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Raffi Hovasapian

Raffi Hovasapian

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Table of Contents

Section 1: Classical Thermodynamics Preliminaries
The Ideal Gas Law

46m 5s

Intro
0:00
Course Overview
0:16
Thermodynamics & Classical Thermodynamics
0:17
Structure of the Course
1:30
The Ideal Gas Law
3:06
Ideal Gas Law: PV=nRT
3:07
Units of Pressure
4:51
Manipulating Units
5:52
Atmosphere : atm
8:15
Millimeter of Mercury: mm Hg
8:48
SI Unit of Volume
9:32
SI Unit of Temperature
10:32
Value of R (Gas Constant): Pv = nRT
10:51
Extensive and Intensive Variables (Properties)
15:23
Intensive Property
15:52
Extensive Property
16:30
Example: Extensive and Intensive Variables
18:20
Ideal Gas Law
19:24
Ideal Gas Law with Intensive Variables
19:25
Graphing Equations
23:51
Hold T Constant & Graph P vs. V
23:52
Hold P Constant & Graph V vs. T
31:08
Hold V Constant & Graph P vs. T
34:38
Isochores or Isometrics
37:08
More on the V vs. T Graph
39:46
More on the P vs. V Graph
42:06
Ideal Gas Law at Low Pressure & High Temperature
44:26
Ideal Gas Law at High Pressure & Low Temperature
45:16
Math Lesson 1: Partial Differentiation

46m 2s

Intro
0:00
Math Lesson 1: Partial Differentiation
0:38
Overview
0:39
Example I
3:00
Example II
6:33
Example III
9:52
Example IV
17:26
Differential & Derivative
21:44
What Does It Mean?
21:45
Total Differential (or Total Derivative)
30:16
Net Change in Pressure (P)
33:58
General Equation for Total Differential
38:12
Example 5: Total Differential
39:28
Section 2: Energy
Energy & the First Law I

1h 6m 45s

Intro
0:00
Properties of Thermodynamic State
1:38
Big Picture: 3 Properties of Thermodynamic State
1:39
Enthalpy & Free Energy
3:30
Associated Law
4:40
Energy & the First Law of Thermodynamics
7:13
System & Its Surrounding Separated by a Boundary
7:14
In Other Cases the Boundary is Less Clear
10:47
State of a System
12:37
State of a System
12:38
Change in State
14:00
Path for a Change in State
14:57
Example: State of a System
15:46
Open, Close, and Isolated System
18:26
Open System
18:27
Closed System
19:02
Isolated System
19:22
Important Questions
20:38
Important Questions
20:39
Work & Heat
22:50
Definition of Work
23:33
Properties of Work
25:34
Definition of Heat
32:16
Properties of Heat
34:49
Experiment #1
42:23
Experiment #2
47:00
More on Work & Heat
54:50
More on Work & Heat
54:51
Conventions for Heat & Work
1:00:50
Convention for Heat
1:02:40
Convention for Work
1:04:24
Schematic Representation
1:05:00
Energy & the First Law II

1h 6m 33s

Intro
0:00
The First Law of Thermodynamics
0:53
The First Law of Thermodynamics
0:54
Example 1: What is the Change in Energy of the System & Surroundings?
8:53
Energy and The First Law II, cont.
11:55
The Energy of a System Changes in Two Ways
11:56
Systems Possess Energy, Not Heat or Work
12:45
Scenario 1
16:00
Scenario 2
16:46
State Property, Path Properties, and Path Functions
18:10
Pressure-Volume Work
22:36
When a System Changes
22:37
Gas Expands
24:06
Gas is Compressed
25:13
Pressure Volume Diagram: Analyzing Expansion
27:17
What if We do the Same Expansion in Two Stages?
35:22
Multistage Expansion
43:58
General Expression for the Pressure-Volume Work
46:59
Upper Limit of Isothermal Expansion
50:00
Expression for the Work Done in an Isothermal Expansion
52:45
Example 2: Find an Expression for the Maximum Work Done by an Ideal Gas upon Isothermal Expansion
56:18
Example 3: Calculate the External Pressure and Work Done
58:50
Energy & the First Law III

1h 2m 17s

Intro
0:00
Compression
0:20
Compression Overview
0:34
Single-stage compression vs. 2-stage Compression
2:16
Multi-stage Compression
8:40
Example I: Compression
14:47
Example 1: Single-stage Compression
14:47
Example 1: 2-stage Compression
20:07
Example 1: Absolute Minimum
26:37
More on Compression
32:55
Isothermal Expansion & Compression
32:56
External & Internal Pressure of the System
35:18
Reversible & Irreversible Processes
37:32
Process 1: Overview
38:57
Process 2: Overview
39:36
Process 1: Analysis
40:42
Process 2: Analysis
45:29
Reversible Process
50:03
Isothermal Expansion and Compression
54:31
Example II: Reversible Isothermal Compression of a Van der Waals Gas
58:10
Example 2: Reversible Isothermal Compression of a Van der Waals Gas
58:11
Changes in Energy & State: Constant Volume

1h 4m 39s

Intro
0:00
Recall
0:37
State Function & Path Function
0:38
First Law
2:11
Exact & Inexact Differential
2:12
Where Does (∆U = Q - W) or dU = dQ - dU Come from?
8:54
Cyclic Integrals of Path and State Functions
8:55
Our Empirical Experience of the First Law
12:31
∆U = Q - W
18:42
Relations between Changes in Properties and Energy
22:24
Relations between Changes in Properties and Energy
22:25
Rate of Change of Energy per Unit Change in Temperature
29:54
Rate of Change of Energy per Unit Change in Volume at Constant Temperature
32:39
Total Differential Equation
34:38
Constant Volume
41:08
If Volume Remains Constant, then dV = 0
41:09
Constant Volume Heat Capacity
45:22
Constant Volume Integrated
48:14
Increase & Decrease in Energy of the System
54:19
Example 1: ∆U and Qv
57:43
Important Equations
1:02:06
Joule's Experiment

16m 50s

Intro
0:00
Joule's Experiment
0:09
Joule's Experiment
1:20
Interpretation of the Result
4:42
The Gas Expands Against No External Pressure
4:43
Temperature of the Surrounding Does Not Change
6:20
System & Surrounding
7:04
Joule's Law
10:44
More on Joule's Experiment
11:08
Later Experiment
12:38
Dealing with the 2nd Law & Its Mathematical Consequences
13:52
Changes in Energy & State: Constant Pressure

43m 40s

Intro
0:00
Changes in Energy & State: Constant Pressure
0:20
Integrating with Constant Pressure
0:35
Defining the New State Function
6:24
Heat & Enthalpy of the System at Constant Pressure
8:54
Finding ∆U
12:10
dH
15:28
Constant Pressure Heat Capacity
18:08
Important Equations
25:44
Important Equations
25:45
Important Equations at Constant Pressure
27:32
Example I: Change in Enthalpy (∆H)
28:53
Example II: Change in Internal Energy (∆U)
34:19
The Relationship Between Cp & Cv

32m 23s

Intro
0:00
The Relationship Between Cp & Cv
0:21
For a Constant Volume Process No Work is Done
0:22
For a Constant Pressure Process ∆V ≠ 0, so Work is Done
1:16
The Relationship Between Cp & Cv: For an Ideal Gas
3:26
The Relationship Between Cp & Cv: In Terms of Molar heat Capacities
5:44
Heat Capacity Can Have an Infinite # of Values
7:14
The Relationship Between Cp & Cv
11:20
When Cp is Greater than Cv
17:13
2nd Term
18:10
1st Term
19:20
Constant P Process: 3 Parts
22:36
Part 1
23:45
Part 2
24:10
Part 3
24:46
Define : γ = (Cp/Cv)
28:06
For Gases
28:36
For Liquids
29:04
For an Ideal Gas
30:46
The Joule Thompson Experiment

39m 15s

Intro
0:00
General Equations
0:13
Recall
0:14
How Does Enthalpy of a System Change Upon a Unit Change in Pressure?
2:58
For Liquids & Solids
12:11
For Ideal Gases
14:08
For Real Gases
16:58
The Joule Thompson Experiment
18:37
The Joule Thompson Experiment Setup
18:38
The Flow in 2 Stages
22:54
Work Equation for the Joule Thompson Experiment
24:14
Insulated Pipe
26:33
Joule-Thompson Coefficient
29:50
Changing Temperature & Pressure in Such a Way that Enthalpy Remains Constant
31:44
Joule Thompson Inversion Temperature
36:26
Positive & Negative Joule-Thompson Coefficient
36:27
Joule Thompson Inversion Temperature
37:22
Inversion Temperature of Hydrogen Gas
37:59
Adiabatic Changes of State

35m 52s

Intro
0:00
Adiabatic Changes of State
0:10
Adiabatic Changes of State
0:18
Work & Energy in an Adiabatic Process
3:44
Pressure-Volume Work
7:43
Adiabatic Changes for an Ideal Gas
9:23
Adiabatic Changes for an Ideal Gas
9:24
Equation for a Fixed Change in Volume
11:20
Maximum & Minimum Values of Temperature
14:20
Adiabatic Path
18:08
Adiabatic Path Diagram
18:09
Reversible Adiabatic Expansion
21:54
Reversible Adiabatic Compression
22:34
Fundamental Relationship Equation for an Ideal Gas Under Adiabatic Expansion
25:00
More on the Equation
28:20
Important Equations
32:16
Important Adiabatic Equation
32:17
Reversible Adiabatic Change of State Equation
33:02
Section 3: Energy Example Problems
1st Law Example Problems I

42m 40s

Intro
0:00
Fundamental Equations
0:56
Work
2:40
Energy (1st Law)
3:10
Definition of Enthalpy
3:44
Heat capacity Definitions
4:06
The Mathematics
6:35
Fundamental Concepts
8:13
Isothermal
8:20
Adiabatic
8:54
Isobaric
9:25
Isometric
9:48
Ideal Gases
10:14
Example I
12:08
Example I: Conventions
12:44
Example I: Part A
15:30
Example I: Part B
18:24
Example I: Part C
19:53
Example II: What is the Heat Capacity of the System?
21:49
Example III: Find Q, W, ∆U & ∆H for this Change of State
24:15
Example IV: Find Q, W, ∆U & ∆H
31:37
Example V: Find Q, W, ∆U & ∆H
38:20
1st Law Example Problems II

1h 23s

Intro
0:00
Example I
0:11
Example I: Finding ∆U
1:49
Example I: Finding W
6:22
Example I: Finding Q
11:23
Example I: Finding ∆H
16:09
Example I: Summary
17:07
Example II
21:16
Example II: Finding W
22:42
Example II: Finding ∆H
27:48
Example II: Finding Q
30:58
Example II: Finding ∆U
31:30
Example III
33:33
Example III: Finding ∆U, Q & W
33:34
Example III: Finding ∆H
38:07
Example IV
41:50
Example IV: Finding ∆U
41:51
Example IV: Finding ∆H
45:42
Example V
49:31
Example V: Finding W
49:32
Example V: Finding ∆U
55:26
Example V: Finding Q
56:26
Example V: Finding ∆H
56:55
1st Law Example Problems III

