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Raffi Hovasapian

Raffi Hovasapian

Vibration-Rotation

Slide Duration:

Table of Contents

I. 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
II. 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
III. 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
IV. 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
V. 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
VI. 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
VII. 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
VIII. 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
IX. 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
X. 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
SchrÓ§dinger 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
SchrÓ§dinger 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 SchrÓ§dinger 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
XI. 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
XII. 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
XIII. 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
XIV. 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
XV. 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
XVI. 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
XVII. 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
XVIII. 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
XIX. 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
XX. 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
XXI. 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
XXII. 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
XXIII. 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
XXIV. 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
XXV. 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 (3)

2 answers

Last reply by: Kaye Lim
Sat Apr 8, 2017 3:06 PM

Post by Kaye Lim on April 5, 2017

Greeting sir,

You said around 5:20: 'Most of the information that we get from spectroscopy, we actually get from electronic spectroscopy. Electronic spectroscopy allows us to, it is difficult to analyze but everything that we need is there. It gives us information on electronic states, on vibrational states, on rotational states. The rotational spectra, the vibrational spectra tend to be easier, but they do not give is as much information.'

-a UV/Vis spectrum that I obtain from running UV/Vis spectroscopy on a sample gives me maximum absorption wavelength of that sample. And then I can create calibration curve and quantify the concentration of the sample. That is my experiences with UV/Vis spectroscopy which I think of it mostly as a quantitative method.

-From IR spectroscopy, it is more of a qualitative method since I know what functional group presented in my sample. I know IR could also be used as quantitative method as well.
-So to me, I think IR gives me more information then UV/Vis spectroscopy. Why did you say UV/Vis spectroscopy or electronic transitions give more information than IR spectroscopy? Do you mean when you do very high resolution UV/Vis spectroscopy inwhich you can zoom in the UV/Vis absorption peak to get sub-peaks of vibrational transitions and sub-subpeaks of rotational transitions?
-If that is the case, then what is the difference between vibrational and transitional transitions in high energy UV/Vis region compared to those in IR region? are they vibrational and transitional transitions between different electronic states instead of in the same electronic state as in the case of IR?

Vibration-Rotation

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
  • Vibration-Rotation 0:37
    • What is Molecular Spectroscopy?
    • Microwave, Infrared Radiation, Visible & Ultraviolet
    • Equation for the Frequency of the Absorbed Radiation
    • Wavenumbers
    • Diatomic Molecules: Energy of the Harmonic Oscillator
    • Selection Rules for Vibrational Transitions
    • Energy of the Rigid Rotator
    • Angular Momentum of the Rotator
    • Rotational Term F(J)
    • Selection Rules for Rotational Transition
    • Vibration Level & Rotational States
    • Selection Rules for Vibration-Rotation
    • Frequency of Absorption
    • Diagram: Energy Transition
    • Vibration-Rotation Spectrum: HCl
    • Vibration-Rotation Spectrum: Carbon Monoxide

Transcription: Vibration-Rotation

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

Today, we are going to start on the next major broad topic of Physical Chemistry which is molecular spectroscopy.0004

We have done the thermodynamics, we have done the quantum mechanics, and now, we are going to bring0013

the quantum mechanics to bare and talk about probably the most important topic for practicality0016

is concerned for the chemist because pretty much everything you do as a chemist is going to be some spectroscopic technique.0024

We will be discussing the theory behind molecular spectroscopy, let us get started.0031

Molecular spectroscopy studies the interaction between a matter and electromagnetic radiation.0039

Molecular spectroscopy studies the interaction of molecules with electromagnetic radiation.0053

We get molecules with certain amount of radiation in different regions of the electromagnetic spectrum and we see what happens.0079

That information gives us most of the information that we have about what is happening inside the molecules and how molecules behave.0085

We will be concerned with three regions of the E and M spectrum.0095

We are going to be concerned with the microwave region.0115

Microwave region is related to rotations.0120

When a molecule is hit with microwave radiation, there are changes in the rotational quantum state of the molecule.0125

And we are also going to be concerned with the infrared, the infrared vibrations.0133

When a molecule is hit with some infrared radiation, changes in the vibration levels of the molecules take place.0138

We have already done a fair amount of IR spectroscopy from your course in organic chemistry.0147

