Example 10.2
The Boltzmann distribution law
probability distributions from the energy levels for the system under consideration
The Boltzmann distribution law
probability distributions from the energy levels for the system under consideration
Chapter 10. The Boltzmann distribution law
Derivation:
System with:
N particle of a single type
t different energy levels
What to know probability that the system is in each level . are held constant
p
j, 1, 2, 3, ..., .
j
E j = t
j
( , ,T V N)
0 dF = dU βTdS =
1
ln
t
j j
j
S k p p
=
= β
β
t
U = E =
β
p E 1t
p =
The Boltzmann distribution law
probability distributions from the energy levels for the system under consideration
Chapter 10. The Boltzmann distribution law
/ / * / 1 j j j
E kT E kT
j t E kT j
e
e
p
Q
e
β β β ==
=
β
/ 1 j t E kT jQ eβ =
Chapter 10. The Boltzmann distribution law
Application (Example 10.2)
Maxwell-Boltzmann distribution of particle velocities. 2 1 ( ) 2 v mv
Ξ΅
= 2 2 2 1/ 2( )/ / 2
/ 2
( )/ / 2
( )
2
x x
x
x x
v kT mv kT
mv kT
x
v kT mv kT
x x
e e m
p v e
kT
e dv e dv
Ξ΅ Ξ΅
Ο
β β β +β +β β β ββ ββ  ο£Ά = = =  ο£· ο£ ο£Έβ«
β«
2 / axe dx
Ο
a +ββ ββ
=
Chapter 10. The Boltzmann distribution law
Application (Example 10.2)
Maxwell-Boltzmann distribution of particle velocities.
2
1/ 2
/ 2
2 2 2
( )
2
x
mv kT
x x x x x x
m
v v p v dv v e dv
kT
Ο
+β +β β ββ ββ  ο£Ά = =  ο£· ο£ ο£Έβ«
β«
width of the distribution2 x kT v m = 2 1 1
2 m vx = 2 kT
2 2 2 2
x y z
v = v + v + v
2
1 3
2 m v = 2 kT
Velocity components are independent
2
3/ 2
/ 2
( ) ( ) ( ) ( ) m mv kT p v = p v p v p v =  ο£Άο£· eβ
Chapter 10. The Boltzmann distribution law
Chapter 10. The Boltzmann distribution law
Partition function
3
1 2
/ / / / /
1
...
j t
t
E kT E kT E kT E kT E kT
j
Q
e
βe
βe
βe
βe
β=
=
β
=
+
+
+
+
β’ Link between macroscopic thermodynamic properties and microscopic model
β’ Specify how particles are partitioned through the accessible states
3 1 1
2 1 ( )/ ( )/
( )/
1
E E kT E E kT...
Et E kTQ
= +
e
β β+
e
β β+
+
e
β βChapter 10. The Boltzmann distribution law
0 *1
0
j E j j TE
p
Q
t
kT
t
β β βο£Ό
β
β
β
β
β
β
ο£½
ο£Ύ
* 1 1 * 1 1 1 0 01
j j j p E j p TE
Q
kT
= β β β β β β β βο£±
ο£Ό

β
ββ
β
β
β
ο£½
ο£²

ο£Ύ
ο£³
Q: number of states that are effectively accessible for the system
/ 1 j t E kT j
Q
e
β=
Chapter 10. The Boltzmann distribution law
Density of states
( )
W E
Density of states, total number of ways a system can occur in
energy level E
( ) 1
W E > degenerate energy level
max
/ 1
(
)
ll
E kT l
l
Q
W E e
β=
=
β
sum over energy levelsmax / / 1
(
)
(
)
l l E kT l l l E kT l lW E e
p
W E e
β β =
=
Chapter 10. The Boltzmann distribution law
Density of states
( )
W E
Density of states, total number of ways a system can occur in
energy level E
( ) 1
W E > degenerate energy level
max
/ 1
(
)
ll
E kT l
l
Q
W E e
β=
=
β
sum over energy levelsmax / / 1
(
)
(
)
l l E kT l l l E kT l lW E e
p
W E e
β β =
=
Chapter 10. The Boltzmann distribution law
Partition function
N distinguishable particles (atoms in crystals )
N indistinguishable particles (particles in gas )
n
Q
=
q
!
n
q
Q
N
=
Chapter 10. The Boltzmann distribution law
Partition function predicts thermodynamic properties 2 , , , , ,
ln
ln
ln
ln
ln
V N T V T VT N T N
Q
U
kT
T
U
S
k
Q
T
F
U
TS
k
Q
Q
F
Β΅
kT
N
N
F
Q
p
kT
V
V
β