44m 34s

Intro
0:00
Example I
0:15
Example I: Finding the Final Temperature
3:40
Example I: Finding Q
8:04
Example I: Finding ∆U
8:25
Example I: Finding W
9:08
Example I: Finding ∆H
9:51
Example II
11:27
Example II: Finding the Final Temperature
11:28
Example II: Finding ∆U
21:25
Example II: Finding W & Q
22:14
Example II: Finding ∆H
23:03
Example III
24:38
Example III: Finding the Final Temperature
24:39
Example III: Finding W, ∆U, and Q
27:43
Example III: Finding ∆H
28:04
Example IV
29:23
Example IV: Finding ∆U, W, and Q
25:36
Example IV: Finding ∆H
31:33
Example V
32:24
Example V: Finding the Final Temperature
33:32
Example V: Finding ∆U
39:31
Example V: Finding W
40:17
Example V: First Way of Finding ∆H
41:10
Example V: Second Way of Finding ∆H
42:10
Thermochemistry Example Problems

59m 7s

Intro
0:00
Example I: Find ∆H° for the Following Reaction
0:42
Example II: Calculate the ∆U° for the Reaction in Example I
5:33
Example III: Calculate the Heat of Formation of NH₃ at 298 K
14:23
Example IV
32:15
Part A: Calculate the Heat of Vaporization of Water at 25°C
33:49
Part B: Calculate the Work Done in Vaporizing 2 Mols of Water at 25°C Under a Constant Pressure of 1 atm
35:26
Part C: Find ∆U for the Vaporization of Water at 25°C
41:00
Part D: Find the Enthalpy of Vaporization of Water at 100°C
43:12
Example V
49:24
Part A: Constant Temperature & Increasing Pressure
50:25
Part B: Increasing temperature & Constant Pressure
56:20
Section 4: Entropy
Entropy

49m 16s

Intro
0:00
Entropy, Part 1
0:16
Coefficient of Thermal Expansion (Isobaric)
0:38
Coefficient of Compressibility (Isothermal)
1:25
Relative Increase & Relative Decrease
2:16
More on α
4:40
More on κ
8:38
Entropy, Part 2
11:04
Definition of Entropy
12:54
Differential Change in Entropy & the Reversible Path
20:08
State Property of the System
28:26
Entropy Changes Under Isothermal Conditions
35:00
Recall: Heating Curve
41:05
Some Phase Changes Take Place Under Constant Pressure
44:07
Example I: Finding ∆S for a Phase Change
46:05
Math Lesson II

33m 59s

Intro
0:00
Math Lesson II
0:46
Let F(x,y) = x²y³
0:47
Total Differential
3:34
Total Differential Expression
6:06
Example 1
9:24
More on Math Expression
13:26
Exact Total Differential Expression
13:27
Exact Differentials
19:50
Inexact Differentials
20:20
The Cyclic Rule
21:06
The Cyclic Rule
21:07
Example 2
27:58
Entropy As a Function of Temperature & Volume

54m 37s

Intro
0:00
Entropy As a Function of Temperature & Volume
0:14
Fundamental Equation of Thermodynamics
1:16
Things to Notice
9:10
Entropy As a Function of Temperature & Volume
14:47
Temperature-dependence of Entropy
24:00
Example I
26:19
Entropy As a Function of Temperature & Volume, Cont.
31:55
Volume-dependence of Entropy at Constant Temperature
31:56
Differentiate with Respect to Temperature, Holding Volume Constant
36:16
Recall the Cyclic Rule
45:15
Summary & Recap
46:47
Fundamental Equation of Thermodynamics
46:48
For Entropy as a Function of Temperature & Volume
47:18
The Volume-dependence of Entropy for Liquids & Solids
52:52
Entropy as a Function of Temperature & Pressure

31m 18s

Intro
0:00
Entropy as a Function of Temperature & Pressure
0:17
Entropy as a Function of Temperature & Pressure
0:18
Rewrite the Total Differential
5:54
Temperature-dependence
7:08
Pressure-dependence
9:04
Differentiate with Respect to Pressure & Holding Temperature Constant
9:54
Differentiate with Respect to Temperature & Holding Pressure Constant
11:28
Pressure-Dependence of Entropy for Liquids & Solids
18:45
Pressure-Dependence of Entropy for Liquids & Solids
18:46
Example I: ∆S of Transformation
26:20
Summary of Entropy So Far

23m 6s

Intro
0:00
Summary of Entropy So Far
0:43
Defining dS
1:04
Fundamental Equation of Thermodynamics
3:51
Temperature & Volume
6:04
Temperature & Pressure
9:10
Two Important Equations for How Entropy Behaves
13:38
State of a System & Heat Capacity
15:34
Temperature-dependence of Entropy
19:49
Entropy Changes for an Ideal Gas

25m 42s

Intro
0:00
Entropy Changes for an Ideal Gas
1:10
General Equation
1:22
The Fundamental Theorem of Thermodynamics
2:37
Recall the Basic Total Differential Expression for S = S (T,V)
5:36
For a Finite Change in State
7:58
If Cv is Constant Over the Particular Temperature Range
9:05
Change in Entropy of an Ideal Gas as a Function of Temperature & Pressure
11:35
Change in Entropy of an Ideal Gas as a Function of Temperature & Pressure
11:36
Recall the Basic Total Differential expression for S = S (T, P)
15:13
For a Finite Change
18:06
Example 1: Calculate the ∆S of Transformation
22:02
Section 5: Entropy Example Problems
Entropy Example Problems I

43m 39s

Intro
0:00
Entropy Example Problems I
0:24
Fundamental Equation of Thermodynamics
1:10
Entropy as a Function of Temperature & Volume
2:04
Entropy as a Function of Temperature & Pressure
2:59
Entropy For Phase Changes
4:47
Entropy For an Ideal Gas
6:14
Third Law Entropies
8:25
Statement of the Third Law
9:17
Entropy of the Liquid State of a Substance Above Its Melting Point
10:23
Entropy For the Gas Above Its Boiling Temperature
13:02
Entropy Changes in Chemical Reactions
15:26
Entropy Change at a Temperature Other than 25°C
16:32
Example I
19:31
Part A: Calculate ∆S for the Transformation Under Constant Volume
20:34
Part B: Calculate ∆S for the Transformation Under Constant Pressure
25:04
Example II: Calculate ∆S fir the Transformation Under Isobaric Conditions
27:53
Example III
30:14
Part A: Calculate ∆S if 1 Mol of Aluminum is taken from 25°C to 255°C
31:14
Part B: If S°₂₉₈ = 28.4 J/mol-K, Calculate S° for Aluminum at 498 K
33:23
Example IV: Calculate Entropy Change of Vaporization for CCl₄
34:19
Example V
35:41
Part A: Calculate ∆S of Transformation
37:36
Part B: Calculate ∆S of Transformation
39:10
Entropy Example Problems II

56m 44s

Intro
0:00
Example I
0:09
Example I: Calculate ∆U
1:28
Example I: Calculate Q
3:29
Example I: Calculate Cp
4:54
Example I: Calculate ∆S
6:14
Example II
7:13
Example II: Calculate W
8:14
Example II: Calculate ∆U
8:56
Example II: Calculate Q
10:18
Example II: Calculate ∆H
11:00
Example II: Calculate ∆S
12:36
Example III
18:47
Example III: Calculate ∆H
19:38
Example III: Calculate Q
21:14
Example III: Calculate ∆U
21:44
Example III: Calculate W
23:59
Example III: Calculate ∆S
24:55
Example IV
27:57
Example IV: Diagram
29:32
Example IV: Calculate W
32:27
Example IV: Calculate ∆U
36:36
Example IV: Calculate Q
38:32
Example IV: Calculate ∆H
39:00
Example IV: Calculate ∆S
40:27
Example IV: Summary
43:41
Example V
48:25
Example V: Diagram
49:05
Example V: Calculate W
50:58
Example V: Calculate ∆U
53:29
Example V: Calculate Q
53:44
Example V: Calculate ∆H
54:34
Example V: Calculate ∆S
55:01
Entropy Example Problems III

57m 6s

Intro
0:00
Example I: Isothermal Expansion
0:09
Example I: Calculate W
1:19
Example I: Calculate ∆U
1:48
Example I: Calculate Q
2:06
Example I: Calculate ∆H
2:26
Example I: Calculate ∆S
3:02
Example II: Adiabatic and Reversible Expansion
6:10
Example II: Calculate Q
6:48
Example II: Basic Equation for the Reversible Adiabatic Expansion of an Ideal Gas
8:12
Example II: Finding Volume
12:40
Example II: Finding Temperature
17:58
Example II: Calculate ∆U
19:53
Example II: Calculate W
20:59
Example II: Calculate ∆H
21:42
Example II: Calculate ∆S
23:42
Example III: Calculate the Entropy of Water Vapor
25:20
Example IV: Calculate the Molar ∆S for the Transformation
34:32
Example V
44:19
Part A: Calculate the Standard Entropy of Liquid Lead at 525°C
46:17
Part B: Calculate ∆H for the Transformation of Solid Lead from 25°C to Liquid Lead at 525°C
52:23
Section 6: Entropy and Probability
Entropy & Probability I

54m 35s

Intro
0:00
Entropy & Probability
0:11
Structural Model
3:05
Recall the Fundamental Equation of Thermodynamics
9:11
Two Independent Ways of Affecting the Entropy of a System
10:05
Boltzmann Definition
12:10
Omega
16:24
Definition of Omega
16:25
Energy Distribution
19:43
The Energy Distribution
19:44
In How Many Ways can N Particles be Distributed According to the Energy Distribution
23:05
Example I: In How Many Ways can the Following Distribution be Achieved
32:51
Example II: In How Many Ways can the Following Distribution be Achieved
33:51
Example III: In How Many Ways can the Following Distribution be Achieved
34:45
Example IV: In How Many Ways can the Following Distribution be Achieved
38:50
Entropy & Probability, cont.
40:57
More on Distribution
40:58
Example I Summary
41:43
Example II Summary
42:12
Distribution that Maximizes Omega
42:26
If Omega is Large, then S is Large
44:22
Two Constraints for a System to Achieve the Highest Entropy Possible
47:07
What Happened When the Energy of a System is Increased?
49:00
Entropy & Probability II

35m 5s

Intro
0:00
Volume Distribution
0:08
Distributing 2 Balls in 3 Spaces
1:43
Distributing 2 Balls in 4 Spaces
3:44
Distributing 3 Balls in 10 Spaces
5:30
Number of Ways to Distribute P Particles over N Spaces
6:05
When N is Much Larger than the Number of Particles P
7:56
Energy Distribution
25:04
Volume Distribution
25:58
Entropy, Total Entropy, & Total Omega Equations
27:34
Entropy, Total Entropy, & Total Omega Equations
27:35
Section 7: Spontaneity, Equilibrium, and the Fundamental Equations
Spontaneity & Equilibrium I

28m 42s

Intro
0:00
Reversible & Irreversible
0:24
Reversible vs. Irreversible
0:58
Defining Equation for Equilibrium
2:11
Defining Equation for Irreversibility (Spontaneity)
3:11
TdS ≥ dQ
5:15
Transformation in an Isolated System
11:22
Transformation in an Isolated System
11:29
Transformation at Constant Temperature
14:50
Transformation at Constant Temperature
14:51
Helmholtz Free Energy
17:26
Define: A = U - TS
17:27
Spontaneous Isothermal Process & Helmholtz Energy
20:20
Pressure-volume Work
22:02
Spontaneity & Equilibrium II

