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Raffi Hovasapian

The Hydrogen Atom Example Problems II

Slide Duration:

Section 1: Classical Thermodynamics Preliminaries
The Ideal Gas Law

46m 5s

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

46m 2s

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

1h 6m 45s

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

1h 6m 33s

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

1h 2m 17s

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

1h 4m 39s

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

16m 50s

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

43m 40s

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

32m 23s

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

39m 15s

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

35m 52s

Intro
0:00
0:10
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
18:08
18:09
21:54
22:34
Fundamental Relationship Equation for an Ideal Gas Under Adiabatic Expansion
25:00
More on the Equation
28:20
Important Equations
32:16
32:17
Reversible Adiabatic Change of State Equation
33:02
Section 3: Energy Example Problems
1st Law Example Problems I

42m 40s

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

1h 23s

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

44m 34s

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

59m 7s

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

49m 16s

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

33m 59s

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

54m 37s

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

31m 18s

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

23m 6s

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

25m 42s

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

43m 39s

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

56m 44s

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

57m 6s

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

54m 35s

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

35m 5s

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

28m 42s

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

34m 38s

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

30m 50s

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

34m 6s

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

39m 18s

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

19m 40s

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

54m 16s

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

31m 17s

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

45m

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

58m 5s

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

25m 20s

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

34m 25s

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

59m 57s

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

42m 5s

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

30m 26s

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

21m 34s

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

56m 22s

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

45m 24s

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

48m 43s

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

46m 18s

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

39m 28s

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

35m 32s

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

29m 55s

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

54m 25s

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

46m 58s

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

44m 11s

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

35m 33s

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

43m 4s

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

26m 30s

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

41m 10s

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

30m 32s

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

35m 19s

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

33m 48s

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

46m 43s

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

40m

Intro
0:00
The Hydrogen Atom I
1:31
Review of the Rigid Rotator
1:32
Hydrogen Atom & the Coulomb Potential
2:50
Using the Spherical Coordinates
6:33
Applying This Last Expression to Equation 1
10:19
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
28:02
Example: 1s Orbital
28:34
33:46
1s Orbital: Plotting Probability Densities vs. r
35:47
2s Orbital: Plotting Probability Densities vs. r
37:46
3s Orbital: Plotting Probability Densities vs. r
38:49
4s Orbital: Plotting Probability Densities vs. r
39:34
2p Orbital: Plotting Probability Densities vs. r
40:12
3p Orbital: Plotting Probability Densities vs. r
41:02
4p Orbital: Plotting Probability Densities vs. r
41:51
3d Orbital: Plotting Probability Densities vs. r
43:18
4d Orbital: Plotting Probability Densities vs. r
43:48
Example I: Probability of Finding an Electron in the 2s Orbital of the Hydrogen
45:40
The Hydrogen Atom VI

51m 53s

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

34m 29s

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

43m 49s

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

1h 1m 57s

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

48m 33s

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

48m 33s

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

48m 33s

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

59m 18s

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

34m 54s

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

38m 3s

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

57m 49s

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

1h 1m 20s

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

56m 34s

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

18m 24s

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

50m 2s

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

59m 47s

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

46m 22s

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

29m 24s

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

30m 53s

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

1h 1m 33s

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

33m 47s

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

1h 1m 5s

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

33m 31s

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

1h 1m 15s

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

47m 23s

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

54m 9s

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

48m 32s

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

57m 30s

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

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### The Hydrogen Atom Example Problems II

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• Intro 0:00
• Example I: Normalization & Pair-wise Orthogonal 0:13
• Part 1: Normalized
• Part 2: Pair-wise Orthogonal
• 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
• Physical Interpretation of Orbital Angular Momentum in Quantum mechanics

### Transcription: The Hydrogen Atom Example Problems II

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

Today, we are going to continue with our example problems for the hydrogen atom.0004