And then of course, the last is visible and ultraviolet.0152

Visible and ultraviolet range electronic transitions, that is when it is in this range,0157

the energy of the visible and ultraviolet range, where electrons actually move to higher states themselves, electronic states.0165

Rotation vibration electronic, rotational spectroscopy, vibration spectroscopy, and electronic spectroscopy.0175

Let me see how I want to do this.0184

The absorption of microwave radiation, as we just said, we have rotational transitions.0202

Now in the IR, not only do we have vibrational transitions but accompany those vibrational transitions there are also rotational transitions.0212

I’m going to say vibrational + rotational transitions.0224

As you would expect in the visible UV range, we have not only electronic transitions0232

but there is much energy there that that energy causes an electronic transition,0240

it causes vibrational transitions and rotational transitions.0244

+ vibrational + rotational transition, all transitions take place when you are hitting it with energy and the visible UV range.0249

Most of the information that we get from spectroscopy, we actually get from electronic spectroscopy.0263

Electronic spectroscopy allows us to, it is difficult to analyze but everything that we need is there.0268

It gives us information on electronic states, on vibrational states, on rotational states.0275

The rotational spectra, the vibrational spectra tend to be easier, but they do not give is as much information.0280

Electronic spectra give us all the information that we want.0288

Let us see, the frequency of the absorb radiation comes from this δ E = H ν,0297

Planck's constant × the frequency is a change in energy from one energy state to another.0318

If we solve for the frequency, what you would end up with is the frequency of absorption is going to equal0324

the change in the energy between the 2 states divided by Planck's constant,0330

or we can say the energy of the upper state - the energy of the lower state divided by Planck's constant.0335

That gives us the actual frequency that we observe in the spectra.0345

That is what we are going to see on the spectrum.0348

When we see a line of the spectrum or a peak, it is this number right here.0350

That is what that is, it come from this relation.0356

What we are going to do, we would use the energies for the rigid rotator, the harmonic oscillator,0359

and electronic energies, to find this difference.0365

Because we want to find the equation for what is the absorption number.0369

In general, spectroscopy frequencies, I will put in parentheses.0376

Frequencies are listed in something called wave numbers, we have seen them before.0387

Frequencies are given in wave number.0391

That is what we would be working in,0394

are given in wave numbers which is just the inverse of the wavelength to inverse cm.0396

In general, it is going to be inverse cm.0404

The wave number, anything that is in a wave numbers is going to have a ~ over it.0408

That is equal to 1/ λ, or my preference it is equal to the actual frequency0412

that we got from the other equation just divided by the speed of light.0420

If you get a frequency, if you divide that frequency by the speed of light, you are going to end up getting your wave number.0424

We said transitions between vibrational states are accompanied by transitions and rotational states as well.0431

The following discussion is going to apply to diatomic molecules.0480

We will discuss polyatomic molecules but for right now,0484

we are just going to talk about diatomic molecules, homo nuclear and hetero nuclear.0487

The following discussion and for several more lessons, the following discussion applies to diatomic molecules.0495

Let us begin with the energy of the harmonic oscillator.0514

The energy of the harmonic oscillator HO is energy, that space on the quantum number R,0518

the vibrational quantum number is equal to H ν × R + ½, or R takes on the values 0, 1, 2, and so on.0532

When R is equal to 0, we have ½ H ν, that is the energy of the ground state.0545

When the quantum number is 0, put it in the equation for the energy which gives the ground state energy.0551

We see that there is always some vibrational energy, it is never a 0.0557

Where ν, this frequency, is equal to 2 π × the force constant of the molecule divided by the reduced mass ^½.0564

And this ν is called the fundamental vibration frequency.0583

You will sometimes see it as ν sub 0, something like that.0604

K is the force constant of the bond and how springy the bond is.0611

Is it really tight or is it really loose?0621

And μ is the reduced mass, we have seen the reduced mass before.0625

I just want to make sure that we understand what all of the parameters are.0632

There are selection rules for vibrational transition.0638

The selection rules for vibrational transitions and the selections rules are that δ R = + or -1.0642