34m 38s

Intro
0:00
Transformation under Constant Temperature & Pressure
0:08
Transformation under Constant Temperature & Pressure
0:36
Define: G = U + PV - TS
3:32
Gibbs Energy
5:14
What Does This Say?
6:44
Spontaneous Process & a Decrease in G
14:12
Computing ∆G
18:54
Summary of Conditions
21:32
Constraint & Condition for Spontaneity
21:36
Constraint & Condition for Equilibrium
24:54
A Few Words About the Word Spontaneous
26:24
Spontaneous Does Not Mean Fast
26:25
Putting Hydrogen & Oxygen Together in a Flask
26:59
Spontaneous Vs. Not Spontaneous
28:14
Thermodynamically Favorable
29:03
Example: Making a Process Thermodynamically Favorable
29:34
Driving Forces for Spontaneity
31:35
Equation: ∆G = ∆H - T∆S
31:36
Always Spontaneous Process
32:39
Never Spontaneous Process
33:06
A Process That is Endothermic Can Still be Spontaneous
34:00
The Fundamental Equations of Thermodynamics

30m 50s

Intro
0:00
The Fundamental Equations of Thermodynamics
0:44
Mechanical Properties of a System
0:45
Fundamental Properties of a System
1:16
Composite Properties of a System
1:44
General Condition of Equilibrium
3:16
Composite Functions & Their Differentiations
6:11
dH = TdS + VdP
7:53
dA = -SdT - PdV
9:26
dG = -SdT + VdP
10:22
Summary of Equations
12:10
Equation #1
14:33
Equation #2
15:15
Equation #3
15:58
Equation #4
16:42
Maxwell's Relations
20:20
Maxwell's Relations
20:21
Isothermal Volume-Dependence of Entropy & Isothermal Pressure-Dependence of Entropy
26:21
The General Thermodynamic Equations of State

34m 6s

Intro
0:00
The General Thermodynamic Equations of State
0:10
Equations of State for Liquids & Solids
0:52
More General Condition for Equilibrium
4:02
General Conditions: Equation that Relates P to Functions of T & V
6:20
The Second Fundamental Equation of Thermodynamics
11:10
Equation 1
17:34
Equation 2
21:58
Recall the General Expression for Cp - Cv
28:11
For the Joule-Thomson Coefficient
30:44
Joule-Thomson Inversion Temperature
32:12
Properties of the Helmholtz & Gibbs Energies

39m 18s

Intro
0:00
Properties of the Helmholtz & Gibbs Energies
0:10
Equating the Differential Coefficients
1:34
An Increase in T; a Decrease in A
3:25
An Increase in V; a Decrease in A
6:04
We Do the Same Thing for G
8:33
Increase in T; Decrease in G
10:50
Increase in P; Decrease in G
11:36
Gibbs Energy of a Pure Substance at a Constant Temperature from 1 atm to any Other Pressure.
14:12
If the Substance is a Liquid or a Solid, then Volume can be Treated as a Constant
18:57
For an Ideal Gas
22:18
Special Note
24:56
Temperature Dependence of Gibbs Energy
27:02
Temperature Dependence of Gibbs Energy #1
27:52
Temperature Dependence of Gibbs Energy #2
29:01
Temperature Dependence of Gibbs Energy #3
29:50
Temperature Dependence of Gibbs Energy #4
34:50
The Entropy of the Universe & the Surroundings

19m 40s

Intro
0:00
Entropy of the Universe & the Surroundings
0:08
Equation: ∆G = ∆H - T∆S
0:20
Conditions of Constant Temperature & Pressure
1:14
Reversible Process
3:14
Spontaneous Process & the Entropy of the Universe
5:20
Tips for Remembering Everything
12:40
Verify Using Known Spontaneous Process
14:51
Section 8: Free Energy Example Problems
Free Energy Example Problems I

54m 16s

Intro
0:00
Example I
0:11
Example I: Deriving a Function for Entropy (S)
2:06
Example I: Deriving a Function for V
5:55
Example I: Deriving a Function for H
8:06
Example I: Deriving a Function for U
12:06
Example II
15:18
Example III
21:52
Example IV
26:12
Example IV: Part A
26:55
Example IV: Part B
28:30
Example IV: Part C
30:25
Example V
33:45
Example VI
40:46
Example VII
43:43
Example VII: Part A
44:46
Example VII: Part B
50:52
Example VII: Part C
51:56
Free Energy Example Problems II

31m 17s

Intro
0:00
Example I
0:09
Example II
5:18
Example III
8:22
Example IV
12:32
Example V
17:14
Example VI
20:34
Example VI: Part A
21:04
Example VI: Part B
23:56
Example VI: Part C
27:56
Free Energy Example Problems III

45m

Intro
0:00
Example I
0:10
Example II
15:03
Example III
21:47
Example IV
28:37
Example IV: Part A
29:33
Example IV: Part B
36:09
Example IV: Part C
40:34
Three Miscellaneous Example Problems

58m 5s

Intro
0:00
Example I
0:41
Part A: Calculating ∆H
3:55
Part B: Calculating ∆S
15:13
Example II
24:39
Part A: Final Temperature of the System
26:25
Part B: Calculating ∆S
36:57
Example III
46:49
Section 9: Equation Review for Thermodynamics
Looking Back Over Everything: All the Equations in One Place

25m 20s

Intro
0:00
Work, Heat, and Energy
0:18
Definition of Work, Energy, Enthalpy, and Heat Capacities
0:23
Heat Capacities for an Ideal Gas
3:40
Path Property & State Property
3:56
Energy Differential
5:04
Enthalpy Differential
5:40
Joule's Law & Joule-Thomson Coefficient
6:23
Coefficient of Thermal Expansion & Coefficient of Compressibility
7:01
Enthalpy of a Substance at Any Other Temperature
7:29
Enthalpy of a Reaction at Any Other Temperature
8:01
Entropy
8:53
Definition of Entropy
8:54
Clausius Inequality
9:11
Entropy Changes in Isothermal Systems
9:44
The Fundamental Equation of Thermodynamics
10:12
Expressing Entropy Changes in Terms of Properties of the System
10:42
Entropy Changes in the Ideal Gas
11:22
Third Law Entropies
11:38
Entropy Changes in Chemical Reactions
14:02
Statistical Definition of Entropy
14:34
Omega for the Spatial & Energy Distribution
14:47
Spontaneity and Equilibrium
15:43
Helmholtz Energy & Gibbs Energy
15:44
Condition for Spontaneity & Equilibrium
16:24
Condition for Spontaneity with Respect to Entropy
17:58
The Fundamental Equations
18:30
Maxwell's Relations
19:04
The Thermodynamic Equations of State
20:07
Energy & Enthalpy Differentials
21:08
Joule's Law & Joule-Thomson Coefficient
21:59
Relationship Between Constant Pressure & Constant Volume Heat Capacities
23:14
One Final Equation - Just for Fun
24:04
Section 10: Quantum Mechanics Preliminaries
Complex Numbers

34m 25s

Intro
0:00
Complex Numbers
0:11
Representing Complex Numbers in the 2-Dimmensional Plane
0:56
Addition of Complex Numbers
2:35
Subtraction of Complex Numbers
3:17
Multiplication of Complex Numbers
3:47
Division of Complex Numbers
6:04
r & θ
8:04
Euler's Formula
11:00
Polar Exponential Representation of the Complex Numbers
11:22
Example I
14:25
Example II
15:21
Example III
16:58
Example IV
18:35
Example V
20:40
Example VI
21:32
Example VII
25:22
Probability & Statistics

59m 57s

Intro
0:00
Probability & Statistics
1:51
Normalization Condition
1:52
Define the Mean or Average of x
11:04
Example I: Calculate the Mean of x
14:57
Example II: Calculate the Second Moment of the Data in Example I
22:39
Define the Second Central Moment or Variance
25:26
Define the Second Central Moment or Variance
25:27
1st Term
32:16
2nd Term
32:40
3rd Term
34:07
Continuous Distributions
35:47
Continuous Distributions
35:48
Probability Density
39:30
Probability Density
39:31
Normalization Condition
46:51
Example III
50:13
Part A - Show that P(x) is Normalized
51:40
Part B - Calculate the Average Position of the Particle Along the Interval
54:31
Important Things to Remember
58:24
Schrӧdinger Equation & Operators

42m 5s

Intro
0:00
Schrӧdinger Equation & Operators
0:16
Relation Between a Photon's Momentum & Its Wavelength
0:17
Louis de Broglie: Wavelength for Matter
0:39
Schrӧdinger Equation
1:19
Definition of Ψ(x)
3:31
Quantum Mechanics
5:02
Operators
7:51
Example I
10:10
Example II
11:53
Example III
14:24
Example IV
17:35
Example V
19:59
Example VI
22:39
Operators Can Be Linear or Non Linear
27:58
Operators Can Be Linear or Non Linear
28:34
Example VII
32:47
Example VIII
36:55
Example IX
39:29
Schrӧdinger Equation as an Eigenvalue Problem

30m 26s

Intro
0:00
Schrӧdinger Equation as an Eigenvalue Problem
0:10
Operator: Multiplying the Original Function by Some Scalar
0:11
Operator, Eigenfunction, & Eigenvalue
4:42
Example: Eigenvalue Problem
8:00
Schrӧdinger Equation as an Eigenvalue Problem
9:24
Hamiltonian Operator
15:09
Quantum Mechanical Operators
16:46
Kinetic Energy Operator
19:16
Potential Energy Operator
20:02
Total Energy Operator
21:12
Classical Point of View
21:48
Linear Momentum Operator
24:02
Example I
26:01
The Plausibility of the Schrӧdinger Equation

21m 34s

Intro
0:00
The Plausibility of the Schrӧdinger Equation
1:16
The Plausibility of the Schrӧdinger Equation, Part 1
1:17
The Plausibility of the Schrӧdinger Equation, Part 2
8:24
The Plausibility of the Schrӧdinger Equation, Part 3
13:45
Section 11: The Particle in a Box
The Particle in a Box Part I

56m 22s

Intro
0:00
Free Particle in a Box
0:28
Definition of a Free Particle in a Box
0:29
Amplitude of the Matter Wave
6:22
Intensity of the Wave
6:53
Probability Density
9:39
Probability that the Particle is Located Between x & dx
10:54
Probability that the Particle will be Found Between o & a
12:35
Wave Function & the Particle
14:59
Boundary Conditions
19:22
What Happened When There is No Constraint on the Particle
27:54
Diagrams
34:12
More on Probability Density
40:53
The Correspondence Principle
46:45
The Correspondence Principle
46:46
Normalizing the Wave Function
47:46
Normalizing the Wave Function
47:47
Normalized Wave Function & Normalization Constant
52:24
The Particle in a Box Part II

45m 24s

Intro
0:00
Free Particle in a Box
0:08
Free Particle in a 1-dimensional Box
0:09
For a Particle in a Box
3:57
Calculating Average Values & Standard Deviations
5:42
Average Value for the Position of a Particle
6:32
Standard Deviations for the Position of a Particle
10:51
Recall: Energy & Momentum are Represented by Operators
13:33
Recall: Schrӧdinger Equation in Operator Form
15:57
Average Value of a Physical Quantity that is Associated with an Operator
18:16
Average Momentum of a Free Particle in a Box
20:48
The Uncertainty Principle
24:42
Finding the Standard Deviation of the Momentum
25:08
Expression for the Uncertainty Principle
35:02
Summary of the Uncertainty Principle
41:28
The Particle in a Box Part III

48m 43s

Intro
0:00
2-Dimension
0:12
Dimension 2
0:31
Boundary Conditions
1:52
Partial Derivatives
4:27
Example I
6:08
The Particle in a Box, cont.
11:28
Operator Notation
12:04
Symbol for the Laplacian
13:50
The Equation Becomes…
14:30
Boundary Conditions
14:54
Separation of Variables
15:33
Solution to the 1-dimensional Case
16:31
Normalization Constant
22:32
3-Dimension
28:30
Particle in a 3-dimensional Box
28:31
In Del Notation
32:22
The Solutions
34:51
Expressing the State of the System for a Particle in a 3D Box
39:10
Energy Level & Degeneracy
43:35
Section 12: Postulates and Principles of Quantum Mechanics
The Postulates & Principles of Quantum Mechanics, Part I

