chapter 6 boltzmann statistics. boltzmann factor & partition functions u r, s r u, s huge...
TRANSCRIPT
![Page 1: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/1.jpg)
Chapter 6
Boltzmann Statistics
![Page 2: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/2.jpg)
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
![Page 3: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/3.jpg)
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?
![Page 4: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/4.jpg)
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
![Page 5: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/5.jpg)
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
![Page 6: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/6.jpg)
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?
![Page 7: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/7.jpg)
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?
![Page 8: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/8.jpg)
Boltzmann Factor & Partition Functions
Tk
Tk
n
TkZn
Bn
Bn
Bn
e
eZ
eP
/
/
/1)(
Boltzmann Factors
Partition Function
Boltzmann or Canonical Distribution
![Page 9: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/9.jpg)
Boltzmann, Entropy, & Gibbs
)(ln)(
)( /1
nnB
TkZ
total
nn
PPkS
eP Bn
![Page 10: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/10.jpg)
Boltzmann, Entropy, & Gibbs
)(ln)(
)( /1
nnB
TkZ
total
nn
PPkS
eP Bn
![Page 11: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/11.jpg)
Boltzmann, Entropy, & Gibbs
)(ln)(
)( /1
nnB
TkZ
total
nn
PPkS
eP Bn
![Page 12: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/12.jpg)
Boltzmann, Entropy, & Gibbs
)(ln)(
)( /1
nnB
TkZ
total
nn
PPkS
eP Bn
![Page 13: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/13.jpg)
Partition Functions & Hydrogen AtomWhat’s the energy of the electron on a hydrogen atom?
Solution to Rydberg or Bohr Model can be used.
![Page 14: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/14.jpg)
Partition Functions & Hydrogen AtomWhat’s the energy of the electron on a hydrogen atom?
Solution to Rydberg or Bohr Model can be used.
![Page 15: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/15.jpg)
Hydrogen Atom @ 300K
![Page 16: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/16.jpg)
Hydrogen Atom on Sun
![Page 17: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/17.jpg)
Hydrogen Atom on Sun
![Page 18: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/18.jpg)
A System with Smaller Energies
![Page 19: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/19.jpg)
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
![Page 20: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/20.jpg)
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
![Page 21: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/21.jpg)
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?
![Page 22: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/22.jpg)
Averages Values
Be careful about using the proper probability when computing averages.
![Page 23: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/23.jpg)
Rotation of Diatomic Molecules
![Page 24: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/24.jpg)
Rotation of Diatomic Molecules
![Page 25: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/25.jpg)
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
![Page 26: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/26.jpg)
Rotation of Diatomic MoleculesA simplified partition function in the high temperature limit.
![Page 27: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/27.jpg)
Rotation of Diatomic MoleculesHCl rotations
![Page 28: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/28.jpg)
Rotation of Diatomic MoleculesAverage energy and heat capacity.
![Page 29: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/29.jpg)
![Page 30: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/30.jpg)
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
![Page 31: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/31.jpg)
Rotational Partition FunctionB
oltz
man
n F
acto
rs
Bol
tzm
ann
Fac
tors
![Page 32: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/32.jpg)
Rotational Energies
Unlike atomsdistinguishable
like atomsIndistinguishable
BV
V
B
B
NkT
UC
TNkU
TkE
![Page 33: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/33.jpg)
Maxwell Speed Distribution
vx
vy
vz
v
For continuous variables, we talk about probability density function or probability distribution function (pdf).
![Page 34: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/34.jpg)
Maxwell Speed DistributionWhat is Z?
![Page 35: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/35.jpg)
Maxwell Speed Distribution
What is maximum probability speed?
The Maxwell Speed Distribution
![Page 36: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/36.jpg)
Maxwell Speed DistributionWhat is the average speed?
![Page 37: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/37.jpg)
Maxwell Speed Distribution
![Page 38: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/38.jpg)
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)
![Page 39: Chapter 6 Boltzmann Statistics. Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very](https://reader030.vdocument.in/reader030/viewer/2022032517/56649cc25503460f949896b4/html5/thumbnails/39.jpg)
Nitrogen Speed Distribution