N96770
微奈米統計力學
上課地點 : 國立成功大學工程科學系越生講堂
(41X01教室)
2002.11.08
N96770 微奈米統計力學
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OUTLINES

Entropy of Ideal Monatomic Gases

Heat Capacity of Monatomic Solids
References:
D. A. McQuarrie, Statistical Mechanics, Harper & Row Publishers, Inc.,
1976.
C. L. Tien & J. H. Lienhard, Statistical Thermodynamics, Hemisphere
Publishing Corp., 1979.
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Ideal Monatomic Gas
Problem 1
How to calculate the entropy (S) of 1 mole of
argon (Ar) at 298K and 1 atm?
We need to derive the entropy expression in terms of the
number of particles (N) , volume (V), and temperature (T)
for ideal monatomic gas.
Ideal Monatomic Gas
a gas of single-atom
molecules dilute enough that
intermolecular interactions
can be neglected.
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Ideal Monatomic Gas
The Hamiltonian H of a
monatomic gas can be
divided into translational
and internal.
The partition function Z depends
upon the types of energy storage
and can be written as
translational and internal.
H  H trans  H int
Z (V , T )  ztranszint
Partition Function (配分函數)
a function expressing the partition or distribution of energies over
the various energy levels of a system.
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Ideal Monatomic Gas
Translational Partition Function
 2mkBT 
ztrans(V , T )  V 

2
 h

It can be shown
3/ 2
Start from solving the wave equation of a particle m translating
freely in a 1-D motion between the interval 0 and L.
 2 d 2

  x
2
2m dx
h2
2
x 
n
x
8mL2
Expand to 3-D and use Maxwell-Boltzmann Distribution,
then the translational partition function can be obtained.
Maxwell-Boltzmann Distribution
N i e i /( kBT )
where

N
z
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
z   e i /( kBT )
partition function
i 0
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Ideal Monatomic Gas
Internal Partition Function
The only internal mode of
energy storage for a singleatom molecule is electronic.

zelec   gi e i /( kBT )
i 0
gi : degeneracy
It can be shown that the ground energy
level 0 is 0 and its degeneracy g0 is 1,
and higher-order terms vanish at
ordinary temperatures.
zelec  1
Degeneracy
a number of eigenfunctions or states of the system having the
same eigenvalue or energy.
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Ideal Monatomic Gas
Combine translational and internal partition functions :
 2mkBT 
z (V , T )  ztranszint  V 

2
 h

The partition function of the
entire system can be written
in terms of the individual
atomic partition function.
Recall the Helmhotz free energy A :
3/ 2 
 i /( k BT )
g
e
 i
i 0
N

z (V , T )
Z ( N ,V , T ) 
N!
A  k BT ln Z
Ve  2mkBT 3 / 2 

 i /( k BT )
A   NkBT ln  
  gi e

2
N h
 i 0


Using Stirling’s
approximation: ln N!  N ln N  N
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Entropy :
Ideal Monatomic Gas
 A 
S   
 T V ,N
  2mkBT  3 / 2 V  5

S   NkB ln 

   Selec 
2
 N 2
  h



where
Selec  ln  gi e i /( kBT ) 
 i /( k BT )
g

/(
k
T
)
e
 ii B
i 0
i 0

 i /( k BT )
g
e
 i
i 0
h  Plank constant  6.626 1034 J - sec
k B  Boltzmann constant  1.3806 10 23 J - K -1
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Ideal Monatomic Gas
At practical temperatures,
Selec  0
  2mkBT 3 / 2 V  5 
S   NkB ln 

 
2
 N  2 
  h
Sackur-Tetrode Equation
Ideal gas :
pV  Nk BT
  2mkBT  3 / 2 k BT  5 
S   NkB ln 

 
2
p  2 

  h
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Ideal Monatomic Gas
How to calculate the entropy (S) of 1 mole of argon (Ar) at
298K and 1 atm?
  2mkBT  3 / 2 k BT  5 
S   NkB ln 

 
2
p  2 

  h
N  6.02 1023 molecule
M Ar  39.95 g/mole
p  1 atm  1.013 105 N - m-2
T  298 K
M Ar
39.95 g/mole
23
mAr 