Let us get started here.0008

Example number 1, show that S1,0, S1, 1 and S1, -1 are normalized.0014

And that they are pair wise orthogonal.0022

These are the spherical harmonics.0024

They are the angular portion of the hydrogen atom wave function.0027

Let us go ahead and write down what they are so we have them.0033

I can go ahead and work in blue or red.0035

Let us go ahead and work in blue.0040

We have S1, 0 is equal to 3 / 4 π ^½ × cos of θ.0044

S1, 1 is equal to 3/ 8 π ^½ sin θ E ⁺I φ.0056

S1 -1 is equal to 3/8 π ^½ × sin θ E ⁻I φ.0069

We are going to be working in spherical coordinates.0084

It is very easy to forget these extra midterms that we have to put in the integrand.0089

We are working in spherical coordinates.0096

In other words, θ is going to run from 0 to π.0116

And φ is going to run from 0 to 2 π.0122

The integrals look like this.0129

The spherical coordinates has 3 variables, it has R, θ, and φ.0138

In this particular case, we are dealing just with a spherical harmonics so we are just concerned with θ and φ.0144

Since that is the case, we would be working with double integrals.0149

Later, when we start talking about ψ, ψ 211 and ψ 310,0152

the entire hydrogen wave function that is what we are going to be working with R, θ, and φ.0157

That is when it is going to become a triple integral.0163

But for right now, it is going to be double integral, 2 variables.0165

The integrals actually look like this.0167

In general, we are going to be 0 to 2 π, 0 to π, and there is going to be some integrand whenever that happens to be.0171

Then, we are going to have the factors sin θ D θ D φ.0183

This extra thing has to be there.0189

Do not forget this factor, do not forget this sin θ factor when doing integrals of these types.0191

It is very easy to just start working with just D θ D φ to forget that we have0204

to make a little bit of an adjustment because we did a change of variables,0207

when we went from Cartesian coordinates to spherical coordinates.0211

While I’m here, let me go ahead and mention it.0221

Later when we do start talking about R and θ and φ, the integral is going to look like this.0224

Let me write this out.0233

When we later include R, the general integral will look like this.0238

It is going to be the integral from 0 to R, integral from 0 to 2 π, integral from 0 to π, and there is going to be some integrand,0261

whatever that happens to be depending on the functions we are dealing with.0281

Now the factor is going to be R² sin θ D θ D φ DR.0285

It does have to be in this order, in can be in any order you want depending on0295

where does that you are integrating the function that you are dealing with.0299

Just do not forget spherical coordinates, we have to have these factors.0301

Let go ahead and get started with the problem.0308

For S10, normalization looks like this.0311

Our standard normalization integral, we have seen it over and over again.0321

It is going to be the integral of S10 conjugate S10 and we want this which we want to equal 1.0325

We are trying to show that is normalized.0342

We are trying to show that S10 is normalized.0344

It means that when we take the S10 conjugate multiplied by S10 and integrate, that we should get 1.0347

It is equal to 1.0358

Let us go ahead and do S10 conjugate × S10.0360

It is actually going to just equal S10² because in this particular case, the S10 is a real function.0366

It does not have a complex part.0374

Therefore, it is just S10 × S10.0376

You already listed them before, what we are going to end up getting is 3/4 π × the cos² θ.0380

Again, because S10 is a real function, it is real.0390

There is no E ⁺I φ, E ⁻I φ, there is no I in there.0398

Our integral is going to end up looking like 0 to 2 π, 0 to π, 3/ 4 Π, that is this.0408

That is all we are doing, we are forming this integral and we are solving that integral.0419

That is all we are doing.0422

Cos² θ, that is our function.0426

Our factor is sin θ D θ 0 π θ, that is going to be the first integral we will do.0432

That is the integral, and the other integral is φ, we do that afterward.0438

We are going to take care of the inner integral first.0443

We are just going to deal with that one.0448

I'm going to pull the constant out, it is going to be 3/ 4 π the integral from 0 to π.0451

