Physical Chemistry III
01403343
Statistical Mechanics
Piti Treesukol
Chemistry Department
Faculty of Liberal Arts and Science
Kasetsart University : Kamphaeng Saen Campus
1
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2
Introduction
 Macroscopic
picture
 Bulk material
 Thermodynamic & Kinetic
properties
 Microscopic
picture
 Atom, Molecule, Ion
 Position, Energy, Momentum
 Link between micro- and macro
pictures
 Statistical method
3
ประกาศ
 สอบกลางภาค
22 มีนาคม 2557
13:00-16:00 น.
 สอบปลายภาค 22 พฤษภาคม 2557
13:00-16:00 น.
4
Properties
 Mass
 Temperature
 Pressure
 Energy
 Conductivity
 Thermodynamic
properties
 Heat capacity
 Gibbs free energy
 Enthalpy
 Etc.
5
nsive and Intensive prope
Xtotal
X1
 Extensive
Accumulative
Properties
X1
X2
Xtotal
X2
n
X total   X i
i
n
Average
 Intensive
Properties
X total 
m X
i
i
i
n
m
i
i
6
Expectation
values/Measurables
 Internal
Propeties
Temperature
 T = < Ti >
 Ti (t)
 External
Properties
Total Energy
 E = S Ei
 Ei (t)
7
System & Enviroment
Environment T, P, m
Mass
System
Ener
gy
n, N, T, P, V, m,
etc.
8
Energy of a System
 Energy
of a macroscopic
system
depends on …
 Energy of a microscopic
system
depends on …
 A Emacroscopic
system

E

total
i
comprises
of countless
i
 i   n ,l ,m ,m xi ,systems
yi , zi ,   Ei(x10
 23i )H  i
microscopic
l
s
Etotal  E { i }  E T , V , P 
9
E1, T1
Etotal   Ei
i
E2, T2
T1 < T2 then
E1 < E2
Ttotal
1
  Ti
n i
Ttotal   p jT j
j
10
State of a System
 Macroscopic system!!!
 System composes of ???
 State of the system is defined
by a few number of
macroscopic parameters
 Systems with the same state may
be different from each others
 Properties of the system are
either
Acculative property
Average property
or
11
Macroscopic description
 can
be derived statisticaly from
microscopic descriptions of a
collection of microscopic
systems
 Description on average*
 Fluctuation of microscopic
properties
 Microscopic properties depends
on a set of parameters of each
microscopic system
 Macroscopic properties depend
on a small set of macroscopic
12
stribution of Molecular Sta
 Molecules
= Workers of a
department
 Energy level = Salary of each
Population of each level :
position
Configuration = {3,2,0,2,1}
100,000
Total Energy / Expense = ?
50,000
How many configuration is
possible if the total energy
was fixed?
20,000
15,000
10,000
* Nobody wants high salary
(energy) because it has too
much stress!!!
13
Distribution of Molecular S
A
system composed of N
molecules
 IF Total energy (E) is constant
(Equilibrium)
 Posible energy state for each
molecule (ei)
 Molecules in different states (i)
possess different energy levels
 Total energy E = SEj =S (ei ni)
Ej
is fluctuated due to molecular
collision
14
Examples
 Total
particle (N) = 6
{3,1,2,0,0,0}
Etotal = 3x0 + 1x2 + 2x4 = 10
{4,0,1,1,0,0}
Etotal = 4x0 + 1x4 + 1x6 = 10
{3,0,1,2,0,0}
Etotal = 3x0 + 1x4 + 2x6 = 16
15
Configuration and
 Configuration
 Weights
Weights
Conf. 1
e6
e5
e4
e3
e2
e1
e6
e5
e4
e3
e2
e1
Conf.1
Conf. 2
Conf.3 …
Different
configurations
have different
w.1
w. 2
w.3 …
Number of ways in
achieved a
particular
16
stantaneous
Configurati
 Possible energy level (e , e , e
0
…)
 N molecules
 n0 molecules in e0 state
1
2
n1 molecules in e1 state …
 The instantaneous configuration
is {n0,n1,n2…}
 Constraint: n0+n1N+n
+… = N
2
!
N!
W nto
n2 ... 