In other words, if it is going to make a transition from one vibrational state to another,0660

it is going to go from 1 to 2, 2 to 3, 3 to 4, or 4 to 3, 3 to 2, 2 to 1.0664

In the case of emission, absorption up emission down, as far as the harmonic oscillator is concerned,0669

it is only going to go 1 or 2 steps.0677

I’m sorry, 1 step up or 1 step down.0681

Its not going to go from 1 to 5, from 1 to 4.0683

If it is going to make a transition from 1 to 5, it is going to pass to 2, 3, 4, and then to 5.0685

Those are the transitions that are allowed.0691

The other transition rule is the dipole moment of the molecule must vary during a vibration.0694

I would not worry too much about this idea of the selection rule.0715

Now the dipole moment is related to the selection rule, this is often called gross selection rule.0722

The name itself does not really matter.0728

The dipole moment of the molecule must vary during a vibrational transition.0729

This does not mean that the molecule has to have a permanent dipole.0737

It can or cannot have a permanent dipole, in the case of molecule like hydrogen chloride, it has a permanent dipole.0741

In the case of a molecule like N2, they are both the same.0748

It is homo nuclear so there is no permanent dipole.0752

Now, these are actually not the best examples to use but it has to change during the vibrations.0755

It does not have to have a permanent one.0764

But if the molecule, if somewhere during the vibration the dipole shows up then it is capable of a vibrational transition.0766

The one that we would be concert with most of all, is the changes in the quantum numbers itself.0779

In this case, δ R = + or -1, but it is good to know this because some of your classes depending on what they are going to cover,0783

what they are not going to cover, they may actually go into the mathematics behind this.0791

I do not know, we ourselves are not.0794

We are going to be concerned more with just the spectroscopic aspects.0795

As we said earlier, spectroscopy is conducted primarily in wave numbers.0801

Spectroscopy is conducted in inverse cm, in wave numbers.0817

We want to take this equation for energy and convert them to wave numbers.0825

For vibration, the symbol that we use is G.0832

G is a function of R, it is equal to the energy of R that we had from the previous page divided by HZ.0837

The energy that you have, the energy that is given in Joules divided by Planck's constant divided by the speed of light.0847

Any energy in Joules divided by those by HC is going to give you the number in inverse cm.0856

This is called the vibrational term and this is what you will see in the literature.0861

This G of R called the vibrational term and it is just a symbol for the vibrational energy expressed in terms of wave numbers,0868

expressed in inverse cm as opposed to joules or anything else.0882

G as a function of R is equal to, the E of R is H ν R + ½.0884

We will go ahead and divide by HC.0895

The H will cancel, what we end up with is this ν ~.0898

It is the frequency expressed in wave numbers, R + ½.0903

Everything else is the same, R is going to take on the values of 0, 1, 2, 3, and so on.0908

The reason is here we are left with N/ C.0916

That is just equal to N/ C.0921

G of R is equal to ν ~, the fundamental vibration frequency expressed in wave numbers, × R + ½,0930

where ν ~ is equal to this ν/ C, which is equal to 1/ 2 π C.0944

That is it, just a little bit of mathematical manipulation.0955

This is the important thing right here.0958

This is what we have, this gives us the energy of a particular vibrational state depending on the quantum number R.0961

Let me go ahead and put R = 0, 1, 2, and so n.0968

It gives the energy of that particular vibrational state expressed in inverse cm.0974

That is all this.0979

That is the energy of the harmonic oscillator.0983

Let us go ahead and talk about the energy of the rigid rotator.0986

The energy of the rigid rotator, in case you are wondering why we are talking about the harmonic oscillator and the rigid rotator,0994

these things are for spectroscopic movement, spectroscopic transitions.1004

The harmonic oscillator is this way, it is the molecule that is vibrating.1011

The mathematics behind this is what we are doing.1014

The rigid rotator is something that rotates like this.1018

When you have a diatomic molecule that is rotating, we modeled it with the mathematics of the rigid rotator.1021

That is all, that is what we are doing here.1026

The energy of the rigid rotator is, in terms of what it is that we studied earlier, EJ = H ̅² / 2I × J × J + 1,1028

where J is the rotational quantum number that takes on the values 0, 1, 2, and so on.1043

Here I is the rotational inertia of the molecule.1050

It is equal to the reduced mass × the equilibrium bond length, the radius length between the two nuclei.1054