This is going to be cos² sin θ cos² θ sin θ D θ.0458

I’m doing one variable at a time.0465

I do not know if you remember these trigonometric integrals.0467

Again, you can just go ahead and have your software do it, it is not the end of the world.0470

But I thought it would be nice to actually do so by hand, just so we get to refresh our memories0473

because we are actually going to have to be doing this on the quiz and the tests that you take.0478

You can go ahead and do use substitution here.0484

I’m going to set U equal to cos θ and I'm going to set Du is going to be - sin θ D θ.0486

Therefore, when I substitute these back into this here, I'm going to end up getting - 3/,0497

Basically, what happens here is sin θ D θ is equal to – DU.0506

Sin θ D θ, I will put - DU cos², I put U².0512

It is going to be -3/ 4 π, you remember this right.0517

It should not be too long ago.0521

It was 0 to π U² DU.0522

I should change my limits of integration, I generally do not because I tend to just go back and put these functions back in when I solve it.0528

I would like to go ahead and put this into θ to get the lower limit and the π into θ to get the upper limit.0536

I just leave it as 0 to π because I’m going to go back to the θ in just a minute.0544

It is going to be equal -3/ 4 π × U³/ 3 0 to π and of course U is just cos θ.0551

It is going to be -3 / 4 π cos θ³ from 0 to π.0564

What we are going to end up here is getting -3/ 4 π × -1/ 3 -1/ 3, when you do that.0575

You are going to end up actually getting 2/ 4 π or 1/ 2 π, that is just the inner integral.0587

We are not in the outer integral yet.0598

1/2 π is that.0600

Now, we will go ahead and do the outer integral.0610

The outer integral, we have the integral from 0 to π of 1/ 2 π D φ.0620

And that is going to equal 1/ 2 π from 0 to 2 π of D φ which is going to equal 1/ 2 π × 2 π, and it is going to equal 1.0631

Yes, we did get the number 1 when we did the integral.0644

This particular function, this particular spherical harmonic is normalized.0648

Now for the rest, I will set the integral but I’m not going to go ahead and solve the integral.0653

You could do by hand, table, or just use your software.0660

For the rest, I will set up the integral but leave it to you to evaluate.0665

We want to do S11, for S11 the normalization condition is S11 conjugate × S11.0682

We want it to equal 1 when we do that, which we want equaling 1.0701

S11 is equal to 3/ 8 π ^½ sin θ E ⁺I φ.0715

Therefore, S11 conjugate is equal to, this does have something complex in it.0727

It does have a conjugate but is different than the original function.0736

S11 conjugate is equal to 3/ 8 π ^½ × sin θ E ^- I φ.0739

Therefore, S11 conjugate × S11 is equal to this × this.0752

We get 3/ 8 π, sin θ and sin θ is sin² θ, E ⁺I φ × E ⁻I φ is E⁰ which is equal to 1.0762

It is equal to this.0774

The integral of S11 conjugate S11 which is the normalization integral is equal to 0 to 2 π, 0 to π, 3/ 8 π sin² θ D θ.0779

When you solve this integral, do it by hand, you have to do the.0800

I'm sorry, D θ I forgot my D φ.0810

Just like I told you guys not to forget the factor, I forgot the factor.0821

This is the function, another factor is sin θ D θ D φ.0825

I think I should just go slow.0834

The first integral is that one, that is the inner and I will do the outer.0838

Or just have your software do it.0846

And when you do, do this, you are going to get 1.0848

It is also normalized, S11 is normalized.0852

We are confirming this.0856

We are just getting comfortable with the functions, manipulating the functions,0857

dealing with the functions, writing them down, that is what we are doing.0862

Let me go back to red, I actually write red, it is nice.0867

For S1-1 normalization, that normalization is basic S1 -1 conjugate × S1 -1.0884

We want it to equal 1.0901

S1-1 is equal to 3/ 8 π ^½ sin of θ E ^- I φ.0904

Therefore, S1 -1 conjugate is equal to the conjugate of this is 3/ 8 π ^½ sin of θ E ⁺I φ.0918