 # ways
0 , n1 ,achieve
n0!n1!n2!
 ni !
instantaneous conf. (W)
i
17
Examples
 {2,1,1}
4!
24
W 2,1,1 

 12
2!1!1! 2
 {1,0,3,5,10,1}
20!
W 1,0,3,5,10,1 
 9.31108
1!0!3!5!10!1!
18
Principle of Equal a prior
 All
possibilities for the
distribution of energy are
equally probable
 The populations of states
depend on a single parameter,
the temperature.
{0,3,0,0}
{1,1,1,0}
{2,0,0,1}
 If at temperature
T, the total
3
3
3
1
1
1
0
0
0
2
energy is 3
2
 Energy
levels: 0, 1, 2, 32
W=1
W=6
W=3
19
Possible configurations for 5 molecules
State 1 5
State 2
State 3
State 4
State 5
State 6
4
4
1
3
3
2
1
1
1
3
3
2
2
3
1
3
1
1
2
1
1
1
1
1
2
1
1
1
1
1
N
E
5
5
5
6
5
7
5
7
1
1
5
5 5 5 5 5 5 5 5 5
8 12 12 8 11 11 20 17 30
W
1
5
5
10
20
20
20
10
10
60 120 60
1
Energy of state j = j
20
e Dominating Configurat
 Some
specific configuration
have much greater weights
than others
 There is a configuration with so
great a weight that it
overwhelms all the rest
 W is a function of all ni: W(n0, n1,
n2 …)
n  N
 The dominating configuration
i nie i  E
has the values of ni that lead to a
maximum value of W
i
i
21
aximum & Minimum Poi
F
is a function of x : F(x)
Maximum point:
F ’= 0 ; F ’’ < 0
2
1
Minimum point:
F ’= 0 ; F ’’ > 0
3
F(x)
4
5
6
9
7
8
x
22
aximum & Minimum in 3
F(x,y)
23
 Configuration
is defined by a
set of ni, {ni}
 W depends on a set of ni or {ni}
 At a specific condition, several
configurations may be possible
 The configuration with greatest
weight (W) will dominate and that
configuration can be used to
represent the system
Greatest weight
Weight
Configuration
 Other configurations= Dominating
with less
weight
is negligible
Configuration
24
ominating Configuratio
 Weight
of each configuration
 2 energy states
 Possible configurations (6
particles) :
{0,6}, {1,5}, {2,4}, {3,3}, {4,2}, {5,1},
{6,0}
25
ominating Configuratio
 Weight
of each configuration
 3 energy states
10 particles
20 particles
30 particles
 Possible configurations (10
particles) :
{0,0,10}, {0,1,9}, {0,2,8}, … {1,0,9},
{2,0,8}, … {1,1,8} …
26
Maximum Value of W{ni}
 We
are looking for the best set
of ni that yields maximum value
of ln(W)
 Maximum W = W{ni,max}
 Maximum ln W = ln W{ni,max}
{ni,max} = ?
27
Maximum Value of W{ni
 {ni,max}
can be determined by
differentiate
 ln W 
dn  0
d ln W   
n
i


i
i
 Constraints
 Total
particle
dN   dn(N)
 0 is constant
N 
n
i
i
i
i
 Total
E  energy
e n dE   e(E)
dn  0is constant
i i
i
i
i
i
28
Maximum Value of W{ni
 Maximum
ln(W) plus
Constraints
 e dn
i
  ln W 
dni  0
d ln W   
i  ni

i
0
i
 dn
0
i
i
 Method
of undetermined
 ln W 

dn    dn    e dn
d ln W   
multipliers n 
i
i
i
  ln W
  
i  ni
i
i
i
i
i


    e i dni


29
Stirling’s Approximation
 Natural
logarithmic of the
weight
W n , n , n ... 
0
1
2
N!
n0 !n1!n2 !
ln W  ln N !ln n0 ! ln n1! ln n2 !
 ln N ! ln ni !
i
ln x! x ln x  x
If x is large
 Stirling’s Approximation
 The approximation
for

ln W  N ln N  N    n ln
n  nthe
weight  N ln N  n ln n
i

i
i
i
i
i
i
30
ln x! x ln x  x
when x is a large number!
1.67%
31
  ln W
d ln W   
i  ni
Eq. 1 is possible
if (and only if) …
  ln W

 ni


    e i dni  0


Eq. 1

    e i  0

n j ln n j 
  ln W  N ln N 

 


n

n
ni
j
i
i



N ln N   N 
N
 ln N 
 
 ln N  1
ni

n

n
i
 i
N
 1 N 
ln N 

 N 
ni
N

n
i 

j
 n j ln n j 
ni
 n j
  
j  ni
 ln ni  1

  ln n j
 ln n j  n j 

 ni



n
 ln W
 ln ni  1  ln N  1   ln i
ni
N
 ln
n
ni
  e
   e i  0  i  e i
N
N
ni  Ne  e i  N   n j  Ne  e
j
e 
j
1
e
 e j
 e j
ei is relative
energy
j
32
he Boltzmann Distributio
 The
populations in the
configuration of the greatest
weight depend
the energy of
n on
e

N e
the state
 e i
i
 e i
i
***
 The fractionp of
in
n molecules
e

 e i
i
the state i (pi)Nis Z
i
The Molecular
Partition Function
(Z,q,Q)
Z   e  e i
Sum over all states (i)
i
  g je
 e j