46m 18s

Intro
0:00
Postulate I
0:31
Probability That The Particle Will Be Found in a Differential Volume Element
0:32
Example I: Normalize This Wave Function
11:30
Postulate II
18:20
Postulate II
18:21
Quantum Mechanical Operators: Position
20:48
Quantum Mechanical Operators: Kinetic Energy
21:57
Quantum Mechanical Operators: Potential Energy
22:42
Quantum Mechanical Operators: Total Energy
22:57
Quantum Mechanical Operators: Momentum
23:22
Quantum Mechanical Operators: Angular Momentum
23:48
More On The Kinetic Energy Operator
24:48
Angular Momentum
28:08
Angular Momentum Overview
28:09
Angular Momentum Operator in Quantum Mechanic
31:34
The Classical Mechanical Observable
32:56
Quantum Mechanical Operator
37:01
Getting the Quantum Mechanical Operator from the Classical Mechanical Observable
40:16
Postulate II, cont.
43:40
Quantum Mechanical Operators are Both Linear & Hermetical
43:41
The Postulates & Principles of Quantum Mechanics, Part II

39m 28s

Intro
0:00
Postulate III
0:09
Postulate III: Part I
0:10
Postulate III: Part II
5:56
Postulate III: Part III
12:43
Postulate III: Part IV
18:28
Postulate IV
23:57
Postulate IV
23:58
Postulate V
27:02
Postulate V
27:03
Average Value
36:38
Average Value
36:39
The Postulates & Principles of Quantum Mechanics, Part III

35m 32s

Intro
0:00
The Postulates & Principles of Quantum Mechanics, Part III
0:10
Equations: Linear & Hermitian
0:11
Introduction to Hermitian Property
3:36
Eigenfunctions are Orthogonal
9:55
The Sequence of Wave Functions for the Particle in a Box forms an Orthonormal Set
14:34
Definition of Orthogonality
16:42
Definition of Hermiticity
17:26
Hermiticity: The Left Integral
23:04
Hermiticity: The Right Integral
28:47
Hermiticity: Summary
34:06
The Postulates & Principles of Quantum Mechanics, Part IV

29m 55s

Intro
0:00
The Postulates & Principles of Quantum Mechanics, Part IV
0:09
Operators can be Applied Sequentially
0:10
Sample Calculation 1
2:41
Sample Calculation 2
5:18
Commutator of Two Operators
8:16
The Uncertainty Principle
19:01
In the Case of Linear Momentum and Position Operator
23:14
When the Commutator of Two Operators Equals to Zero
26:31
Section 13: Postulates and Principles Example Problems, Including Particle in a Box
Example Problems I

54m 25s

Intro
0:00
Example I: Three Dimensional Box & Eigenfunction of The Laplacian Operator
0:37
Example II: Positions of a Particle in a 1-dimensional Box
15:46
Example III: Transition State & Frequency
29:29
Example IV: Finding a Particle in a 1-dimensional Box
35:03
Example V: Degeneracy & Energy Levels of a Particle in a Box
44:59
Example Problems II

46m 58s

Intro
0:00
Review
0:25
Wave Function
0:26
Normalization Condition
2:28
Observable in Classical Mechanics & Linear/Hermitian Operator in Quantum Mechanics
3:36
Hermitian
6:11
Eigenfunctions & Eigenvalue
8:20
Normalized Wave Functions
12:00
Average Value
13:42
If Ψ is Written as a Linear Combination
15:44
Commutator
16:45
Example I: Normalize The Wave Function
19:18
Example II: Probability of Finding of a Particle
22:27
Example III: Orthogonal
26:00
Example IV: Average Value of the Kinetic Energy Operator
30:22
Example V: Evaluate These Commutators
39:02
Example Problems III

44m 11s

Intro
0:00
Example I: Good Candidate for a Wave Function
0:08
Example II: Variance of the Energy
7:00
Example III: Evaluate the Angular Momentum Operators
15:00
Example IV: Real Eigenvalues Imposes the Hermitian Property on Operators
28:44
Example V: A Demonstration of Why the Eigenfunctions of Hermitian Operators are Orthogonal
35:33
Section 14: The Harmonic Oscillator
The Harmonic Oscillator I

35m 33s

Intro
0:00
The Harmonic Oscillator
0:10
Harmonic Motion
0:11
Classical Harmonic Oscillator
4:38
Hooke's Law
8:18
Classical Harmonic Oscillator, cont.
10:33
General Solution for the Differential Equation
15:16
Initial Position & Velocity
16:05
Period & Amplitude
20:42
Potential Energy of the Harmonic Oscillator
23:20
Kinetic Energy of the Harmonic Oscillator
26:37
Total Energy of the Harmonic Oscillator
27:23
Conservative System
34:37
The Harmonic Oscillator II

43m 4s

Intro
0:00
The Harmonic Oscillator II
0:08
Diatomic Molecule
0:10
Notion of Reduced Mass
5:27
Harmonic Oscillator Potential & The Intermolecular Potential of a Vibrating Molecule
7:33
The Schrӧdinger Equation for the 1-dimensional Quantum Mechanic Oscillator
14:14
Quantized Values for the Energy Level
15:46
Ground State & the Zero-Point Energy
21:50
Vibrational Energy Levels
25:18
Transition from One Energy Level to the Next
26:42
Fundamental Vibrational Frequency for Diatomic Molecule
34:57
Example: Calculate k
38:01
The Harmonic Oscillator III

26m 30s

Intro
0:00
The Harmonic Oscillator III
0:09
The Wave Functions Corresponding to the Energies
0:10
Normalization Constant
2:34
Hermite Polynomials
3:22
First Few Hermite Polynomials
4:56
First Few Wave-Functions
6:37
Plotting the Probability Density of the Wave-Functions
8:37
Probability Density for Large Values of r
14:24
Recall: Odd Function & Even Function
19:05
More on the Hermite Polynomials
20:07
Recall: If f(x) is Odd
20:36
Average Value of x
22:31
Average Value of Momentum
23:56
Section 15: The Rigid Rotator
The Rigid Rotator I

41m 10s

Intro
0:00
Possible Confusion from the Previous Discussion
0:07
Possible Confusion from the Previous Discussion
0:08
Rotation of a Single Mass Around a Fixed Center
8:17
Rotation of a Single Mass Around a Fixed Center
8:18
Angular Velocity
12:07
Rotational Inertia
13:24
Rotational Frequency
15:24
Kinetic Energy for a Linear System
16:38
Kinetic Energy for a Rotational System
17:42
Rotating Diatomic Molecule
19:40
Rotating Diatomic Molecule: Part 1
19:41
Rotating Diatomic Molecule: Part 2
24:56
Rotating Diatomic Molecule: Part 3
30:04
Hamiltonian of the Rigid Rotor
36:48
Hamiltonian of the Rigid Rotor
36:49
The Rigid Rotator II

30m 32s

Intro
0:00
The Rigid Rotator II
0:08
Cartesian Coordinates
0:09
Spherical Coordinates
1:55
r
6:15
θ
6:28
φ
7:00
Moving a Distance 'r'
8:17
Moving a Distance 'r' in the Spherical Coordinates
11:49
For a Rigid Rotator, r is Constant
13:57
Hamiltonian Operator
15:09
Square of the Angular Momentum Operator
17:34
Orientation of the Rotation in Space
19:44
Wave Functions for the Rigid Rotator
20:40
The Schrӧdinger Equation for the Quantum Mechanic Rigid Rotator
21:24
Energy Levels for the Rigid Rotator
26:58
The Rigid Rotator III

35m 19s

Intro
0:00
The Rigid Rotator III
0:11
When a Rotator is Subjected to Electromagnetic Radiation
1:24
Selection Rule
2:13
Frequencies at Which Absorption Transitions Occur
6:24
Energy Absorption & Transition
10:54
Energy of the Individual Levels Overview
20:58
Energy of the Individual Levels: Diagram
23:45
Frequency Required to Go from J to J + 1
25:53
Using Separation Between Lines on the Spectrum to Calculate Bond Length
28:02
Example I: Calculating Rotational Inertia & Bond Length
29:18
Example I: Calculating Rotational Inertia
29:19
Example I: Calculating Bond Length
32:56
Section 16: Oscillator and Rotator Example Problems
Example Problems I

33m 48s

Intro
0:00
Equations Review
0:11
Energy of the Harmonic Oscillator
0:12
Selection Rule
3:02
Observed Frequency of Radiation
3:27
Harmonic Oscillator Wave Functions
5:52
Rigid Rotator
7:26
Selection Rule for Rigid Rotator
9:15
Frequency of Absorption
9:35
Wave Numbers
10:58
Example I: Calculate the Reduced Mass of the Hydrogen Atom
11:44
Example II: Calculate the Fundamental Vibration Frequency & the Zero-Point Energy of This Molecule
13:37
Example III: Show That the Product of Two Even Functions is even
19:35
Example IV: Harmonic Oscillator
24:56
Example Problems II

46m 43s

Intro
0:00
Example I: Harmonic Oscillator
0:12
Example II: Harmonic Oscillator
23:26
Example III: Calculate the RMS Displacement of the Molecules
38:12
Section 17: The Hydrogen Atom
The Hydrogen Atom I

40m

Intro
0:00
The Hydrogen Atom I
1:31
Review of the Rigid Rotator
1:32
Hydrogen Atom & the Coulomb Potential
2:50
Using the Spherical Coordinates
6:33
Applying This Last Expression to Equation 1
10:19
Angular Component & Radial Component
13:26
Angular Equation
15:56
Solution for F(φ)
19:32
Determine The Normalization Constant
20:33
Differential Equation for T(a)
24:44
Legendre Equation
27:20
Legendre Polynomials
31:20
The Legendre Polynomials are Mutually Orthogonal
35:40
Limits
37:17
Coefficients
38:28
The Hydrogen Atom II

35m 58s

Intro
0:00
Associated Legendre Functions
0:07
Associated Legendre Functions
0:08
First Few Associated Legendre Functions
6:39
s, p, & d Orbital
13:24
The Normalization Condition
15:44
Spherical Harmonics
20:03
Equations We Have Found
20:04
Wave Functions for the Angular Component & Rigid Rotator
24:36
Spherical Harmonics Examples
25:40
Angular Momentum
30:09
Angular Momentum
30:10
Square of the Angular Momentum
35:38
Energies of the Rigid Rotator
38:21
The Hydrogen Atom III

36m 18s

Intro
0:00
The Hydrogen Atom III
0:34
Angular Momentum is a Vector Quantity
0:35
The Operators Corresponding to the Three Components of Angular Momentum Operator: In Cartesian Coordinates
1:30
The Operators Corresponding to the Three Components of Angular Momentum Operator: In Spherical Coordinates
3:27
Z Component of the Angular Momentum Operator & the Spherical Harmonic
5:28
Magnitude of the Angular Momentum Vector
20:10
Classical Interpretation of Angular Momentum
25:22
Projection of the Angular Momentum Vector onto the xy-plane
33:24
The Hydrogen Atom IV

