Thermodynamics and Statistical
Mechanics
Entropy
Thermo & Stat Mech Spring 2006 Class 17
1
Thermodynamic Probability
N!
N!
w

N1! N 2 ! N 3!     N i !
n
N
j 1
j
N
n
N 
j 1
j
j
U
Thermo & Stat Mech - Spring 2006
Class 17
2
Distribution
N=4
3
2
1
0
w
k
U = 3
1
1
3
4
2
3
1
1
2
12
3
1
4
Thermo & Stat Mech - Spring 2006
Class 17
3
Combining Systems
Consider two systems.
System A: Number of arrangements: wA
System B: Number of arrangements: wB
Combined systems: wA × wB
Thermo & Stat Mech - Spring 2006
Class 17
4
Entropy
S = k ln w
SA = k ln wA
SB = k ln wB
SA+B = k ln(wA × wB) = k ln wA + k ln wB
SA+B = SA + SB
Thermo & Stat Mech - Spring 2006
Class 17
5
Wave Equation
  k  0
2
2
k
2

 ( x, y, z)   x ( x) y ( y) z ( z)
d x
2
2  kx x  0
dx
d y
d z
2
 k z z  0
2
dz
k  k k k
2
2
2
dy
2
k y 0
2
Thermo & Stat Mech - Spring 2006
Class 17
2
y
2
x
2
y
2
z
6
Boundary Conditions
k  k k k
2
2
x
2
y
2
z
n
n
n
  2
2
2
k  2  2  2    nx  n y  nz
L
L
L
L
2
2
x
2
2
y
2
2
z
2
Thermo & Stat Mech - Spring 2006
Class 17
2


7
Energy of Particles
 k
   2
2
2


  nx  n y  nz
2m 2m  L 
3
13
V  L so L  V
2
2
2
2


(  )
2
2
2
2 (  )

nx  n y  nz  n j
23
23
2mV
2mV
2
2
2
2
n j  nx  n y  nz
2


Thermo & Stat Mech - Spring 2006
Class 17
2
8
Density of States
The allowed values of k can be plotted in k
space, and form a three dimensional cubic
lattice. From this picture, we can see that each
allowed state occupies a volume of k space
equal to,
3
 
Vs   
L
Thermo & Stat Mech - Spring 2006
Class 17
9
Density of States
All the values of k that have the same
magnitude fall on the surface of one octant of a
sphere in k space, since nx, ny, and nz are
positive. The volume of that octant is given by,
14
1
3
3
Vk 
k  k
83
6
Thermo & Stat Mech - Spring 2006
Class 17
10
Density of States
Then, the volume of a shell that extends from k
to k + dk can be obtained by differentiating the
expression for Vk,
1
 2
2
dVk   3k dk  k dk
6
2
Thermo & Stat Mech - Spring 2006
Class 17
11
Density of States
If we divide this expression by the volume
occupied by one state, we will have an
expression for the number of states between
k and k + dk.
 2
k
dk
3
dVk
L
V
2
2
2
dN 

3 
2 k dk 
2 k dk
Vs
2
2

 
 L
Thermo & Stat Mech - Spring 2006
Class 17
12
Density of States
V 2
g (k )dk  dN  
k dk
2
2
 is the number of states with the same k,
or the number of particles that one k can hold.
Thermo & Stat Mech - Spring 2006
Class 17
13
Density of States
In terms of energy of a particle:
2
2
 k

2m
2m
k

V 2m
g ( )d  
2
2
2 
2m 1
dk 
d
 2 
2m 1
d
 2 
V  2m 


g ( )d  
2  2 
4   
3/ 2
Thermo & Stat Mech - Spring 2006
Class 17
 d
14
Free Electrons
V
g ( ) d  
2
4
 2m 
 2 
 
3/ 2
 d
 2
V
g ( ) d 
2
2
 2m 
 2 
 
3/ 2
Thermo & Stat Mech - Spring 2006
Class 17
 d
15