Sum over energy level (j)
1
kT
j
degeneracy
Boltzmann constant =
1.38x10-23 J/K
33
Molecular Partition Func
Z   g je
j
 e j
1

kT
 An
interpretation of the
partition functionlim e  0 lim Z  g
 at very low T ( T0)
 e i
T 0

T 0
0
lim e
 1 lim Z  
∞
 at very high T ( T∞)
 0
 The molecular partition function
gives an indication of the
average number of states that
are thermally accessible to a
 e i
T 0
T 0
34
Uniform Energy Levels

Equally spaced non-degenerate energy
levelse0 0 e1 e e2 2e e3 3e …
e
3
 Finite
n
Z   e  e i
number
i
e2
e1
e
e0
 Infinite number

Z   e  e i
Infinite # of
energy levels
Si  1  x  x  
2
xS
e  e0 x 
e xe21 
 ex3e2 

 S 1
 1  e  e S 
e 12 e  e 3e  
S
 e
     e   
 1 e
x e
1

1  e  e
 e 2
 e 3
Finite # of
energy levels
35
 What
are the possible states of
particles at high temperature?
 High-energy states?
 Low-energy states?
 All states?
36
The Possibility *
 The
possibility of molecules in
the state with energy
Z of infinitee
#iof(p
energy
i) levels*
e  e i
pi 
 1  e  e e  e i
Z


 The
possibilities of molecules
in the 2-level system
1
p0 
1  e  e
e  e
p1 
1  e  e




As T   the populations of
all states (pi’s) are equal.
37
 The
possibilities of molecules
in the infinite-level system*


p  1  e e
p  1  e e
p0  1  e  e
 e
 e
 e
  2e
1
2
As T  
the populations of
all states are equal.
38
Temperature
39
Examples
 Vibration
of I2 in the ground, firstand second excited states (Vibrational
wavenumber is 214.6 cm-1)
for v  0,1, 2 and T  298.15 K
kT
 207.226 cm 1
hc
hc
214.6 cm 1
e 

 1.036
1
kT 207.226 cm
Relative
energy
pv  (1  e  e )e  ve
p0  0.645
p1  0.229
p2  0.081
40
roximations and Factoriza
 In
general, exact analytical
expression for partition
functions cannot be obtained.
 Closed approximation
expressions to estimate the
value of the partition functions
are required for neach
systems
h
E 
n  1, 2, 
 Energy levels of
a8mXmolecule
in a
box of length X E  h  e  0
2
2
n
2
2
1
1
8mX 2


h2
e n  n 1 e e 
8mX 2
2
Relative
energy
41
nslational
Partition
Func
Partition function of a molecule in a box of

length X


2
h
2
e n  n 1 e e 
8mX 2

qX   e 
n  1, 2, 

 n 2 1 e
n 1
 The translation energy levels are very close together, therefore the
sum can be approximated by an integral.

Transitional partition
function
qX   e 

 dn  e n e dn

 n 1 e
2
1
2
0
 Make substitution: x2=n2e and dn = dx/(e)1/2

qX 
  e
1 1/ 2
e
0
x
2
dx 
 
1 1/ 2
e


 2
1/ 2
1/ 2
  2m 
   2  X
 h  
42
 When
the energy of a molecule
arises from several different
independent sources
 E = Ex+Ey+Ez
 q = qxqyqz
e
 e in
e 3-d
 e box
 A molecule





(X )
nx
nx ,n y , nz
q    e
 nx
 qx q y qz
 e n( X )
 2m 
q   2 
h  
x
  e
 n
 y
(Y )
ny
 e n( Y )
y
(Z )
nz
  e  e nz 

 n
z



(Z )
3/ 2
XYZ
43
3/ 2
 2m 
q   2  XYZ
h  
V
q 3

  
  h

2

m


1/ 2

h
2mkT 1/ 2

is called the thermal
wavelength
 The partition function
increases with
 The mass of particle (m3/2)
 The volume of the container (V)
 The temperature (T3/2)
44

Example
Calculate the translational partition
function of an H2 molecule in 100 cm3
vessel at 25C


h
2mkT 1/ 2
6.626 10 34 Js
2  2.016 1.6605 10
 27

kg 1.38 10 23 JK 1  298 K

1/ 2
 7.12 10 11 m
 About 1026 quantum states are
thermally
at 26room
V
1.00accessible
104 m3
q 3 
 2.77 10
temperature
3