ο£Ά
=

ο£·
β
ο£
ο£Έ
=
+
=
β
= β
β
β

ο£Ά

ο£Ά
=

ο£·
= β

ο£·
β
β
ο£
ο£Έ
ο£
ο£Έ
β
β

ο£Ά

ο£Ά
= β

ο£·
=

ο£·
β
β
ο£
ο£Έ
ο£
ο£Έ
Internal energy EntropyChemical potential Β΅
Helmholtz free energy
Chapter 10. The Boltzmann distribution law
Partition function predicts thermodynamic properties 2 , , , , ,
ln
ln
ln
ln
ln
V N T V T VT N T N
Q
U
kT
T
U
S
k
Q
T
F
U
TS
k
Q
Q
F
Β΅
kT
N
N
F
Q
p
kT
V
V
β

ο£Ά
=

ο£·
β
ο£
ο£Έ
=
+
=
β
= β
β
β

ο£Ά

ο£Ά
=

ο£·
= β

ο£·
β
β
ο£
ο£Έ
ο£
ο£Έ
β
β

ο£Ά

ο£Ά
= β

ο£·
=

ο£·
β
β
ο£
ο£Έ
ο£
ο£Έ
Internal energy EntropyChemical potential Β΅
Helmholtz free energy
Pressure
/
j
t
E kT
Q
e
β=
Chapter 11. Statistical mechanics of simple gases and solids
Chapter 11. Statistical mechanics of simple gases and solids
π»π»ππ = πΈπΈππ
Wave function
ππ = ππ(π₯π₯,π¦π¦,π₯π₯)
ππ2
SchrΓΆdinger equation
Spatial probability distribution
Only certain function ππ can satisfy Eq (11.4) if E is constant.
There are multiple πΈπΈππ, eigenvalues of the equation. Those are energy levels we need.
ππ - quantum number.
π»π» = 2ππππ2 + ππ π₯π₯ ππ2 = ββ4ππ22 πππ₯π₯ππ22
π»π» = 8ββππ2ππ2 ππ2πππ₯π₯ππ(2π₯π₯) + ππ π₯π₯ ππ(π₯π₯)
ββ2
8ππ2ππ
ππ2ππ(π₯π₯)
Chapter 11. Statistical mechanics of simple gases and solids
Model for translation motion (particle in a box)
ππ π₯π₯ = 0 0 < π₯π₯ < πΏπΏ ββ2
8ππ2ππ
ππ2ππ(π₯π₯)
πππ₯π₯2 + ππ π₯π₯ ππ π₯π₯ = πΈπΈππ π₯π₯
ππ2ππ(π₯π₯)
πππ₯π₯2 + πΎπΎππ π₯π₯ = 0 πΎπΎ =
8ππ2ππππ β2
ππ π₯π₯ = π΄π΄sinπΎπΎπ₯π₯ + π΅π΅cosπΎπΎπ₯π₯ ππ2 0 = ππ2 0 = 0
ππ β single particle
Chapter 11. Statistical mechanics of simple gases and solids
Model for translation motion (particle in a box)
ππππ = 8πππΏπΏππβ 22
ππ π₯π₯ = 2πΏπΏ
1 2
sin πππππ₯π₯πΏπΏ
energy levels
wave function
Chapter 11. Statistical mechanics of simple gases and solids
Model for translation motion (particle in a box)
Chapter 11. Statistical mechanics of simple gases and solids
Model for translation motion (particle in a box)
ππ π₯π₯ = 2πΏπΏ
1 2
Chapter 11. Statistical mechanics of simple gases and solids
Model for translation motion (particle in a box)
ππππ = 8πππΏπΏππβ 22
πππ‘π‘π‘π‘π‘π‘πππ‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ππ = οΏ½
ππ=1 ππ=β
ππβππππππππ = οΏ½ ππ=1 ππ=β
ππ βππ
2β2
8πππΏπΏ2ππππ
πππ‘π‘π‘π‘π‘π‘πππ‘π‘ = 8πππΏπΏββ22ππ transition temperature
πππ‘π‘π‘π‘π‘π‘πππ‘π‘
ππ βͺ 1 sum can be approximated as an integral
πππ‘π‘π‘π‘π‘π‘πππ‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ππ = οΏ½
ππ=0 ππ=β
ππ βππ
2β2
8πππΏπΏ2ππππ ππππ = 2ππππππππ
β2
1/2
Chapter 11. Statistical mechanics of simple gases and solids
Model for translation motion (particle in a box)
ββ2
8ππ2ππ
ππ2 πππ₯π₯2 +
ππ2 πππ¦π¦2 +
ππ2
πππ§π§2 ππ π₯π₯,π¦π¦,π§π§ + ππ π₯π₯,π¦π¦,π§π§ ππ π₯π₯,π¦π¦,π§π§ = β°ππ π₯π₯,π¦π¦,π§π§
ππ π₯π₯,π¦π¦,π§π§ = ππ π₯π₯ + ππ π¦π¦ + ππ(π§π§)
π»π»π₯π₯πππ₯π₯ = β°π₯π₯πππ₯π₯ π»π»π¦π¦πππ¦π¦ = β°π¦π¦πππ¦π¦ π»π»π§π§πππ§π§ = β°π§π§πππ§π§ ππ = πππ₯π₯πππ¦π¦πππ§π§
β°πππ₯π₯,πππ¦π¦,πππ§π§ = 8ππβ 2 πππππ₯π₯22 + πππππ¦π¦22 + πππππ§π§22
ππ
π‘π‘π‘π‘π‘π‘πππ‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ππ=
