure Fluid
Also:
  ln  
N  kT 




T ,V
 N
Ei  N ,V  

kT   
kT 1  kT

kT
N

Ne
e






   T ,V kT  N 
energy states i
for N molecules


 N
Ei  N ,V  

 e kT

kT
e



energy states i
for N molecules

  Np T ,V ,   N
N


N

 N
Ei  N ,V  
N

 e kT

kT
e
N  energystates i


for N molecules


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Grand canonical Ensemble for a Pure Fluid
The average energy U in the system is:
 N
Ei  N ,V  

 e kT

kT
E
N
,
V
e



i


N 
energy states i
for N molecules


U E
 V , T ,  
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Grand canonical Ensemble for a Pure Fluid
Also:
1   
  ln  

  

 T V ,    T V , 

 N
Ei  N ,V   

   e kT

kT
e

 

 
N 
energy states i

1
for N molecules


 



T






V , 

Ei  N ,V  
N

  Ne kT
e kT 



energy states i
1
1
for N molecules


 2
 2
kT N

kT

N
kT
2


E
1

U  N
2
2
kT
kT
 N
Ei  N ,V  

 e kT
Ei e kT 



energy states i
for N molecules


N


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Grand canonical Ensemble for a Pure Fluid
Also:
  ln  
U   N  kT 


T

V , 


2
  ln  
  ln  
2   ln  
U  kT 
   N  kT 
   kT 

 T V , 
 T V ,
  T ,V
2
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Grand canonical Ensemble for a Pure Fluid
The total derivative of the natural logarithm of the
grand canonical partition function is:
  ln  
  ln  
  ln  
d ln  T ,V ,    
dT

dV





 d
 T V , 
 V T , 
 T T ,V
Using the results of the previous slides, we have that:
U  N
N
  ln  
d ln  T ,V ,   
dT  
d
 dV 
2
kT
kT
 V T , 
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Grand canonical Ensemble for a Pure Fluid
U  N
N
  ln  
d ln  T ,V ,   
dT  
d
 dV 
2
kT
kT
 V T , 
We also have that:
 N 
N

N
d
dN 
d
   2 dT 
kT
kT
kT
 kT 
Combining these two equations:
 N  
U
  ln  
d ln  T ,V ,    2 dT  
dN

 dV  d 
kT
 V T , 
 kT  kT
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Grand canonical Ensemble for a Pure Fluid
 N  
U
  ln  
d ln  T ,V ,    2 dT  
dN

 dV  d 
kT
 V T , 
 kT  kT
Note that:
U
d
 kT
U
1
U
1

U 
dU  2 dT 
dU  d 
   2 dT 

kT
kT
kT
kT
kT



Combining these two equations:
 N  
1
 U    ln  
d ln  T ,V ,   
dU  d    
dN

 dV  d 
kT
 kT   V T ,
 kT  kT
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Grand canonical Ensemble for a Pure Fluid

U
dU  kTd  ln  T , V ,    

 kT

  N 
  ln  
   kT 

 dV   d N
  kT  
 V T , 
Compare now with the expression for dU for a pure
substance from classical thermodynamics:
dU  TdS  PdV   dN
They can only be equal if the coefficients of the
differential forms are equal:
  ln  
P  kT 


V

T , 

U
S  k  ln  T ,V ,    

 kT

  N 
 

  kT  
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Grand canonical Ensemble for a Pure Fluid

U
S  k  ln  T ,V ,    

 kT

  N 
 

  kT  
But a few slides before, we showed that:
  ln  
  ln  
U  kT 
   kT 

 T V ,
  T ,V
2
Then:

 N 
  ln  
  ln  
S  k  ln  T ,V ,    T 









T


kT




V
,



T
,
V


25
Grand canonical Ensemble for a Pure Fluid

 N 
  ln  
  ln  
S  k  ln  T ,V ,    T 









T


kT




V
,



T
,
V


But a few slides before, we showed that:
  ln  
N  kT 

  T ,V
Then:
  ln  
S  k ln  T ,V ,    kT 


T

V , 
26
Grand canonical Ensemble for a Pure Fluid
Let us now use another expression that comes from
classical thermodynamics: the Euler relation for a
pure substance:
U PV N 
U  TS  PV   N  S  

T
T
T
In the previous slides, we obtained expressions for
the several properties that appear in the Euler
relation. The next step is to replace them in Euler’s
relation.
27
Grand canonical Ensemble for a Pure Fluid
U PV N 
S 

T
T
T
  ln  
k ln  T ,V ,    kT 
 
 T V , 
  ln  
  ln  
  ln  
  ln  
kT 
  k 
  k 
  kV 


T



V



V , 

T , 

T ,V

T ,V
Then:
  ln  
ln  T ,V ,    V 


V

T , 
28
Grand canonical Ensemble for a Pure Fluid
Using that:
  ln  
ln  T ,V ,    V 


V

T , 
The expression for the pressure then becomes:
kT
P
ln  T , V ,  
V
PV  kT ln  T ,V ,  
There is a simple connection between the product PV
and the partition function of the grand canonical
29
ensemble.
Grand canonical Ensemble for a Pure Fluid
It is also possible to show that for a system of fixed
volume V (refer to the textbook for details):
S  k

p  E ,V , N  ln p  E , V , N 
states of energy E
number of molecules N
in volume V
30
Isothermal-Isobaric Ensemble for a Pure Fluid
In the isothermal-isobaric ensemble, the system is
subject to the following specifications:
• constant mole numbers;
• constant pressure;
• constant temperature.
This ensemble involves specifications very common in
chemical engineering design problems
31
Isothermal-Isobaric Ensemble for a Pure Fluid
The isothermal-isobaric partition function is:

Ei  N ,V 
PV


 T , P, N      e kT e kT

volume V  energy states i
 for N molecules
PV



kT
 Q  N ,V , T  e 

volume V 





fixed number of molecules in a container with flexible walls that allow
heat transfer from an infinite bath of fixed T and P
32
Isothermal-Isobaric Ensemble for a Pure Fluid
There is a direct connection between the Gibbs
energy and the partition function in the isothermalisobaric ensemble:
G T , P, N   kT ln  T , P, N 
33
Restricted Grand or Semi-Grand Canonical
Ensemble for a Pure Fluid
In this ensemble, the system is subject to the
following specifications:
rigid, thermally
conductive
walls, permeable
only to species 1
• constant mole numbers of all substances
numbered from 2 and upwards;
•constant chemical potential of substance 1
• constant volume;
• constant temperature.
1, N2, N3, N4…,
T, P, 1
This ensemble involves specifications common in
osmotic equilibrium problems
34
Restricted Grand or Semi-Grand Canonical
Ensemble for a Pure Fluid
The semi-grand canonical partition function is:

Ei  N1 , N 2 ,...,V  N11 


kT
kT 
 T ,V , 1 , N 2 , N3 ,...  
e
e




N1  energy states i
 for N molecules

N11


kT
 Q  N ,V , T  e 

N1 

35
Restricted Grand or Semi-Grand Canonical
Ensemble for a Pure Fluid
It can be shown that:
A  1 N1  kT ln  T ,V , 1 , N2 , N3 ,...
36