6
.
636

10
g/molecule
23
N 0 6.02  10 molecule/m ole
SAr  154.68 J
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mole - K
(experimental = 154.72)
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Monatomic Solids
Problem 2
How to determine the heat capacity (Cv) of
crystalline solids, such as copper?
— Start from lattice dynamics.
— Need to determine the partition function for monatomic solids.
— Einstein theory and Debye theory.
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Monatomic Solids
A crystalline solid can be represented by a
system of regularly spaced springs and masses,
in which atoms vibrate with small amplitude
about their equilibrium positions.
Minimum total potential energy U for N atoms :
1 N
U  U (0;  )   kij i j
2 i , j 1
U(0;) : total potential energy for atoms at rest
 = V/N : lattice density
 j : displacement at the j-th atom
k ij : force constant between the i-th & j-th atoms
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Monatomic Solids
The complete partition
function is related to
lattice vibration only.
 V  U ( 0; ) kBT
( j)
Z  ,T   e
z

vibr
N


j 1
3N
It can be shown that the vibrational partition function for the
j-th atom is given by :
 h
( j)
z vibr
2k T
e j B

h k T
1 e j B
vj : vibrational frequency of the j-th oscillator
total partition function
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 h
2k T
3N
e j B
 V  U ( 0; ) kBT
Z  ,T   e

h j k BT
1

e
N 
j 1
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Monatomic Solids
Introducing a function g(v) that gives the number
of frequencies between v and v+dv :

U (0;  )  
h 
h k BT

 ln Z 
  ln 1  e
g ( )d

0
k BT
2k B T 


where
 g ( )d  3N
0
if g() is known
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A  k BT ln Z
Helmhotz free energy
 A 
S   
 T V ,N
entropy
 S 
CV  T  
 T V ,N
heat capacity
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Monatomic Solids
Einstein Theory of Solids
• Assuming there are 3N independent quantum oscillators.
• Each oscillator having the same vibrating frequency.
• Applying the Planck theory of quantized oscillators.
g ( )  3N (  E )
vE : vibrational frequency assigned to all 3N oscillators
(Einstein frequency)
2
 h E 
e h E kBT

CV  3NkB 
h E k BT 2
k
T

 B  1  e
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Monatomic Solids
Define
E 
h E
kB
Einstein temperature
e E T
 E 
CV  3NkB 
 E T
2
T

 e
 1
2
  E  E
lim CV  3NkB 
 e
T 0
 T 
2
T
★ Drawback on the Einstein model :
Heat capacity approaches 0 too quickly as T  0
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Monatomic Solids
Debye Theory of Solids (to improve Einstein model)
• Lattice vibrating at low frequencies at low temperatures.
• Corresponding wavelengths much longer than atomic spacing.
• Assuming crystal as a continuous elastic body.
• Treating lattice vibration as elastic vibration.
• Applying the concept of phonons.
4V 2
g ( ) 
v3
where
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v  
v : velocity of the elastic wave
 : wavelength of the elastic wave
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Monatomic Solids
An elastic wave can be de-coupled into 2 transverse
and 1 longitudinal waves :
 2

1
2

g ( )  

4

V

v 3 v 3
long 
 trans
vtrans : velocity of the transverse wave
vlong : velocity of the longitudinal wave
Define average velocity as
12V 2
g ( ) 
3
v0
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3
2
1


3
3
3
v0
v trans v long
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Monatomic Solids
Define maximum frequency D such that
D
 g ( )d  3N
0
13
 3N 
D  
 v0
 4V 
Debye frequency
9 N 2 ν D 3 (0     D )

g ( )  
0
(   D )

 T 

CV  9 NkB 
 D 
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3

D T
0
x 4e x
e
x
 1
2
dx where
N96770 微奈米統計力學
h D
D 
kB
h
x
k BT
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Monatomic Solids
 T 
12

lim CV 
NkB 
T 0
5
 D 
4
T3
law :
3
meaning at low temperatures Cv behaves as T3
★ Improvement on the Debye model :
Heat capacity is in good agreement with experiment for
solids at low temperatures.
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