33m 55s

Intro
0:00
The Hydrogen Atom IV
0:09
The Equation to Find R( r )
0:10
Relation Between n & l
3:50
The Solutions for the Radial Functions
5:08
Associated Laguerre Polynomials
7:58
1st Few Associated Laguerre Polynomials
8:55
Complete Wave Function for the Atomic Orbitals of the Hydrogen Atom
12:24
The Normalization Condition
15:06
In Cartesian Coordinates
18:10
Working in Polar Coordinates
20:48
Principal Quantum Number
21:58
Angular Momentum Quantum Number
22:35
Magnetic Quantum Number
25:55
Zeeman Effect
30:45
The Hydrogen Atom V: Where We Are

51m 53s

Intro
0:00
The Hydrogen Atom V: Where We Are
0:13
Review
0:14
Let's Write Out ψ₂₁₁
7:32
Angular Momentum of the Electron
14:52
Representation of the Wave Function
19:36
Radial Component
28:02
Example: 1s Orbital
28:34
Probability for Radial Function
33:46
1s Orbital: Plotting Probability Densities vs. r
35:47
2s Orbital: Plotting Probability Densities vs. r
37:46
3s Orbital: Plotting Probability Densities vs. r
38:49
4s Orbital: Plotting Probability Densities vs. r
39:34
2p Orbital: Plotting Probability Densities vs. r
40:12
3p Orbital: Plotting Probability Densities vs. r
41:02
4p Orbital: Plotting Probability Densities vs. r
41:51
3d Orbital: Plotting Probability Densities vs. r
43:18
4d Orbital: Plotting Probability Densities vs. r
43:48
Example I: Probability of Finding an Electron in the 2s Orbital of the Hydrogen
45:40
The Hydrogen Atom VI

51m 53s

Intro
0:00
The Hydrogen Atom VI
0:07
Last Lesson Review
0:08
Spherical Component
1:09
Normalization Condition
2:02
Complete 1s Orbital Wave Function
4:08
1s Orbital Wave Function
4:09
Normalization Condition
6:28
Spherically Symmetric
16:00
Average Value
17:52
Example I: Calculate the Region of Highest Probability for Finding the Electron
21:19
2s Orbital Wave Function
25:32
2s Orbital Wave Function
25:33
Average Value
28:56
General Formula
32:24
The Hydrogen Atom VII

34m 29s

Intro
0:00
The Hydrogen Atom VII
0:12
p Orbitals
1:30
Not Spherically Symmetric
5:10
Recall That the Spherical Harmonics are Eigenfunctions of the Hamiltonian Operator
6:50
Any Linear Combination of These Orbitals Also Has The Same Energy
9:16
Functions of Real Variables
15:53
Solving for Px
16:50
Real Spherical Harmonics
21:56
Number of Nodes
32:56
Section 18: Hydrogen Atom Example Problems
Hydrogen Atom Example Problems I

43m 49s

Intro
0:00
Example I: Angular Momentum & Spherical Harmonics
0:20
Example II: Pair-wise Orthogonal Legendre Polynomials
16:40
Example III: General Normalization Condition for the Legendre Polynomials
25:06
Example IV: Associated Legendre Functions
32:13
The Hydrogen Atom Example Problems II

1h 1m 57s

Intro
0:00
Example I: Normalization & Pair-wise Orthogonal
0:13
Part 1: Normalized
0:43
Part 2: Pair-wise Orthogonal
16:53
Example II: Show Explicitly That the Following Statement is True for Any Integer n
27:10
Example III: Spherical Harmonics
29:26
Angular Momentum Cones
56:37
Angular Momentum Cones
56:38
Physical Interpretation of Orbital Angular Momentum in Quantum mechanics
1:00:16
The Hydrogen Atom Example Problems III

48m 33s

Intro
0:00
Example I: Show That ψ₂₁₁ is Normalized
0:07
Example II: Show That ψ₂₁₁ is Orthogonal to ψ₃₁₀
11:48
Example III: Probability That a 1s Electron Will Be Found Within 1 Bohr Radius of The Nucleus
18:35
Example IV: Radius of a Sphere
26:06
Example V: Calculate <r> for the 2s Orbital of the Hydrogen-like Atom
36:33
The Hydrogen Atom Example Problems IV

48m 33s

Intro
0:00
Example I: Probability Density vs. Radius Plot
0:11
Example II: Hydrogen Atom & The Coulombic Potential
14:16
Example III: Find a Relation Among <K>, <V>, & <E>
25:47
Example IV: Quantum Mechanical Virial Theorem
48:32
Example V: Find the Variance for the 2s Orbital
54:13
The Hydrogen Atom Example Problems V

48m 33s

Intro
0:00
Example I: Derive a Formula for the Degeneracy of a Given Level n
0:11
Example II: Using Linear Combinations to Represent the Spherical Harmonics as Functions of the Real Variables θ & φ
8:30
Example III: Using Linear Combinations to Represent the Spherical Harmonics as Functions of the Real Variables θ & φ
23:01
Example IV: Orbital Functions
31:51
Section 19: Spin Quantum Number and Atomic Term Symbols
Spin Quantum Number: Term Symbols I

59m 18s

Intro
0:00
Quantum Numbers Specify an Orbital
0:24
n
1:10
l
1:20
m
1:35
4th Quantum Number: s
2:02
Spin Orbitals
7:03
Spin Orbitals
7:04
Multi-electron Atoms
11:08
Term Symbols
18:08
Russell-Saunders Coupling & The Atomic Term Symbol
18:09
Example: Configuration for C
27:50
Configuration for C: 1s²2s²2p²
27:51
Drawing Every Possible Arrangement
31:15
Term Symbols
45:24
Microstate
50:54
Spin Quantum Number: Term Symbols II

34m 54s

Intro
0:00
Microstates
0:25
We Started With 21 Possible Microstates
0:26
³P State
2:05
Microstates in ³P Level
5:10
¹D State
13:16
³P State
16:10
²P₂ State
17:34
³P₁ State
18:34
³P₀ State
19:12
9 Microstates in ³P are Subdivided
19:40
¹S State
21:44
Quicker Way to Find the Different Values of J for a Given Basic Term Symbol
22:22
Ground State
26:27
Hund's Empirical Rules for Specifying the Term Symbol for the Ground Electronic State
27:29
Hund's Empirical Rules: 1
28:24
Hund's Empirical Rules: 2
29:22
Hund's Empirical Rules: 3 - Part A
30:22
Hund's Empirical Rules: 3 - Part B
31:18
Example: 1s²2s²2p²
31:54
Spin Quantum Number: Term Symbols III

38m 3s

Intro
0:00
Spin Quantum Number: Term Symbols III
0:14
Deriving the Term Symbols for the p² Configuration
0:15
Table: MS vs. ML
3:57
¹D State
16:21
³P State
21:13
¹S State
24:48
J Value
25:32
Degeneracy of the Level
27:28
When Given r Electrons to Assign to n Equivalent Spin Orbitals
30:18
p² Configuration
32:51
Complementary Configurations
35:12
Term Symbols & Atomic Spectra

57m 49s

Intro
0:00
Lyman Series
0:09
Spectroscopic Term Symbols
0:10
Lyman Series
3:04
Hydrogen Levels
8:21
Hydrogen Levels
8:22
Term Symbols & Atomic Spectra
14:17
Spin-Orbit Coupling
14:18
Selection Rules for Atomic Spectra
21:31
Selection Rules for Possible Transitions
23:56
Wave Numbers for The Transitions
28:04
Example I: Calculate the Frequencies of the Allowed Transitions from (4d) ²D →(2p) ²P
32:23
Helium Levels
49:50
Energy Levels for Helium
49:51
Transitions & Spin Multiplicity
52:27
Transitions & Spin Multiplicity
52:28
Section 20: Term Symbols Example Problems
Example Problems I

1h 1m 20s

Intro
0:00
Example I: What are the Term Symbols for the np¹ Configuration?
0:10
Example II: What are the Term Symbols for the np² Configuration?
20:38
Example III: What are the Term Symbols for the np³ Configuration?
40:46
Example Problems II

56m 34s

Intro
0:00
Example I: Find the Term Symbols for the nd² Configuration
0:11
Example II: Find the Term Symbols for the 1s¹2p¹ Configuration
27:02
Example III: Calculate the Separation Between the Doublets in the Lyman Series for Atomic Hydrogen
41:41
Example IV: Calculate the Frequencies of the Lines for the (4d) ²D → (3p) ²P Transition
48:53
Section 21: Equation Review for Quantum Mechanics
Quantum Mechanics: All the Equations in One Place

18m 24s

Intro
0:00
Quantum Mechanics Equations
0:37
De Broglie Relation
0:38
Statistical Relations
1:00
The Schrӧdinger Equation
1:50
The Particle in a 1-Dimensional Box of Length a
3:09
The Particle in a 2-Dimensional Box of Area a x b
3:48
The Particle in a 3-Dimensional Box of Area a x b x c
4:22
The Schrӧdinger Equation Postulates
4:51
The Normalization Condition
5:40
The Probability Density
6:51
Linear
7:47
Hermitian
8:31
Eigenvalues & Eigenfunctions
8:55
The Average Value
9:29
Eigenfunctions of Quantum Mechanics Operators are Orthogonal
10:53
Commutator of Two Operators
10:56
The Uncertainty Principle
11:41
The Harmonic Oscillator
13:18
The Rigid Rotator
13:52
Energy of the Hydrogen Atom
14:30
Wavefunctions, Radial Component, and Associated Laguerre Polynomial
14:44
Angular Component or Spherical Harmonic
15:16
Associated Legendre Function
15:31
Principal Quantum Number
15:43
Angular Momentum Quantum Number
15:50
Magnetic Quantum Number
16:21
z-component of the Angular Momentum of the Electron
16:53
Atomic Spectroscopy: Term Symbols
17:14
Atomic Spectroscopy: Selection Rules
18:03
Section 22: Molecular Spectroscopy
Spectroscopic Overview: Which Equation Do I Use & Why

50m 2s

Intro
0:00
Spectroscopic Overview: Which Equation Do I Use & Why
1:02
Lesson Overview
1:03
Rotational & Vibrational Spectroscopy
4:01
Frequency of Absorption/Emission
6:04
Wavenumbers in Spectroscopy
8:10
Starting State vs. Excited State
10:10
Total Energy of a Molecule (Leaving out the Electronic Energy)
14:02
Energy of Rotation: Rigid Rotor
15:55
Energy of Vibration: Harmonic Oscillator
19:08
Equation of the Spectral Lines
23:22
Harmonic Oscillator-Rigid Rotor Approximation (Making Corrections)
28:37
Harmonic Oscillator-Rigid Rotor Approximation (Making Corrections)
28:38
Vibration-Rotation Interaction
33:46
Centrifugal Distortion
36:27
Anharmonicity
38:28
Correcting for All Three Simultaneously
41:03
Spectroscopic Parameters
44:26
Summary
47:32
Harmonic Oscillator-Rigid Rotor Approximation
47:33
Vibration-Rotation Interaction
48:14
Centrifugal Distortion
48:20
Anharmonicity
48:28
Correcting for All Three Simultaneously
48:44
Vibration-Rotation

59m 47s

Intro
0:00
Vibration-Rotation
0:37
What is Molecular Spectroscopy?
0:38
Microwave, Infrared Radiation, Visible & Ultraviolet
1:53
Equation for the Frequency of the Absorbed Radiation
4:54
Wavenumbers
6:15
Diatomic Molecules: Energy of the Harmonic Oscillator
8:32
Selection Rules for Vibrational Transitions
10:35
Energy of the Rigid Rotator
16:29
Angular Momentum of the Rotator
21:38
Rotational Term F(J)
26:30
Selection Rules for Rotational Transition
29:30
Vibration Level & Rotational States
33:20
Selection Rules for Vibration-Rotation
37:42
Frequency of Absorption
39:32
Diagram: Energy Transition
45:55
Vibration-Rotation Spectrum: HCl
51:27
Vibration-Rotation Spectrum: Carbon Monoxide
54:30
Vibration-Rotation Interaction