This E stands for its equilibrium², that is it.1066

This is the moment of inertia.1069

This I is the moment of inertia or the rotational inertia of the molecule.1071

Our E is the equilibrium bond length.1080

One of the parameters that you will see when you look at a table of constants for spectroscopic data is you are going to see the R sub E.1092

You are going to see the equilibrium bond length for that molecule.1099

The rotational say this gives the energy of the rotational state, the degeneracy of each level.1104

In other words, the number of levels that actually has this energy.1111

The degeneracy is equal to 2 J + 1.1115

When we discussed the rigid rotator earlier, I do not think I explained why this degeneracy exists.1121

I may have, but I do not believe that I did.1144

I think I just throw it out there as a number.1146

I do not think I explained where this degeneracy in the rotational states comes from,1149

explained that nature of this degeneracy.1155

Here is what is going on, I will tell you and then I will write it all out.1170

Let us say J is 1, that is going to have certain energy.1175

It is going to rotate with certain energy.1182

However, for each quantum number, in this case 1, there are 2 × 1 + 1.1186

There are going to be three actual orientations in space, fundamental orientations where the rotation is going to have that energy.1194

That is what degeneracy means.1202

It is going to be a particular quantum state that has that same energy, that is what the degeneracy.1204

We know that by definition.1209

The orientation in space of the molecule does not affect its rotational energy.1211

In other words, if I have a certain molecule that is oriented this way and it is rotating like this,1215

or if it is this way rotating like this, or if it is this way rotating like this, they have the same energy.1221

That is what this degeneracy means.1228

In the case of J = 1, it can be this way, it can be this way, it can be this way.1230

If J = 2, that is 2 × 2 + 1.1238

It means there are going to be 5 fundamental orientations in space that give you that same energy.1241

It is going to be 1, 2, 3, 4, 5.1246

For 3, you are going to end up with 7 levels of degeneracy, 7 fundamental orientations in space that all have the same energy.1255

That is the nature of this degeneracy of the rotational states.1262

Let us write it all out and give you a little bit of a quantitative aspect of it.1265

In molecules orientation in space has no affect on this rotational energy.1274

J is the quantum number that represents the angular momentum of the rigid rotator.1298

In other words, we know that anything that rotates has an angular momentum.1325

That angular momentum is going to be perpendicular to the direction of rotation.1329

If the molecule is rotating like this, its angular momentum is that way.1332

The magnitude of that angular momentum that is what J is.1337

J² actually.1340

That is what it represents.1343

It represents the magnitude of the angular momentum.1345

If it is rotating this way, the angular momentum vector is pointing that way.1347

If it is rotating this way, it is pointing that way.1351

This is the Z axis, it can be rotating like this, like this like this.1356

Angular momentum pointing that way, angular momentum vector pointing this way,1363

angular momentum vector pointing this way.1366

For each value of J, there is a J sub Z.1369

It is the component of the angular momentum vector along the Z axis, J sub Z is, we already seen this before.1381

For angular momentum, for rotational angular, we have seen this for spin angular momentum already.1391

JZ is the component of the angular momentum vector along the Z axis.1395

JZ takes on the values 0, + or -1, + or -2, all way to + or - J.1422

If J is 1 then what we have is 1, 0, -1.1444

If J is 2, we have 2, 1, 0, -1, 2.1449

If J is 3, we have 3, 2, 1, 0, -1, -2, -3.1454

It is the projection of the angular momentum vector along the Z axis.1461

All of those orientations in space carry the same energy given by the,1466

Here is where 2J + 1 come from.1473

There are 2J + 1 value in J sub Z.1475

For each J, or each J, there are 2J + 1.1478

I will just call that fundamental orientation with the same energy.1490

When we discuss the hydrogen atom, we call the J we called it L, the angular momentum quantum number.1519

The rotational quantum number.1537

And we called this J sub Z, we call it M sub L.1544

This was the magnetic quantum number, that is all that is going on here.1551

Remember for each value of L, you have 0, + or -1, + or – 2, all the way to ± L.1558

That is the magnetic quantum number which takes on the values 0, + or -1, + or -2, all the way to + or – L.1567