Therefore, S1 -1 conjugate × S1 -1 is equally the same as before.0937

What we just did is equal to 3/8 π sin² θ.0945

Of course, the φ goes away.0955

The integral S1-1 conjugate S1 -1 is equal to 0 to 2 π, 0 to π, 3/8 π sin² θ, that is the function.0960

And the factor is sin θ D θ D φ, this is the integral that you would enter to your software.0977

And when you do so, you are going to get that is = 1.0985

That takes care of the normalization.0990

All of those are normalized, now we are going to deal with the pair wise orthogonality process.0995

Let us take a look and deal with that.1001

Let us see if I should start in a new page or not.1005

That is fine, I will just go ahead and continue here.1009

For pair wise orthogonality.1014

The integral of the integrals that we would be looking at, the orthogonality condition1028

is going to be the the integral of S sub LM conjugate × S sub L primer M prime,1038

these are different because now we are taking 2 functions that are actually different.1047

We want this, which we want equal to 0 that will confirm that they are orthogonal.1053

Let us see what we have got for S10 and S11.1068

S10 is equal to 3/ 4 π ^½ cos of θ and S11 is equal to 3/ 8 π ^½ sin θ E ⁺I φ.1086

Therefore, the integral of S10 conjugate S11 is going to equal the integral from 0 to 2 π.1107

The integral from 0 to π of this function × the conjugate of this function × this function.1125

The conjugate of this function is the same because this is real.1134

There is no I part to it.1138

It is going to be 3/4 π ^½ cos θ × 3/8 π ^½ sin θ E ⁺I φ.1140

And it is going to be sin θ D θ D φ.1172

We are going to get of 3/ 4 π √ 2, when we take care of the constants.1180

It is going to be 0 to 2 π, 0 to π.1189

We are going to have cos θ.1194

I’m going to put this sin θ and that sin θ together, sin² θ E ⁺I φ D θ D φ.1201

Here is something we can do which is really nice.1213

Now that we have a function which is a function of both θ and φ, we can actually separate the functions out.1217

We can write the integral like this, it is very convenient.1223

3/4 π √ 2, 0 to 2 π.1227

The φ, we can just take this function E ⁺I φ D φ.1232

Then, we can take the θ portion, cos θ sin² θ D θ.1240

We can separate this out and do it this way because again these are individual functions of 1 variable1251

and we are integrating 1 variable at a time.1256

You do have a lot of integrals that actually end up looking like this.1261

With this E ⁺I φ, E ⁺2I φ, E ⁺3I φ, things like that.1264

For integrals that look like this, this is a good thing to know.1269

For integrals that look like this, it is good to know the following.1275

It is good to know the following.1292

We are just making our life easier because the integral, you have seen over and over again.1297

We do not want to keep evaluating them and good to know the following.1300

The integral from 0 to 2 π of E ^+ or - I φ is actually always going to be equal to 0.1308

In fact, it is true for any multiple of the I φ.1327

In fact, the integral from 0 to 2 π of E 6+ or – IN φ D φ is equal to 0.1333

I forgot my D φ which I often forget.1347

This integral is automatically equal to 0.1350

It saves me from having to actually solve the rest of the integral, very nice.1352

This integral right here, because this is true, this part is 0.1359

0 × whatever it is does not matter, it is going to equal 0.1365

Therefore, this integral, 3/ 4 π √ 2 integral from 0 to 2 π E ⁺I φ D φ 0 to π cos θ sin² θ D θ is equal to 0,1368

which means that they are orthogonal.1398

Which is what we wanted, very nice and convenient property.1400

Now we do S10 and S1 -1.1405

The integral for that is going to be the integral of S10 conjugate S1 -1, we want it to equal 0.1417

This integral S10 conjugate S1 -1, the integral is going to turn out to be 3/4 π √ 2 the integral from 0 to 2 π,1425

this time E ^- I φ D φ, because this S1 -1 that was that particular function 0 to π.1456