46m 22s

Intro
0:00
Vibration-Rotation Interaction
0:13
Vibration-Rotation Spectrum: HCl
0:14
Bond Length & Vibrational State
4:23
Vibration Rotation Interaction
10:18
Case 1
12:06
Case 2
17:17
Example I: HCl Vibration-Rotation Spectrum
22:58
Rotational Constant for the 0 & 1 Vibrational State
26:30
Equilibrium Bond Length for the 1 Vibrational State
39:42
Equilibrium Bond Length for the 0 Vibrational State
42:13
Bₑ & αₑ
44:54
The Non-Rigid Rotator

29m 24s

Intro
0:00
The Non-Rigid Rotator
0:09
Pure Rotational Spectrum
0:54
The Selection Rules for Rotation
3:09
Spacing in the Spectrum
5:04
Centrifugal Distortion Constant
9:00
Fundamental Vibration Frequency
11:46
Observed Frequencies of Absorption
14:14
Difference between the Rigid Rotator & the Adjusted Rigid Rotator
16:51
Adjusted Rigid Rotator
21:31
Observed Frequencies of Absorption
26:26
The Anharmonic Oscillator

30m 53s

Intro
0:00
The Anharmonic Oscillator
0:09
Vibration-Rotation Interaction & Centrifugal Distortion
0:10
Making Corrections to the Harmonic Oscillator
4:50
Selection Rule for the Harmonic Oscillator
7:50
Overtones
8:40
True Oscillator
11:46
Harmonic Oscillator Energies
13:16
Anharmonic Oscillator Energies
13:33
Observed Frequencies of the Overtones
15:09
True Potential
17:22
HCl Vibrational Frequencies: Fundamental & First Few Overtones
21:10
Example I: Vibrational States & Overtones of the Vibrational Spectrum
22:42
Example I: Part A - First 4 Vibrational States
23:44
Example I: Part B - Fundamental & First 3 Overtones
25:31
Important Equations
27:45
Energy of the Q State
29:14
The Difference in Energy between 2 Successive States
29:23
Difference in Energy between 2 Spectral Lines
29:40
Electronic Transitions

1h 1m 33s

Intro
0:00
Electronic Transitions
0:16
Electronic State & Transition
0:17
Total Energy of the Diatomic Molecule
3:34
Vibronic Transitions
4:30
Selection Rule for Vibronic Transitions
9:11
More on Vibronic Transitions
10:08
Frequencies in the Spectrum
16:46
Difference of the Minima of the 2 Potential Curves
24:48
Anharmonic Zero-point Vibrational Energies of the 2 States
26:24
Frequency of the 0 → 0 Vibronic Transition
27:54
Making the Equation More Compact
29:34
Spectroscopic Parameters
32:11
Franck-Condon Principle
34:32
Example I: Find the Values of the Spectroscopic Parameters for the Upper Excited State
47:27
Table of Electronic States and Parameters
56:41
Section 23: Molecular Spectroscopy Example Problems
Example Problems I

33m 47s

Intro
0:00
Example I: Calculate the Bond Length
0:10
Example II: Calculate the Rotational Constant
7:39
Example III: Calculate the Number of Rotations
10:54
Example IV: What is the Force Constant & Period of Vibration?
16:31
Example V: Part A - Calculate the Fundamental Vibration Frequency
21:42
Example V: Part B - Calculate the Energies of the First Three Vibrational Levels
24:12
Example VI: Calculate the Frequencies of the First 2 Lines of the R & P Branches of the Vib-Rot Spectrum of HBr
26:28
Example Problems II

1h 1m 5s

Intro
0:00
Example I: Calculate the Frequencies of the Transitions
0:09
Example II: Specify Which Transitions are Allowed & Calculate the Frequencies of These Transitions
22:07
Example III: Calculate the Vibrational State & Equilibrium Bond Length
34:31
Example IV: Frequencies of the Overtones
49:28
Example V: Vib-Rot Interaction, Centrifugal Distortion, & Anharmonicity
54:47
Example Problems III

33m 31s

Intro
0:00
Example I: Part A - Derive an Expression for ∆G( r )
0:10
Example I: Part B - Maximum Vibrational Quantum Number
6:10
Example II: Part A - Derive an Expression for the Dissociation Energy of the Molecule
8:29
Example II: Part B - Equation for ∆G( r )
14:00
Example III: How Many Vibrational States are There for Br₂ before the Molecule Dissociates
18:16
Example IV: Find the Difference between the Two Minima of the Potential Energy Curves
20:57
Example V: Rotational Spectrum
30:51
Section 24: Statistical Thermodynamics
Statistical Thermodynamics: The Big Picture

1h 1m 15s

Intro
0:00
Statistical Thermodynamics: The Big Picture
0:10
Our Big Picture Goal
0:11
Partition Function (Q)
2:42
The Molecular Partition Function (q)
4:00
Consider a System of N Particles
6:54
Ensemble
13:22
Energy Distribution Table
15:36
Probability of Finding a System with Energy
16:51
The Partition Function
21:10
Microstate
28:10
Entropy of the Ensemble
30:34
Entropy of the System
31:48
Expressing the Thermodynamic Functions in Terms of The Partition Function
39:21
The Partition Function
39:22
Pi & U
41:20
Entropy of the System
44:14
Helmholtz Energy
48:15
Pressure of the System
49:32
Enthalpy of the System
51:46
Gibbs Free Energy
52:56
Heat Capacity
54:30
Expressing Q in Terms of the Molecular Partition Function (q)
59:31
Indistinguishable Particles
1:02:16
N is the Number of Particles in the System
1:03:27
The Molecular Partition Function
1:05:06
Quantum States & Degeneracy
1:07:46
Thermo Property in Terms of ln Q
1:10:09
Example: Thermo Property in Terms of ln Q
1:13:23
Statistical Thermodynamics: The Various Partition Functions I

47m 23s

Intro
0:00
Lesson Overview
0:19
Monatomic Ideal Gases
6:40
Monatomic Ideal Gases Overview
6:42
Finding the Parition Function of Translation
8:17
Finding the Parition Function of Electronics
13:29
Example: Na
17:42
Example: F
23:12
Energy Difference between the Ground State & the 1st Excited State
29:27
The Various Partition Functions for Monatomic Ideal Gases
32:20
Finding P
43:16
Going Back to U = (3/2) RT
46:20
Statistical Thermodynamics: The Various Partition Functions II

54m 9s

Intro
0:00
Diatomic Gases
0:16
Diatomic Gases
0:17
Zero-Energy Mark for Rotation
2:26
Zero-Energy Mark for Vibration
3:21
Zero-Energy Mark for Electronic
5:54
Vibration Partition Function
9:48
When Temperature is Very Low
14:00
When Temperature is Very High
15:22
Vibrational Component
18:48
Fraction of Molecules in the r Vibration State
21:00
Example: Fraction of Molecules in the r Vib. State
23:29
Rotation Partition Function
26:06
Heteronuclear & Homonuclear Diatomics
33:13
Energy & Heat Capacity
36:01
Fraction of Molecules in the J Rotational Level
39:20
Example: Fraction of Molecules in the J Rotational Level
40:32
Finding the Most Populated Level
44:07
Putting It All Together
46:06
Putting It All Together
46:07
Energy of Translation
51:51
Energy of Rotation
52:19
Energy of Vibration
52:42
Electronic Energy
53:35
Section 25: Statistical Thermodynamics Example Problems
Example Problems I

48m 32s

Intro
0:00
Example I: Calculate the Fraction of Potassium Atoms in the First Excited Electronic State
0:10
Example II: Show That Each Translational Degree of Freedom Contributes R/2 to the Molar Heat Capacity
14:46
Example III: Calculate the Dissociation Energy
21:23
Example IV: Calculate the Vibrational Contribution to the Molar heat Capacity of Oxygen Gas at 500 K
25:46
Example V: Upper & Lower Quantum State
32:55
Example VI: Calculate the Relative Populations of the J=2 and J=1 Rotational States of the CO Molecule at 25°C
42:21
Example Problems II

57m 30s

Intro
0:00
Example I: Make a Plot of the Fraction of CO Molecules in Various Rotational Levels
0:10
Example II: Calculate the Ratio of the Translational Partition Function for Cl₂ and Br₂ at Equal Volume & Temperature
8:05
Example III: Vibrational Degree of Freedom & Vibrational Molar Heat Capacity
11:59
Example IV: Calculate the Characteristic Vibrational & Rotational temperatures for Each DOF
45:03
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Lecture Comments (1)

0 answers

Post by Van Anh Do on December 14, 2015

Can an electron transition from v''=0 to v'=0 of E0 to E1? Thank you.

Electronic Transitions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Electronic Transitions 0:16
    • Electronic State & Transition
    • Total Energy of the Diatomic Molecule
    • Vibronic Transitions
    • Selection Rule for Vibronic Transitions
    • More on Vibronic Transitions
    • Frequencies in the Spectrum
    • Difference of the Minima of the 2 Potential Curves
    • Anharmonic Zero-point Vibrational Energies of the 2 States
    • Frequency of the 0 → 0 Vibronic Transition
    • Making the Equation More Compact
    • Spectroscopic Parameters
    • Franck-Condon Principle
  • Example I: Find the Values of the Spectroscopic Parameters for the Upper Excited State 47:27
  • Table of Electronic States and Parameters 56:41

Transcription: Electronic Transitions

Hello, welcome back to www.educator.com, welcome back to Physical Chemistry.0000

In the last four lessons, we have been discussing vibration spectroscopy, rotational spectroscopy, vibration rotation spectroscopy.0004

Today, we will talk about electronic transitions.0011

Let us get started.0016

Let us go ahead and do blue today.0020

Diatomic molecules absorbing radiation in the visible ultraviolet range.0032

They experience transitions to excited electronic states.0053

We call these electronic transactions.0072

We send the electronic up to a higher level of energy.0081

Electronic transmissions, remember when we did vibrational transition,0087

we had just rotational transitions that is the microwave range.0094

The infrared range, we have vibrational transitions.0098

But with the vibrational, you got the rotational also.0101

With electronic transitions, you get the vibrational and rotational also.0104

Electronic transitions are accompanied by both vibrational and rotational transitions.0111

In general, the rotational transitions we are not going to worry about because there are reasonably insignificant.0136

It is the vibration transitions that we are going to be concerned with.0141

Each electronic state has its own potential energy curve.0146

That is what you see here.0163

This is the ground state, this is the first excited state, or they say E1 could be any of the excited states.0164

Anything above the ground state has a potential energy curve that has is a set of vibrational energy states.0171

And the excited electronic state, this one right here on top, it has its own set of vibrational energy levels.0180

Each electronic state has its own potential energy curve.0188

They do not necessarily need to look alike.0194

One is not a copy of the other.0201

It might look like it here but they are not.0204

Let me see, should I do it on this page?0213

The total energy than of the molecule, the diatomic molecule leaving off the transitional energy,0223

leaving off the energy of motion.0237

We are just going to be concerned with the electronic energy, the vibrational energy, the rotational energy.0240

The molecule is a total, it = the electronic energy + the vibrational energy + the rotational energy.0246

