thermo & stat mech - spring 2006 class 17 1 thermodynamics and statistical mechanics entropy
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Thermo & Stat Mech - Spring 2006 Class 17
1
Thermodynamics and Statistical Mechanics
Entropy
Thermo & Stat Mech - Spring 2006 Class 17
2
Thermodynamic Probability
UN
NN
N
N
NNN
Nw
n
jjj
n
jj
i
1
1
321 !
!
!!!
!
Thermo & Stat Mech - Spring 2006 Class 17
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Distribution
N = 4 U = 3 k 1 2 3321w 4 12 4
Thermo & Stat Mech - Spring 2006 Class 17
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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
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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
6
Wave Equation
2 2 0 k k 2
( , , ) ( ) ( ) ( )x y z x y zx y z
022
2
yyy k
dy
d
k k k kx y z2 2 2 2
d
dxkx
x x
2
22 0
022
2
zzz k
dz
d
Thermo & Stat Mech - Spring 2006 Class 17
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Boundary Conditions
2222
2
22
2
22
2
222
zyxzyx nnn
LL
n
L
n
L
nk
k k k kx y z2 2 2 2
Thermo & Stat Mech - Spring 2006 Class 17
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Energy of Particles
2222
32
22222
32
2
313
2222222
2
)(
2
)(
so
22
zyxj
jzyx
zyx
nnnn
mVnnnn
mV
VLLV
nnnLmm
k
Thermo & Stat Mech - Spring 2006 Class 17
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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
LVs
Thermo & Stat Mech - Spring 2006 Class 17
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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,
V k kk 1
8
4
3
1
63 3
Thermo & Stat Mech - Spring 2006 Class 17
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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,
dV k dk k dkk 1
63
22 2
Thermo & Stat Mech - Spring 2006 Class 17
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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.
dNdV
V
k dk
L
Lk dk
Vk dkk
s
2
2 2
2
3
3
22
22
Thermo & Stat Mech - Spring 2006 Class 17
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Density of States
dkkV
dNdkkg 222
)(
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
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Density of States
In terms of energy of a particle:
m
k
2
22 k
m
2
dkm
d2 1
2
dmmV
dg2
122
2)(
22
dmV
dg2/3
22
2
4)(
Thermo & Stat Mech - Spring 2006 Class 17
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Free Electrons
dmV
dg
dmV
dg
2/3
22
2/3
22
2
2)(
2
2
4)(