Again, you are going to get cos θ sin² θ D θ, that is equal to 0.1467

Therefore, the whole integral is equal to 0.1478

Again, S10 and S1 -1 are orthogonal, very nice.1480

Let us go ahead and see the last one we got.1490

I think we got one more.1493

You have S11 and S1 -1.1494

For S11 and S1 -1, our integral is going to be S11 conjugate, S1 -1, we want the integral to be equal to 0.1499

S11 conjugate is going to equal 3/ 8 π ^½ sin θ E ⁻I φ.1514

And S1-1 is equal to 3/ 8 π ^½ sin θ E ⁺I φ.1529

When we multiply those 2 together, we are going to get the integral of S11 conjugate S1 -1 is going to equal 3/ 8 π,1547

the integral 0 to 2 π, the integral 0 to π, it is going to be sin² θ E ⁻I 2 φ sin θ D θ D φ, which we can separate.1567

It is going to be 3/ 8 π × the integral from 0 to 2 π E ⁻I 2 φ D φ.1589

The integral from 0 to φ of sin³ θ D θ.1601

This integral is equal to 0, therefore this is equal to 0.1609

Therefore, we have shown that they are orthogonal.1612

And of course, it is generally true.1617

All of the spherical harmonics are pair wise orthogonal.1619

Let us see what we have got, lot of extra pages here.1626

In the previous exercise, we use the fact that 0 to 2 π E ⁺IN φ D φ equal 0 for any integer N.1633

We want to actually have you show explicitly that this is true.1641

Show that this is true rather than just using it.1644

It should not be a problem.1647

E ⁺IN φ, we are going to use the formula that we now.1651

We are going to separate it into its real and complex parts.1657

We are going to factor the cos θ and sin θ.1661

This is cos of N φ + I × the sin of N φ.1664

Therefore, the integral from 0 to 2 π of E ⁺IM φ D φ is going to equal1673

the integral from 0 to 2 π of cos N φ D φ + I × the integral from 0 to 2 π of sin N φ D φ.1682

We just separate it out so we can actually solve the normal integrals.1702

This is going to equal 1/ N × sin N φ from 0 to 2 π + I ×.1706

It is going to be actually ±I because when we integrate the sin, it is going to be a negative cos.1720

Cos N φ 0 to 2 π and you are going to get 1/ N 0 -0 + - I/ N, sorry I forgot the N here.1727

This is going to be 1 -1 0, there you go.1742

We just show explicitly that this integral is always equal to 0 and this integral will come up a lot.1749

You can just, on a test and on a quiz, it is equal to 0.1755

You do not have to go ahead and actually evaluate the θ portion of the integral.1759

Let us see what we have got next.1767

In spherical coordinates, the angular momentum operator for the X direction of the angular momentum,1771

the X component of the angular momentum operator is this expression right here.1779

This is in spherical coordinates and we have already seen previously in Cartesian coordinates,1787

we are not going to go through the exercise of partial differentiation to change the variables or to actually turn it into this,1792

but this is what it looks like when you expressed in spherical coordinates.1801

Using the spherical harmonics for L = 1, in other words S10 S11 S1 -1,1805

show that the average value of the angular momentum in the X direction is 0.1814

Using the spherical harmonic S11, for L = 1 N = 1, 0 and -1, we deal with S10 S11 and S1-1.1828

For these 3, we want to show that the average value of the X component of the angular momentum is going to be 0.1855

Let us take a look, for S10 the average value integral looks like this.1865

It is going to be S10 conjugate and you actually put the operator in between and then you operate on S10.1878

You operate on S10 first and then you multiply on the left by S10 and you integrate that.1887

We want to see if this is actually going to equal 0.1897

Let us go ahead and take care of this part first.1901

Let us go ahead and actually operate, its form L sub X of S10, let us operate on it using this operator.1904