11

7.12 10
m

45
Internal Energy and Ent
 The
molecular partition function
contains all information needed to
calculate the thermodynamic
properties of a system of
independent particles
 q  Thermal wave function
e
Boltzmann
distribution

n

N

 ne
 TheE Internal
Energy **
q
 e i
i i
i
i
N
E   e i e  e i
q i
e i e e
i
de  e i

d
N
d  e i
N d
E  e 
q i d
q d
N dq
E
q d
 e i
e

i
46
Relative
energy
 Total
E   nie i  
energy
i
N dq
q d
e3
3e
e2
2e
 ei is relative energy (e0=0) e
 E is internal energy relativee
1
e
0
to its value at T=0
 The
(U)
e
0
conventional Internal Energy
U  U (0)  E
U  U (0) 
N  q 


q   V
A system with N
independent molecules
• q=q(T,X,Y,Z,…)
  ln q 

U  U (0)  N 
  V
Only the partition function is required to determine the internal energy
relative to its value at T=0.
***
47
Example
two-level partition function
N dq
 N
E
 
 e
q d
1

e

 d
 e


1

e

 d
Nee  e
Ne
E

 e
1 e
1  e e
0 .5
 At T = 0 : E  0
0 .4
all are in lower state (e=0)
0 .3
E /Ne
 The
 As T   : E  ½ Ne
0 .2
two levels become equally
populated
0 .1
0
0 .0 0
0 .5 0
1 .0 0
1 .5 0
2 .0 0
2 .5 0
3 .0 0
3 .5 0
4 .0 0
kT/e
48
The value of 
 The
internal energy of
N  q 
monatomic
U  U (0)    ideal
 U (0)  nRTgas
q   
3
2
V
V
q 3

 q  the
  Vtranslational

 1
V d partition
 For

  
  V
 3
  V
3
   V
function
 3
4 d
d
d  h 1/ 2 
1
h







d d  2m1/ 2  2 1/ 2 2m1/ 2 2
 q

 

3V
  
2  3
V
 3   3V 
3N

U  U (0)  N  

U
(
0
)

3
2
 V  2 V
3nRT 3N

2
2
N
nN A 1



nRT nRT kT
R
k
NA
This result is also true
for general cases.
49
1 amu = ? g
 12C
 12C
 12C
1 mol = 12 g
1 atom = 12 amu
1 mol = 6.02x1023 atom
amu = 1g/6.02x1023
=1.66x10-27 kg
1
50
mperature and Populatio
 When
a system is heated,

 The energy levels are
 The populations are changed

2
h
(X )
2
e

n
1
n
unchanged
8mX 2
HEAT
e10
e9
e8
e7
e6
e5
e4
e3
e2
e1
e0
0
Increase T
0.2
0.4
0.6
0.8
e10
e9
e8
e7
e6
e5
e4
e3
e2
e1
e0
0
0.2
0.4
0.6
0.8
51
Volume and Populations
Translational energy levels
2
h
X)
When work is done on ae n(system,
 n 2  1
2
8
mX
 The energy levels are changed
 The populations are changed


WORK
e10
e9
e8
e7
e6
e5
e4
e3
e2
e1
e0
0
e5
e4
decrease V e3
e2
e1
e0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
52
The Statistical Entropy
 The
partition function contains
all thermodynamic information.
 Entropy is related to the disposal
of energy
 Partition function is a measure of
the number of thermally
accessible states
S  k ln W
***
 Boltzmann formula for the
entropy
As
T  0, W  1 and S  0
53
Entropy and Weight
A
change in internal energy
U  U (0)   nie i  dU  dU (0)   ni de i   e i dni
i
i
i
 When the system is heated at
constant V, the energy levels do
dU   e dn
not change.
i
i
dU  dq  TdS
  e dn
From
thermodynamics
 ln W ,
dS  k 
dn  k d ln W 
n
i
rev
i
i
i
dS 
dU
 k  e i dni
T
i
 ln W
   e i  0
ni
dS  k 
i
 ln W
dni  k  dni
ni
i
i
i
i
S  k ln W
U  U 0
S
 Nk ln q
T
54
Calculating the Entropy
 Calculate
the entropy of N
independent harmonic oscillators
1
q

for I2 vapor at 25ºC
1 e
 Molecular partition function:
N  q 
Nee
Ne
 e
U  U ( 0) 
 The internal energy:
 e
  
 e
 e
q   V 1  e
e 1
E ntropy
 The
U entropy:
U (0)
S
 Nk ln q
35
30
-1

20
-1

25
S(J K mol )
T
 e

S  Nk  e
 ln 1  e  e 
 e 1

15
10
5
0
0
1000
2000
3000
4000
5000
T(K)
55
Entropy and Temperatur
What do we know from the
graph?
 T increases, S increases
 What else?
E ntropy
35
30
-1
S(J K mol )
25
-1
20
15
10
5
0
0
1000
2000
3000
4000
5000
T(K)
56