That simple.0267

E total = electronic + ν sub E × R + ½ - X sub E ν sub E × R + ½².0274

This is the vibrational energy under the anharmonic oscillator + B × J × J + 1 - D × J² × J + 1².0295

This is the rotational energy in non rigid rotator that accounts for the centrifugal distortion.0320

This one right here, E electronic is the energy at the minimum of the potential energy curve of the potential energy curve.0331

In other words, in the ground vibrational state is that energy right there.0362

Whatever that happens to be.0367

In the first excited state, it is that right there.0368

It is the energy at the minimum of the potential energy curve.0372

That is what the electronic energy is.0376

Transitions between the vibrational states during electronic transitions,0388

in other words during the transition from one electronic state to another, are called vibronic transitions.0405

In general, we will ignore the rotational term.0429

It makes our equations a little bit easier to deal with.0442

Ignore the rotational terms in the above equation.0449

The reason is because on the scale of electronic energies, when we are talking about 10⁻¹⁸ J,0467

the rotational energies are insignificant.0486

The rotational energies 10⁻²⁴ J are very small.0494

For the most part, we can just ignore them.0504

We have E total = electronic E × R + ½ ν sub E × R + ½².0532

This is the total energy of molecule.0557

Our selection rule for the vibronic transitions, δ R = + or -1, + or -2, + or -3, and so on.0562

You can jump from 0 to 1, 0 to 2, 0 to 3.0586

You can jump from 1 to 2, 1 to 3, 1 to 4, 1 to 5.0595

You have to go just from one level at the time.0601

As we said before, at normal temperatures most molecules are in the R = 0 vibrational state.0608

Most of the vibronic transitions would happen from the ground state, the R = 0.0639

It happen from the 0 vibrational state.0648

Most of the vibronic transitions originate there.0654

In other words, R = 0.0670

You are going to have the 0 to 1 transition.0673

You are going to have 0 to 2.0676

You are going to have 0 to 3.0678

You are going to have 0 to 1, 0 to 2, 0 to 3, 0 to 4.0682

Those are the transitions that are going to take place.0690

As we said before, each electronic state has its own potential energy curve.0705

That is the potential energy curve for the upper state.0710

That is the potential energy curve for the lower state.0714

In general, the ground state.0717

It is not necessary to write this down.0733

I will just go ahead and tell you.0735

The upper states are usually designated with a single prime.0738

You are going to see it right there.0762

The lower energy states, they are usually designated with a double prime.0768

Personally, I prefer the designations U for upper and L for lower.0786

The primes tend to confuse me especially when you start taking the difference between energy levels.0791

All of the sudden you have got single primes and double primes.0796

We have already seen symbolism heavy.0799

Primes and double primes, it is just a personal thing.0802

In general, in your book more than likely you are going to see single primes and double primes.0805

Single for upper and double for lower.0809

For my notation, I will just use U and L.0811

Sometimes I will go ahead and use 0 for the ground state.0815

We are going to investigate the vibronic transitions from the lower R value of 0 to the upper R value 1, 2, 3, and so on.0821

We are going to go from the ground vibrational state up.0858

In the upper electronic state, we would be hitting the 1, 2, 3, 4, 5 vibrational states.0861

That is what we are going to actually look at.0867

Each transition 0 to 1, 0 to 2, 0 to 3, 0 to 4, represents a line in the spectrum.0869

Each transition is a line in the spectrum.0878

Actually in the case of electronic spectrum, you are not going to see necessarily a line.0889

What you are going to see is a peak.0893

In electronic spectrum, under visible UV spectroscopy, you are going to see just a bunch of peaks.0895

Those represent the lines, the absorption, the transmission, things like that.0905

This set of transitions 0 to 1, 0 to 2, 0 to 3, 0 to 4, 0 to 5, it is called a progression.0912

This collection of transitions 0 to 1, 0 to 2, 0 to 3, there a lot of them by the way, not just 1, 2, 3, 4, 5.0930

We are talking 60 or 70 sometimes.0943

This collection of transitions is called a progression.0947

0 to 1, 0 to 2, 0 to 3, 0 to 4, 0 to 5, that is a question that we see on the actual spectrum that we take.0961

Each one of those peaks represents a vibronic transition.0972

When you look in your spectrum, that is what you are going to be seeing.0976

We have the energy total is equal to the electronic energy + that × R + ½ - X sub E ν sub E R + ½², that gives us the energy.0981

The frequencies that we observe in the spectrum, the peaks that we see,1006

the frequencies we see in the electronic spectrum of a particular diatomic molecule is going to be,1012

We will call it ν observed.1024

Ν observed, I will specify that we are talking about vibronic, although we should know that because that is the lesson we are in.1029

Again, the observed frequency that we see in the spectrum is going to be the difference between one energy level and another.1037

The upper - the lower.1043

It is going to be the total energy of the upper level - the total energy of the lower level.1045

In this particular case, it is the ground state 0.1055

Sometimes, I will use 0 instead of L but again we know what it is going to be upper – lower.1059

The energy of the upper, that is that one.1067

The energy of the lower, that is that one.1069

Let us go ahead and put these values in.1072

We are moving the lower level, it is the ground state.1074

It is going to be R = 0 vibrational state.1078

In this particular case, R is going to equal 0.1086

A lot of symbolism here, I apologize.1090

The electronic energy, the upper state + ν sub E upper × R upper + ½ -1099

X sub E upper ν sub E upper × R upper + ½² - the lower energy.1120

This is the upper energy - the lower energy.1137

The lower energy R value is equal to 0.1140

That is going to be the energy electronic 0 state + ½ ν 0 – X sub E 0 ν sub E.1143

I will stick with upper and lower so I’m not going to use the 0.1170

This is going to be the energy of the electronic of the lower state + ν sub E lower state × ½ - R 0 ½² is ¼.1173

Let us go ahead and do a little bit of algebra here.1215

I’m going to separate out the terms, multiply these out, put some terms together.1219

It is going to be, our ν observed for vibronic is going to be the electronic energy of1227

the upper state - the electronic energy of the lower state.1235

That takes care of the electronic energies.1240

+ ν sub E upper × R in the upper + ½ ν sub E of the upper.1244

It is not necessary for me to actually go through this algebra.1257

I could just write down the equation, but I think it is nice to go through them.1264

It is part of your scientific and your mathematical literacy here with physical chemistry1268

in quantum mechanics spectroscopy, whatever it is.1277

I just think it is nice to go through the mathematics, it makes it a lot more clear1280

instead of dropping down like a stone.1284

Some equation being dropped in your lab.1286

All we are doing is we are taking the upper energy - lower energy.1288

The rest is just very very careful algebra with this insane symbolism.1291

I apologize for that.1295

- X sub E upper ν sub E upper × R upper² – X sub E upper ν sub E upper.1300

That is crazy, I have no idea how they kept all of it straight all those years.1320

In some of the problems, you might be asked to derive these equations, - ½ ν sub E lower + ¼.1331

The negative × negative is positive.1351

¼ X sub E upper ν sub E upper.1355

Our final equation comes down to this.1366

When I put some things together here, I'm going to get ν observed equal1372

to the upper electronic state - the lower electronic state +,1381

I'm going to combine different terms, ½ ν sub E upper - 1/4 X sub E upper ν sub E upper - ½ ν sub E lower1395

- ¼ X sub E lower ν sub E lower + ν sub E upper × R - X sub E upper ν sub E upper.1421

This X sub E ν sub E, this is a single parameter.1445

We just write it together.1447

× R × R + 1.1450

This equation right here, this gives us the frequency of the line that we see.1454

Let us break this down even further.1461

The frequency of the transition 0 to 1, 0 to 2, things like that.1463

1, 2, 3, 4, 5, 6, there are 6 terms in this equation.1475

The difference of the first two terms, in other words the E upper - E lower electronic is sometimes called T sub E.1490

It is the difference of the minima of the two potential energy curves.1540

In other words, it is going to be this energy - this energy.1565

That is TE.1571

If you want to put a little line there, a little line here, go like this.1575

The difference between been the minima.1582

The third and fourth terms of the equation that we just had,1585

they are just the anharmonic 0 point vibrational energies of the two states.1598

In other words, the third term in that equation, that represents the energy of that level.1623

The fourth term represents the energy of that level.1629

The 0 point energy of the two states.1635

The ground state, in other words.1638

For the first 4 terms taken together, the difference between the energy minima and1657

the difference between these two ground state energies,1664

Let me go ahead and write this down.1669

The first 4 terms taken together represents the frequency of the 0 to 0 vibronic transition.1671

The transition that goes from this level 0 in the lower electronic state to the R = 0 of the upper electronic state.1715

That, the frequency of that transition, that is what those 4 terms taken together represent.1727

The E upper, the E lower, and that third term and that 4th term.1735

We often symbolize this as ν 00 or sometimes ν 0 to 0, with a little arrow.1741

Some variation, thereof.1754

You put a comma, you do not put a comma, it is up to you.1756

Again, it represents the vibronic transition from the R = 0 state, ground state to the R = 0 state of the upper state.1761

The ground state of the upper electronic state, that is what that represents.1770

If we use the symbolism, either that one or this one, if we use the symbolism1776

to make our 6 term equation more compact, we get the following.1796

We get that the observed frequency of transition is equal to this ν 00 + ν sub E upper × R – X sub E upper ν sub E upper.1810

I think I should just put that is as 1, that is okay.1831

× R × R + 1.1834

Here, R is equal to 1, 2, 3, and so on.1838

Here, R is the vibration quantum number of the upper state, the one in that electronic state.1845

If we set R = 0 in this equation, we get the observed frequency of the 0 to 0 transition.1871

That is what we actually get.1902

The 0 to 0 vibronic transition, when we set R = 0 in this equation.1907

Let us see what we have got here.1925

Let us go ahead and go to blue.1933

Let me write it over here.1941

This is very very important, please make sure you understand that each electronic state, each potential energy curve,1944

it has its own set of spectroscopic parameters.1966

In other words, it has its own ν sub E, X sub E, ν sub E.1993

It has its own B sub E and so on.2003

It is very important.2006

Those are the parameters that we are actually going to be solving for many of the problems.2007

Let us talk about this thing called the Franck-Condon principle, which actually is what this image really represents.2013

We see a lower electronic state, we see an upper electronic state.2019

We should have the vibration levels but all the vibration levels, we also have the actual wave functions.2022

These wave functions right here.2031

This image shows the wave functions.2034

If we were to take the square of the wave function, ψ², what we would get is the probability density.2037

The only difference between the wave function of a probability density is the same exact picture.2051

All of these curves, they would all be above the axes.2055

In other words, like this one right here, it would be curved up.2059

Everything would be above the axis because you square something, you end up getting something positive.2066

Now, let us talk about the Franck Condon principle.2073

Let me go ahead and do this in red.2085

We see that each electronic state has its own potential energy curve.2090

The minima of each state, the minima of the various states, in this particular case2098

I have 2 electronic states but I will say various because it is more than one electronic state, many of them.2114

The minima of the various electronic states do not necessarily lie on top of each other.2120

In other words, you notice this minima is right here.2162

This minima is right here.2166

There is a difference between them.2169

That difference is very important and you will see in a minute.2172

Do not necessarily lie on top of each other.2174

In other words, the R sub E for this state is different than the R sub E for this state.2180

Again, we know that already, they have different parameters.2189

The Franck Condon principle says that the electronic transitions happens very fast2193

because the electronic transitions happen in time frames that are instantaneous,2216

compared to the motion of the nuclei of the atoms involved.2241

Let us go ahead and say that.2269

The much more massive nuclei, in other words the electrons can move a lot faster and2274

move to other states a lot faster than the nuclei can actually adjust to the new state.2280