ππ
π₯π₯ππ
π¦π¦ππ
π§π§=
2ππππππππ
β
23/2
ππππππ
=
2ππππππππ
β
23/2
Chapter 11. Statistical mechanics of simple gases and solids
Model for vibrations (Harmonic oscillator)
ππ(π₯π₯) = πππ‘π‘2π₯π₯2 square-law or parabolic potential
ββ2
8ππ2ππ
ππ2ππ(π₯π₯)
πππ₯π₯2 +
πππ‘π‘π₯π₯2
2 ππ π₯π₯ = πππ£π£ππ π₯π₯
πππ£π£ = π£π£ + 12 βππ π£π£ = 0,1,2,3, β¦ β¦
ππ = 21ππ πππππ‘π‘
1/2
Chapter 11. Statistical mechanics of simple gases and solids
Chapter 11. Statistical mechanics of simple gases and solids
Chapter 11. Statistical mechanics of simple gases and solids
Model for vibrations (Harmonic oscillator)
ππ = 2ππ1 πππππ‘π‘
1/2
β ππ = 21ππ πππππ‘π‘
1/2
ππ = ππππ1ππ2
1 + ππ2
πππ£π£π‘π‘π£π£π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ππ = οΏ½
π£π£=1 ππ=β
ππβππβπ£π£ππππ = 1 + ππββπ£π£ππππ + ππβ2βπ£π£ππππ + ππβ3βπ£π£ππππ + β― = 1 + π₯π₯ + π₯π₯2 + β―
0 < π₯π₯ < 1
ππ
π£π£π‘π‘π£π£π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ππ=
1
Chapter 11. Statistical mechanics of simple gases and solids
Model for rotations. Rigid rotor.
πππ‘π‘ = ππ ππ8+ 1ππ2πΌπΌβ2 πΌπΌ = πππ π 2 moment of inertia
For each ππ there is degeneracy 2ππ + 1
πππ‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ππ = οΏ½
π‘π‘=1 π‘π‘=β
(2ππ + 1)ππβππππππππ
πππ‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ππ = 8ππππβ2πΌπΌππππ2
ππ nuclear and rotation symmetry factor
ππ
π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ππ=
πππΌπΌ
π‘π‘πΌπΌ
π£π£πΌπΌ
ππ1 2
ππ
8
ππ
2ππππ
β
23 2
Chapter 11. Statistical mechanics of simple gases and solids
The electronic partition function
πππππ‘π‘πππππ‘π‘π‘π‘π‘π‘πππ‘π‘ππ = πππ‘π‘ + ππ1ππβΞππππππ1 + ππ2ππβΞππππππ2 + β―
ππ1 electronic degeneracies
βΞππππ
ππ β 104-105K
Chapter 11. Statistical mechanics of simple gases and solids
Total partition function
πππ‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ππ = πππΌπΌπ‘π‘πΌπΌπ£π£πΌπΌππ
1 2
ππ
8ππ2ππππ β2
3 2
πππ£π£π‘π‘π£π£π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ππ = 1 1 β ππββππππππ
πππ‘π‘π‘π‘π‘π‘πππ‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ππ = 2ππππππππβ2
3/2
ππ
πππ‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ = πππ‘π‘π‘π‘π‘π‘πππ‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ππ + πππ‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ππ + πππ£π£π‘π‘π£π£π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ππ + πππππ‘π‘πππππ‘π‘π‘π‘π‘π‘πππ‘π‘ππ πππ‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ = πππ‘π‘π‘π‘π‘π‘πππ‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘πππππ£π£π‘π‘π£π£π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘πππππ‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘πππππππ‘π‘πππππ‘π‘π‘π‘π‘π‘πππ‘π‘ππ
πππ‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ππ = 1 linear diatomic molecular
ππ
π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘=
2ππππππππ
β
23/2
ππ
8
ππ
ππβ
2πΌπΌππππ
21
Chapter 11. Statistical mechanics of simple gases and solids
Ideal gas free energy