We will do the rotational term.1589

We went ahead and expressed the vibrational energy in terms of wave numbers.1596

We are going to express the rotational energy in terms of wave numbers.1600

The rotational term is symbolized as F of J.1604

F of J is equal to E of J divided by HZ equal to H ̅² / 2I HZ × J × J + 1.1612

H ̅ is equal to H/ 2 π, that implies that H ̅² is equal to H²/ 4 π².1630

When we put all of this back in to here, we get that the rotational term,1644

in other words the rotational energy expressed in terms of wave numbers is going to be equal to H² / 8 I π² HZ × J × J + 1.1648

We get some cancellation with the H and one of these.1668

What we are left with is F of J is equal to H/ 8I π² ψ × J × J + 1.1671

And again, J takes on the value 0, 1, 2, 3, 4.1690

I will circle the whole thing not just the bottom part.1696

This whole thing this is called the rotational constant and is symbolized as a B with a ~ on it.1699

This is another one of the spectroscopic parameters that you find in a table of spectroscopic data.1722

Just like you see in the equilibrium bond length, you will also see the rotational constant.1730

And as we go on with the lessons, you will see that there are more and more constants that are actually tabulated.1734

Since that is symbolized that way, we will go ahead and write it as F of J is equal to B ~ × J × J + 1,1742

where J takes on the values of 0, 1, 2, and so on.1758

And B is of course what we just said F of J is now in inverse cm.1763

The selection rules for rotational transitions δ J = + or -1,1771

that means it can only go from one rotational state to the next, either up or down.1790

It is not going to jump 5 levels.1793

In this case, the molecule must have a permanent dipole.1795

In the case of a rotational transition, it has to have a permanent dipole.1807

In the case of the vibrational transition, there has to be a change in the dipole moment during the vibration, during the transition.1810

If the transition is to happen, it must have a permanent dipole.1821

That is the difference between the two.1825

We have the rigid rotator energy, we have the harmonic oscillator energy, therefore like we said,1828

the transitions in the infrared, the vibrational transitions are accompanied by rotational transitions.1835

The combined energy of the transition is going to be the energy of the rotation + the energy of the vibration.1841

The harmonic oscillator rigid rotator approximation for the energy of the molecule is1849

therefore, the sum of the vibrational rotational energy.1873

Therefore, the energy R J is equal to the vibrational term + the rotational term.1892

The vibrational energy + the rotational energy.1901

Let me make my J a little bit more clear so it is not connected like that.1903

E sub RJ is equal to, this one is ν prime R + ½ +, now we have B~ J × J + 1.1908

Here R takes on the values 0, 1, 2, and so on.1924

J takes on the values 0, 1, 2, and so on.1928

Notice, there are two quantum numbers here in the total expression for the energy.1932

Once again, this ν ~ is equal to 1/ 2 π Z × the force constant divided by the reduced mass ^½.1937

This is the fundamental frequency of the vibration.1949

B~ is equal to Planck's constant divided by 8I π² Z.1966

All the parameters are taken care of, this is the equation of the harmonic oscillator rigid rotator approximation.1976

The mathematical equation that approximates what we see when we look at vibration rotation spectra is this.1983

The energy level diagram looks like this.1997

Let me draw this one by hand, actually.2001

What we have is my harmonic oscillator R = 0, R = 1, R = 2.2004

Remember the spacing between energy levels is the same for the harmonic oscillator.2021

R = 3, this is R = 0.2027

R = 1, R = 2, R = 3.2030

Within each vibrational level, there is a series of rotational levels.2036

We have J = 0, J = 1, J = 2, J = 3, and so on.2042

Here we have this one, this one, this one, this one, this one.2053

For R = 2, for each vibrational level there is a series of rotational levels.2060

The spacing of the rotation levels is not the same, the energies.2072

Between each vibration level, or for each vibrational level2080

which is the vibrational quantum number R there is a progression of rotational states.2090

When a photon of infrared is absorbed,2119

not only does a vibrational transition take place from R to R + 12140

but several rotational transitions take place.2155

Several rotational transitions take place from J to either J + 1 or to J-1.2165

Right now, we are talking just about absorption.2182

When we are talking about absorption, we are going to go from R to R + 1.2184

If we are talking about emission, we would be talking about going from some level R to R -1.2189