That is what is happening, the electron is so much smaller than a nucleus.2286

When it moves to a higher electronic state, it is there in a minute.2291

It is going to take a lot longer, relatively speaking, for the nuclei to adjust to that new electronic state2295

compared to the motion of much more massive nuclei.2302

Because of that, we can represent vibronic transitions as vertical lines.2309

This is one electronic state, this is another electronic state.2314

It is already been adjusted.2317

One electron actually move from one state to the other.2321

It is just going to jump straight up.2325

This electronic state, these wave functions represent the different vibrational levels of that state.2329

Here, the wave functions represent the different vibration levels of that state.2338

When electron makes a jump to a higher electronic state, it is just going to jump time wise because it happened quickly.2342

Relative to the motion of the nuclei, we can represent them as just a vertical leap.2348

Graphically, we represent it as just a straight vertical line from the ground state .2353

Therefore, on a diagram like the one above, like the ones that you see in your book,2367

the transition from the lower state to the upper electronic state is represented vertically.2383

When we represent a vibronic transition, we are representing it vertically.2397

There is a state, there is another state, it is going to go this way.2401

Where it lands have a relative, based on the wave function is the extent to which we are actually going to see that line of the spectrum.2405

Therefore, on the diagram, the transition is represented vertically.2415

Let me write up here.2426

Each curve in these diagrams shows the wave function for each value of R, the probability density ψ² look the same.2433

Except all the shadings are above the X axis.2476

Nothing that we do not know from our previous work in quantum mechanics.2486

What the Franck Condon principle does is, it gives us the relative intensities of the vibronic transitions.2495

Let us say the vibronic transition lines.2535

The lines that we see, some of them are going to be very strong lines.2537

Some of them are going to be very weak lines.2541

The strength and the weakness of those lines depends on the probability density of the electron2543

is going to be in that particular state there.2551

Here is what is going on.2563

We would be going from, let us say to 0 to 1 transition.2565

You look over here, the 0 transition from the ground state R = 0 up to level 1.2569

First of all, notice that this particular transition.2582

Because this in this particular image, for this state E1 and E0, the R value of E1 is if a significantly larger than the RE value of the E sub 0.2587

They are not on top of each other.2606

The transition that takes place, the vibronic transition we said it was vertically,2607

it actually ends up passing the one level and go straight to the two level.2611

At the two level, notice where it hits.2617

It actually hits where the density is rather high.2620

In this particular case, we might not even see a line for the 0 to 1 transition.2623

Because it does not even touch the potential energy curve from here, the place of maximum density2629

and the 0 vibration state for the lower electronic state to a place of maximum density.2636

For the upper actually ends up hitting for the level 2, that is where it hits.2642

That particular line is going to be very intense.2649

Maximum intensity, maximum intensity.2652

Here, you may or may not see a line for the 0 to 1 transition.2656

You are definitely not going to see one for the 0 to 0 transition.2660

You may or may not.2662

Again, a little bit of density out here but it is outside of the potential energy curve so you might not see anything at all.2665

What about the 0 to 3 transition?2672

The 0 to 3 transition, if we go straight up, we hit right about there.2675

It is a place of minimum density.2682

We will still probably see one but it may not be very strong.2684

It may not be very intense.2689

How about the 0 to 4 transition?2691

Let us see, where 0 to 4?2693

Over here, 0 to 4 transition we would go straight up.2695

Sorry about that, the 0 to 4 transition straight up.2699

It is probably going to be a little bit more intense than the 0 to 3 but not quite as intense as the 0 to 2.2703

And that is what the Franck Condon principal says.2709

When you have one electronic state, you have another electronic state,2712

the transitions are going to take place vertically on these diagrams.2715

The intensity of the transition 0 to 1, 0 to 2, 0 to 3, depends on where you are actually going to hit2719

maximum probability density, minimum probability density, or somewhere in between.2727

You are just going to get a series of lines that have different intensities.2733

That is all the Franck Condon principle.2739

Let us see here.2745

Let me remind you.2752

It can happen that the upper states R sub E is significantly larger than the R sub E for the lower state.2757

In other words, the upper state can lie much further, not over but in a shifted away from the lower state,2791

such that the 00 transition may not even appear.2804

Sometimes the 0 to 1, 0 to 2 transitions do not even appear.2823

Sometimes the first transition that you see in the spectrum is maybe 0 to 3, 0 to 4, 0 to 5, and so on.2826

The relative intensities of each of those lines is going to depend on the probability density at that particular position.2833

Let us go ahead and do example and see if we can make sense.2844

The following data table lists the observed frequencies of the first 3 vibronic transitions of hydrogen gas2850

to a certain excited electronic state.2857

We see a line at 121 to 76 inverse cm for the 0 to 0 transition.2863

We see a line at 123 to 70 for the 0 to 1 transition.2869

And we see 124 to 438 for the 0 to 2 transition.2874

In this case, we do see 3 lines.2878

We do not know what the relative intensities are.2879

At this point, that is a separate problem, I do not know.2882

We see the 0 to 0, 0 to 1, 0 to 5 vibronic transitions.2885

These are the frequencies that we see, that we observe on the lines on the spectrum.2889

Use this data to find the values of the spectroscopic parameters, ν sub E upper and X sub E ν for the upper excited state.2894

We are going from 0 to 0, 0 to 1, 0 to 2.2907

We want you to use this information, these 3 lines on this electronic spectrum to actually2910

find spectroscopic parameters for the upper electronic state, and this is how we do it.2917

Let me see our equation.2926

Let me go ahead and do this in red, I think.2928

I’m getting really tired of writing here.2935

I apologize if my writing is sloppier.2937

Our equation for the observed frequencies, the vibronic transitions is ν observed is equal to ν 00 +2940

ν sub E upper × R – X sub E upper ν sub E upper × R × R + 1.2955

Let us go ahead and take R = 0, 1, 2.2968

When R is equal to 0, it represents the 0 to 0 transition.2971

In other words, just the ν sub 00.2975

Let us go ahead and do the 0 to 0 transition.2979

The 0 to 0 transition, let me actually write down each one so we have everything.2981

When R is equal to 0 that represents the 0 to 0 vibronic transition.2989

Our ν observed is going to equal ν sub 00 +, if R is 0 this term is 0.2996

If R is 0, this term is 0. 0 + 0, we end up with ν observed = ν sub 00 that is equal to 120,176.3005

This one of the equations that we want.3025

We will call it equation 1.3028

Let us go ahead and deal with the R = 1 case.3032

This represents the transition from 0 to 1.3035

In this particular case, ν observed is equal to ν 00 +,3038

We are putting 1 now, R into the equation.3046

+ ν sub E × 1 - X sub E upper ν sub E upper × 1 × 1 + 1.3051

We end up with ν observed = ν 00 + ν sub E upper -2 X sub E upper ν sub E upper.3066

This one was equal to 122,370 inverse cm.3084

This is our second equation that we have.3090

Let us go ahead and find the R = 2.3096

This represents the transition from the 0 to 2 vibronic line.3100

Here we have ν observed = ν 00.3107

Ν sub E upper × 2 – X sub E upper ν sub E upper × 2 × 2 + 1.3116

The equation that we get is ν 00 + 2 × ν sub E upper -6 × X sub E ν sub E.3133

Upper upper ~ ~, and that one, the table said is 124,438 inverse cm.3147

In order to solve this, I have 3 equations and a couple unknowns.3158

I'm going to go ahead and take equation number 2.3162

This is this one right here, it is going to be equation number 3.3168

I’m going to take the equation 2 - equation 1.3173

When I take equation 2 - equation 1, I end up with the following.3180

I end up with ν sub E upper -2 X sub E upper ν sub E upper is equal to 2194 inverse cm.3190

We will call this one equation A.3207

When I take equation 3 - equation 1, I end up with 2 ν sub E upper -6 X sub E ν sub E both upper, both ~, and I end up with 4262.3210

This one is going to be my equation B.3242

I’m going to solve equation A and equation B simultaneously.3247

I'm not going to keep writing out these X sub E and ν sub E stuff.3252

I’m just going to call it S and T.3258

I’m going to let S equal to this ν sub E upper and I’m going to let T = this X sub E upper ν sub E upper.3261

Remember that is a single a parameter taken together.3273

What I end up with is the following equation.3278

I get S -2 T = 2194 and I get 2S - 6T = 4262.3280

2S - 4T multiply the top by 2, I get 4388.3295

I will not do this for you but what the hell.3303

-6 T = 4262.3306

I subtract and I end up with 2T = 126.3312

T = 63.3318

I get S - 2 × 63 is equal to 2194.3325

I want to make sure my numbers are right here.3343

I get S is equal to 2320.3345

There we go, we said that S was equal to ν sub E, that is equal to 2320 inverse cm.3351

This is ν sub E upper.3360

That is what we are doing. We are finding the parameters for the upper state and T is equal to X sub E upper ~ ν sub E upper ~.3363

It is a single parameter, that is equal to 263 inverse cm.3374

There you go.3382

We finished this problem, I thought you guys might like to see what a particular table of parameters of states actually looks like,3385

if you happen to be interested.3395

If not, not a big deal.3397

If so, this is what it looks like.3398

This is from the NIST website, the National Institute of Standards and Technology.3403

They have a bunch of databases, a bunch of spectroscopic databases, all kinds of things.3409

You should check it out.3416

If you want to see for yourself, basically what you are going to do is you are going to go to,3418

web book.NIST.gov/chemistry.3428

Under general search, click formula or however you want to search.3444

I generally just click formula.3456

Enter molecule in the box on line 1.3462

Check off the box that says constants of diatomic molecules and click search.3478

After that, you are going to scroll down to give you some information and they will give you this very long table.3502

I have only taken a section of this table and they will go all the way down.3510

Scroll down until you see the table and we are interested in the ground state, it is at the bottom of the table.3515

This is the bottom of the table photograph that I actually took.3538

The ground state is going to be represented by something like this.3548

You are going to see an X, you are going to see this singlet sigma +,3551

Do not worry about the term symbol for the electronic state,3558

I will be explaining what those mean in subsequent lessons but you want to look for this X and3561

you want to see this 0 here for the P sub E.3566

It is the ground state electronic energy, we set that equal to 0.3572

Notice, the first column, this is for the NIST website, this table is ω E.3579

For our purposes, this is our ν sub E.3585

Ν sub E, ω E, in this particular table you will also see it with an ω.3589

That is 29946.3594

This O sub E X sub E, this is the X sub E ν sub E.3598

That is that for the ground state.3605

Do not worry about that, here the B sub E that is the rotational constant.3609

There is the α sub E, that was the constant that had to do with the vibration rotation interaction.3614

Here is the dissociate energy.3623

Do not worry about that, do not worry about that.3626

Here is the R sub E, the equilibrium bond length.3630

This is the transition as represented.3633

And here is the ν sub 00.3636

Very important, that was the difference between energies of the ground vibrational state in lower electronic state3640

and the ground vibrational state of the upper electronic state.3646

That is what those columns mean.3650

For the states are concerned, here is the ground state, you might jump up to let us say that excited electronic state.3653

You have a whole different set of parameters for each electronic state, for each potential energy curve.3665

I hope that helps.3678

This is not something that you need, most of the information is going to be provided for you in your problems.3680

You have tables in your books but I figured if you want to see, you can go ahead and see for yourself.3684

Thank you so much for joining us here at www.educator.com.3689

We will see you next time, bye.3692

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