πΉπΉ = βππππlnππ = βππππln ππππππ! = βππππln ππππππ ππ = βππππππln ππππππ
ππ = ππππππ!
Stirlingβs approximation ππ! β ππππ ππ
ππ = ππ0ππ
πΉπΉ = βππππππlnππ β ππππππln ππππππ0
Ideal gas presure
ππ = β πππΉπΉππππ
ππ,ππ
Chapter 11. Statistical mechanics of simple gases and solids
Ideal gas
ππ = β πππΉπΉππππ
ππ,ππ
= ππππππππ Ideal gas law
ππ = 32ππππππ Ideal gas internal energy πππ‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ππ(πππ‘π‘πππ‘π‘π‘π‘πππππ‘π‘π‘π‘) = 32ππππππ
πππ‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ππ(π‘π‘π‘π‘πππππ‘π‘π‘π‘) = ππππππ
Ideal gas entropy (Sackur-Tetrode equation)
ππ = ππππ ln 2ππππππππβ2
3/2 ππ5/2
ππ ππ βππ = ππππ ln ππ1 ππ2
Chapter 11. Statistical mechanics of simple gases and solids
Ideal gas chemical potential
ππ = ππππ ln ππππ
π‘π‘πππ‘π‘π‘π‘
πππ‘π‘πππ‘π‘π‘π‘ = ππππ 2ππππππππβ2
3/2
Chapter 11. Statistical mechanics of simple gases and solids
The equipartition theorem
Average energies are integral multipliers of ππππ
2
ππ = 32ππππππ πππ‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ππ(πππ‘π‘πππ‘π‘π‘π‘πππππ‘π‘π‘π‘) = 32ππππππ
Ideal gas internal energy
ππ = β«ββ
+βππ(π₯π₯)ππβππππππ(π₯π₯)πππ₯π₯
β«ββ+βππβππππππ(π₯π₯)πππ₯π₯
π₯π₯ degree of freedom
ππ = βπ₯π₯ππ(π₯π₯)ππ
βππ(π₯π₯) ππππ
βπ₯π₯ππβππππππ(π₯π₯)
Chapter 11. Statistical mechanics of simple gases and solids
The equipartition theorem
ππ = β«ββ
+βππ(π₯π₯)ππβππππππ(π₯π₯)πππ₯π₯
β«ββ+βππβππππππ(π₯π₯)πππ₯π₯
Energy stored in every degree of freedom is 1
2ππππ
ππ = β«ββ
+β
πππ₯π₯2ππβπππ₯π₯ππππ2πππ₯π₯
β«ββ+βππβπππ₯π₯ππππ2πππ₯π₯
= 12ππππ
in many cases ππ π₯π₯ = πππ₯π₯2
ππ = ππ π₯π₯2 = 12ππππ π₯π₯2 = ππππ2ππ
πππππ‘π‘π‘π‘π‘π‘πππ‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ππ = 8πππΏπΏβ2 2 ππ2 πππ‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ππ = β2
8ππ2πΌπΌ ππ(ππ + 1) fluctuations
Chapter 11. Statistical mechanics of simple gases and solids
The equipartition theorem. Vibrations
πππ£π£ = π£π£ + 12 βππ linear dependence on quantum number
ππ = β«ββ
+β
πππ₯π₯ππβπππ₯π₯ππππ πππ₯π₯
β«ββ+βππβπππ₯π₯ππππ2πππ₯π₯
= ππππ
At low ππ assumption might not be valid !
Chapter 11. Statistical mechanics of simple gases and solids
Chapter 11. Statistical mechanics of simple gases and solids
The Einstein model of solids
If there are ππ atoms in solid and each vibration should contribute ππππ to the energy.
Chapter 11. Statistical mechanics of simple gases and solids
The Einstein model of solids
Consider 3ππ distinguishable independent oscillators
πππ£π£π‘π‘π£π£π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ππ = 1 1 β ππββππππππ
this obtained from QD consideration!
ππ = β1πππππ½π½ππππ = βπ£π£ ππβπ½π½βππ
1 β ππβπ½π½βππ π½π½ =
1
ππππ
If there are ππ atoms in solid and each vibration should contribute ππππ to the energy.
The heat capacity should be πΆπΆπ£π£ = 3ππππ, independent of ππ
πΆπΆππ = 3ππππ ππππππ = 3ππππ ππππβππ
2 ππββππ/ππππ
Chapter 11. Statistical mechanics of simple gases and solids
The Einstein model of solids
πΆπΆππ = 3ππππ ππππππ = 3ππππ ππππβππ
2 ππββππ/ππππ