For the sake of absorption, we are going up one vibrational level but the J level can actually go down or up.2194

You can go from J1 to J2 or J1 to J0, that is what this means.2202

But several rotation transitions take place when a photon of IR not only does the transition take place2211

from R to R + 1 but several rotational transitions take place from J to J + 1.2219

The selection rules, this is called vibration rotation.2230

When we to look a spectrum, it is called a vibration rotation spectrum.2242

We can get pure rotational spectra, we can get information of pure vibrational spectra2246

but when we run out of vibrational spectrum what we will get is a vibration rotation spectrum.2253

They are the combination of the R jump and the J jumps, that is what is happening.2256

The selection rules for vibration rotation are2267

δ R = + or -1, δ J = + or -1.2285

If R is + 1 that means it is going from a lower or to a higher state of absorption.2293

If R is -1, it means it is coming from a higher to a lower state, that is emission.2298

In the case of δ J + or -1, in the case of absorption, because you are going from one vibrational state to another vibrational state,2304

the rotational state might go up 1 or down 1.2316

But it is still absorption because it is actually going up an entire vibrational level.2322

You are still looking at a higher level.2331

Let us look at what the mathematics behind absorption.2337

What we want to do now is to derive an equation for the frequencies, for the spectra that we see.2341

Let us look at absorption, in the case of absorption δ R = + or -1.2352

Δ R is + 1, it is absorption, sorry about that.2365

Δ J is + or -1.2370

For R = + 1 and J = + 1, we would have two cases, 1 + 1 and 1 -1, + 1, the frequency of absorption.2374

The frequency of absorption is the difference between two energy levels.2389

The frequency of absorption is ν, what we observe.2399

It is the energy of the R + 1 J + 1 state - the lower state which is the energy of the R and J.2408

Let us go ahead and do the mathematics here.2434

This is going to equal, this energy term – this.2437

What we are going to have is prime × R + 1 + ½ + B~ × J + 1 × J + 2.2441

This is the upper state - the lower state which is ν~ × R + ½ + B ~ × J × J + 1.2455

I will not actually go through all the algebra here.2471

I think I actually will, I will do it for this one, that is not a problem.2481

This is equal to ν × R + 3/2 + B ~ × J² + 3J + 2 - ν × R + ½ - B ~ × J² + J,2484

I will distribute the - or both, what you end up with is ν ~ R + 3/2 ν ~ + B~ J² + 3 B ~ J + 2 B ~ – ν R - ½ ν - B ~ J.2511

This is just algebra, that is all it is.2538

It is always the worst part of mathematics.2540

Algebra has always been the worst and it will always be the worst.2542

Do not let it get to you.2546

A combined term, like for example I can cancel this one and this one.2548

I can cancel BJ² and that one I can combine the 3/2 ν - ½ ν.2553

When I'm left with is ν ~ + 2B ~ J + 2B ~.2559

Let me simplify this a little bit.2570

ν observed is going to equal ν + 2B.2572

I’m going to factor out the 2B ~, that is going to be J + 1, where J is going to take on the values 0, 1, 2, 3.2580

This is very important, J here is the value of the lower rotational state.2590

J is the value of the lower rotational state.2598

In other words, the smaller lower quantum number.2608

The quantum number of lower state.2611

It is the value of the quantum number in the lower state.2613

This is one of the equations.2637

This equation for the different values that J takes on.2640

What I'm going to see is the spectrum but I expect to see in the spectrum is this.2644

I expect to see a frequency at this number.2649

Its fundamental vibrational frequency + 2 × this rotational constant × whatever the J value happens to be in the lower state.2655

I expect to see a line there.2663

For the other case, for R = + 1.2668

This time J = -1, the observed frequency that I expect is going to be the energy of the R + 1 state J-1,2674

that is the upper state is J -1 – ERJ.2688

I go through the same algebra and what I end up with is, ν observe is going to equal ν -2 BJ.2697

Here, J takes on the values 1, 2, 3.2710

And again, J here is the value of the quantum number in the lower rotational state.2716

