chapter 6 boltzmann statistics. boltzmann factor & partition functions u r, s r u, s huge...
TRANSCRIPT
Chapter 6
Boltzmann Statistics
Boltzmann Factor & Partition Functions
UR, SR
U, S
Huge reservoir
System
Changes in energy of the reservoir are very small compared to its total energy.
Say the system has 5 atoms and 2 units of energy.
)!1(!
!1),(
Nq
NqqN
Boltzmann Factor & Partition FunctionsSay the system has 5 atoms and 2 units of energy.
)!1(!
!1),(
Nq
NqqNtotal
Atom 1 Atom 2 Atom 3 Atom 4 Atom 5
2 0 0 0 0
0 2 0 0 0
0 0 2 0 0
0 0 0 2 0
0 0 0 0 2
1 1 0 0 0
1 0 1 0 0
1 0 0 1 0
1 0 0 0 1
0 1 1 0 0
0 1 0 1 0
0 1 0 0 1
0 0 1 1 0
0 0 1 0 1
0 0 0 1 1
)0(
)1(
)2(
P
P
P
What is the probability of finding a particular atom with 2, 1, or 0 units of energy?
Boltzmann Factor & Partition FunctionsSay the system has 10 atoms and 4 units of energy.
715
)!1(!
!1),(
Nq
NqqNtotal
692.0715/495)0(
231.0715/165)1(
0629.0715/45)2(
0126.0715/9)3(
00140.0715/1)4(
P
P
P
P
P
What is the probability of finding a particular atom with 4, 3, 2, 1, or 0 units of energy?
UR, SR
U, S
Huge reservoir
System
Boltzmann Factor & Partition FunctionsSay the system has 10 atoms and 4 units of energy.
715
)!1(!
!1),(
Nq
NqqNtotal
692.0715/495)0(
231.0715/165)1(
0629.0715/45)2(
0126.0715/9)3(
00140.0715/1)4(
P
P
P
P
P
What is the probability of finding a particular atom with 4, 3, 2, 1, or 0 units of energy?
UR, SR
U, S
Huge reservoir
System
Boltzmann Factor & Partition FunctionsSay the system has 10 atoms and 4 units of energy.
715
)!1(!
!1),(
Nq
NqqNtotal
692.0715/495)0(
231.0715/165)1(
0629.0715/45)2(
0126.0715/9)3(
00140.0715/1)4(
P
P
P
P
P
What is the probability of finding a particular atom with 4, 3, 2, 1, or 0 units of energy?
Boltzmann Factor & Partition FunctionsSay the system has 10 atoms and 4 units of energy.
715
)!1(!
!1),(
Nq
NqqNtotal
n
eP
n
TkZ
total
nn
Bn
/1)(
What is the probability of finding a particular atom with 4, 3, 2, 1, or 0 units of energy?
Boltzmann Factor & Partition Functions
Tk
Tk
n
TkZn
Bn
Bn
Bn
e
eZ
eP
/
/
/1)(
Boltzmann Factors
Partition Function
Boltzmann or Canonical Distribution
Boltzmann, Entropy, & Gibbs
)(ln)(
)( /1
nnB
TkZ
total
nn
PPkS
eP Bn
Boltzmann, Entropy, & Gibbs
)(ln)(
)( /1
nnB
TkZ
total
nn
PPkS
eP Bn
Boltzmann, Entropy, & Gibbs
)(ln)(
)( /1
nnB
TkZ
total
nn
PPkS
eP Bn
Boltzmann, Entropy, & Gibbs
)(ln)(
)( /1
nnB
TkZ
total
nn
PPkS
eP Bn
Partition Functions & Hydrogen AtomWhat’s the energy of the electron on a hydrogen atom?
Solution to Rydberg or Bohr Model can be used.
Partition Functions & Hydrogen AtomWhat’s the energy of the electron on a hydrogen atom?
Solution to Rydberg or Bohr Model can be used.
Hydrogen Atom @ 300K
Hydrogen Atom on Sun
Hydrogen Atom on Sun
A System with Smaller Energies
A System with Smaller Energies
0 0.1 0.2 0.3 0.4 0.5 0.60
0.2
0.4
0.6
0.8
1
Energy (eV)
Pro
ba
bil
ity
A System with Smaller Energies
0 0.1 0.2 0.3 0.4 0.5 0.60
0.10.20.30.40.50.60.70.80.9
Energy (eV)
Pro
ba
bil
ity
Average ValuesSay the system has 10 atoms and 4 units of energy.
What is the average energy of the system if4 atoms have n=03 atoms have n=12 atoms have n=21 atom has n=30 atoms have n=4?
Averages Values
Be careful about using the proper probability when computing averages.
Rotation of Diatomic Molecules
Rotation of Diatomic Molecules
Rotation of Diatomic Molecules
0
/)(
0
/)1( 2
1212j
Tkjj
j
Tkjjrot
BB ejejZ
This can be calculated as an integral if dj is small compared to kBT/e (high temperature limit).
djejZ Tkjjrot
B 0
/)( 2
12
Rotation of Diatomic MoleculesA simplified partition function in the high temperature limit.
Rotation of Diatomic MoleculesHCl rotations
Rotation of Diatomic MoleculesAverage energy and heat capacity.
Rotational Partition Function• For diatoms with unlike atoms
• For diatoms with like atoms
Tk
ejZ B
j
Tkjjrot
B
0
/)1()12(
20
/)1(2)12( TkeZ B
j
Tkjjjrot
B
Unlike atomsdistinguishable
like atomsIndistinguishable
Bol
tzm
ann
Fac
tors
Rotational Partition FunctionB
oltz
man
n F
acto
rs
Bol
tzm
ann
Fac
tors
Rotational Energies
Unlike atomsdistinguishable
like atomsIndistinguishable
BV
V
B
B
NkT
UC
TNkU
TkE
Maxwell Speed Distribution
vx
vy
vz
v
For continuous variables, we talk about probability density function or probability distribution function (pdf).
Maxwell Speed DistributionWhat is Z?
Maxwell Speed Distribution
What is maximum probability speed?
The Maxwell Speed Distribution
Maxwell Speed DistributionWhat is the average speed?
Maxwell Speed Distribution
Example: Nitrogen @ 300KFor Monday:
(a) Show the maximum probability speed is 517 m/s.
(b) Show the average speed is 476 m/s.
(c) Show the rms speed is 422 m/s.
(d) Calculate the probability of a molecule moving faster than 1000 m/s. (Need Maple or Mathematica)
Nitrogen Speed Distribution