That is why there is a difference between these two.2725

Let us actually see what this looks like.2730

This is the other equation that I'm interested in right here.2731

This gives me one set of lines for different values of J.2739

This gives me another set of lines for different values of J.2742

Let us go ahead and go to a picture of the energy transitions to see what is happening first,2757

then we will take a look at the spectrum.2761

These right here, the blue lines, this blue level lower vibrational state,2765

Let me actually erase this.2771

Most books tend to use the symbol V or sort of a variation on ν as the vibrational quantum number.2774

I do not like that because it looks a lot like the frequency and it tends to get really confusing2782

which is why I use R for the vibrational quantum number.2788

This is R = 0, the 0 vibrational quantum state.2792

Here, this is the R = 1 vibrational quantum state.2795

Notice, within each vibration level, there are several rotational levels.2801

J = 0, J = 1, J = 2, J = 3, 4, 5.2806

And of course, in the upper level it has its own rotational quantum states 0, 1, 2, 3, 4, 5.2810

What we see in the spectrum is a series of lines.2817

For right now, let us not worry about with this Q branch is.2821

I will tell you in a second.2825

What is happening is R, the vibrational state is going up by 1.2827

We are jumping up from this vibrational state to this vibrational state.2832

Let us go to the R branch first, the R branch of the spectrum.2837

Here, the δ R = + 1 and the δ J = + 1.2845

That is fine, I will stick with blue.2856

I'm going from the J = 0, this one right here, this black line is J equal 0 to the J = 1.2859

The vibrational quantum state is going up by 1.2866

The rotational quantum state is going up by 1.2871

The molecules that are in the state of J = 1, they transition to the J = 2.2877

The 2 go up to 3, the 3 go up to 4, the 4 go up to 5.2885

That is the R branch.2894

The L branch represents δ R = + 1, we are going up a vibrational state.2896

But for the δ J = -1, in other words on the spectrum, I see a line for this transition,2905

a line for this transition, a line for this transition, and a line for this transition.2912

Now for the P branch, sorry about that.2918

It represents a vibrational state of going from a lower vibration to a higher vibration R = + 1,2926

but the rotational state drops by 1.2931

Here we are going from 1 to 0, J = 2 to J = 1.2933

J = 3 to J = 2, J = 4 to J = 3, J = 5 to J = 4.2942

There is a line for each of these transitions.2950

What we see is 1, 2, 3, 4, 5 and so on lines.2954

To the left we see 1, 2, 3, 4, 5, and so on lines.2958

This Q branch, notice where the lines go.2963

We are jumping up from a vibrational state to a vibrational state R = 0 to R = 1.2967

That is fine, that is not a problem.2971

The selection rule can handle that but we said that δ J has to be + or -1.2973

Δ J cannot be 0, that is a forbidden transition.2977

Notice this line right here, that is going from J = 0 to J = 0.2983

J = 1 to J = 1, because δ J = 0 is a forbidden transition, these transitions we do not see them in the spectrum.2988

What you see in the spectrum is this, no line at ν, the fundamental vibration frequency.2999

Remember what we had just a second ago, we had a couple of equations.3009

We said that for the R branch, we would end up seeing a bunch of lines at ν + 2 B × J + 1.3014

And for the P branch, we would see a bunch at ν – 2BJ.3028

There are going to be lines to the left, lines to the right of this number.3037

But we do not actually see a line in there because the transition from the δ J = 0 transition is forbidden.3042

There is no actual Q branch.3051

There are molecules with Q branch, they will show up3053

Your teacher may or may not decide not to talk about it.3056

We, ourselves, we will not talk about it.3060

It was not altogether that important, at least for what we are doing.3061

But know that there is a P branch, there is a Q branch, there is an R branch.3064

But in general, for the vibration rotation spectra of diatomic molecule,3068

because δ G = 0 is a forbidden transition, δ J has to equal + or -1, we do not see a Q branch.3072

It is just that nothing there.3079

Let us go ahead and take a look at the vibration rotation spectra.3082

This is a vibration rotation spectrum for HCL.3089

Do not worry about this, it should not be in here.3093

This is a vibration rotation of HCL.3097

You will sometimes see them this way, in terms of the peaks to be pointing down.3101

Other times you are going to see them when the peaks are pointing up.3107

The R branch here, right here, see these lines?3110

This right here, this little gap, this is the ν sub 0.3113

This is the fundamental frequency.3122

It to this point where the J = 0 to J = 0 transitions should happen.3123

But because they are forbidden, it is not going to happen, but that is where we see it.3128

If we know what ν 0 is, reading of the spectrum we just find that middle point and we go down and we mark it.3134

That is our ν sub 0, that is our fundamental vibration frequency.3143

The R branch we said is where you do,3148

Let us write it down.3156

It is equal to that, + 2B × J + 1.3159

Notice, it is where the absorption frequency is going to be increasing from the fundamental frequency.3168

The fundamental frequency + a certain amount.3175

The fundamental frequency + a certain amount.3178

The fundamental frequency + a certain amount + a certain amount, this is a second transition.3180

+ a certain amount, this is the third transition.3184

+ a certain amount, this is the fourth transition.3186

+ a certain amount, it is the fifth transition.3187

As the frequencies go up, that is the R branch.3190

This right here is the R branch.3193

Over here, the frequencies go down.3196

The fundamental frequency of the P branch is the fundamental frequency - some value as J increases.3199

- - -, these are going down.3206

This is the P branch.3209

Like I said, sometimes you see it the other way.3212

When you look at the spectra, you are not going to look at left and right.3214

You are looking at R and P.3220

R is for the frequencies that increase, P is for the frequencies that decrease from the fundamental frequency.3222

That is what is happening here.3228

Let us look at these lines, the fundamental frequency is here.3230

This one represents the transition from 0 to J = 1.3233

This one represents J = 1 to J = 2.3237

This one represents J = 2 to J = 3.3240

J = 3 to J = 4, and so on.3244

Here represents J = 1 to J = 0, J = 2 to J = 1, J = 3 to J = 2.3246

This is the δ J = + 1, here is the δ J = -1.3260

Let us look at the spectra that actually look the other way.3267

This is the spectrum for carbon monoxide.3271

Let me write it down here.3274

This is the spectrum from carbon monoxide.3276

In this particular case, the peaks are pointing up.3278

This is interesting.3280

The left is actually increasing, to the right it is actually decreasing.3288

Here is the fundamental frequency increase.3291

This is the R branch decreasing, that is the P branch.3294

That is all that is going on here.3299

Because the equation R ν + 2B × J + 1 and ν - 2 BJ, the difference in the spacing, this difference right here,3303

the difference between the lines is 2B.3322

The difference between them is 2B.3328

We see absorptions.3334

That is what is going on there.3340

Let us go ahead and see what else we have to say about this.3342

Notice there is a nonexistent Q branch here.3349

Let us talk about the intensities.3359

Notice that all of these lines they have different intensities.3361

Some are very very intense, some not so intense.3364

This is the last page.3370

Let me go ahead and write it over here and continue it down here.3374

The intensities of the transitions are related to the populations of each J level.3386

In other words, if I had a bunch of molecules, let us just say I have 100 molecules in level 2.3405

A hundred molecules in level 2, the more molecules that you have, the more transitions are going to take place.3413

The more molecules you have, each one of those molecules is going to make a transition.3428

Each one of those transitions is represented by, it is going to contribute to the size of the peak.3432

If you only have 10 molecules making the transition, let us say from level 3 to level 4,3437

it is going to have a certain height of a peak.3443

If you have 100 molecules making the transition from level 2 to level 3, that peak is going to be higher.3445

In this particular case, notice that the level 2 to level 3 to level 4, those tend to be the peaks of highest intensity.3452

That tells us that the rotational levels 2, 3, 4, those are the ones that are most highly populated at normal temperatures.3461

As far as the rotational state level molecule, most molecules are not in there,3472

0, 1, 2 rotational states at room temperature, most of them tend to be in the 3, 4, 5 rotational states.3476

That is where most of the molecules are.3483

Therefore, the transitions that are going to take place, they are going to have the ones of higher intensity.3485

Let me say that again.3491

The intensities of the transitions are related to the populations of each J level.3492

The greater the population of the level, the greater the number of transitions,3504

therefore, the more intense the line.3532

We see from these spectra that the J = 3, 4, 5, are the most populated.3542

Once again, notice that the Q branch does not exist because δ J = 0 is a forbidden transition.3574

Thank you so much for joining us here at www.educator.com.3582

We will see you next time for a continuation of molecular spectroscopy, bye.3584

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