This is going to equal - I H ̅ - sin φ DD θ - cot θ cos φ DD φ.1914

We are going to be operating on S10 or S10 is 3/ 4 π ^½ cos θ.1937

This is going to equal ,we are going to distribute and operate on this.1950

In this particular case DD φ, there is no φ in this S10 spherical harmonic.1957

Therefore, this term just goes to 0 so this is the only one that matters.1962

I’m going to go ahead and pull these out, the constants - I H ̅ × 3/ 4 π ^½ × - sin of φ × DD θ of cos θ, which is -sin θ.1966

When I put this, I do this to this.1989

DD φ of this is just 0 so it is going to be -0.1997

This is just going to equal - I H ̅ × 3/ 4 π ^½ × O.2001

This is - × - × -, one of the – stays, this becomes sin φ sin θ.2014

That takes care of just this part.2029

Now, we are going to multiply on the left, multiply what we got on the left by S10 conjugate.2032

S10 conjugate × what it is that we just got which was L sub X of S10, that is going to be 3/ 4 π ^½ cos θ × – I H ̅ 3/4 π ^½ sin φ sin θ.2048

This is going to equal -3 IH ̅/ 4 π × cos θ sin φ sin θ.2085

The integral of S10 conjugate L sub X operating on S10, the average value integral2105

is actually going to equal -3 I H ̅/ 4 π the integral from 0 to 2 π.2119

Let me go ahead and separate these integrals out.2132

I think I’m going to write the whole thing first.2135

0 to π this thing, cos θ sin θ, sin θ that is the function and the factor, sin θ D θ D φ.2138

One thing at a time, I’m going to go ahead and separate these out.2157

It is going to be -3 I H ̅/ 4 π the integral from 0 to 2 π sin of φ D φ × the integral from 0 to π cos θ sin² θ cos sin D θ.2160

The integral from 0 to 2 π of sin φ D φ, let us just deal with this right here.2201

Recall the graph of sin φ, the graph looks like this.2210

From 0 to 2 Π of the sin function goes like this, here is 2 π.2227

The integral from 0 to 2 π of the sin function gives us that area and that area.2234

This is going to end up being positive, this is going to end up being negative.2242

The integral / this is going to equal 0.2245

The integral of the sin of φ sin of θ sin of X does not matter.2248

The sin φ from 0 to 2 π is equal to 0.2253

Let me write this out.2258

The integral from 0 to 2 π sin φ D φ is the shaded region.2261

This integral is equal to 0 because above the X axis is positive and below the X axis is negative.2272

The magnitudes of the area, if they ask for the total area,2286

you just take the absolute value of this particular portion and just get twice that of the integral itself is equal to 0.2291

Therefore, this is equal to 0 which means that the average value of L sub X which was this integral, is equal to 0.2300

For S10, we have the average value is equal to 0.2329

Let us go ahead and take a look at S11.2335

For S11, S11 is equal to 3/ 8 π ^½ sin θ E ⁺I φ.2344

Therefore, the LX of S11 is equal to -I H ̅ - sin φ DD θ – cot θ cos of φ DD φ.2364

Of course, we are operating on this function 3/ 8 π ^½ sin θ E ⁺I φ.2393

If you operate on this, you operate on this, and you are going to get some long expression.2406

Let us see what this actually looks like here.2411

We have – IH ̅ × 3/ 8 π ^½.2416

Hopefully, I got everything right here.2429

This is - sin φ the derivative of this is cos θ × cos θ.2432

And of course, I have to include that.2450

The product function × E ⁺I φ – cot of θ, the cos of φ × DD φ of this which is sin θ × IE ⁺I φ.2453

I will just apply this to that.2481

It is going to equal – I H ̅ 3/ 8 π ^½ × - sin of φ cos of θ E ⁺I φ – cot θ/ sin θ, this is cos φ,2485

this is sin θ × IE ⁺I φ, sin θ and sin θ cancel and I'm left with - I H ̅ 3/ 8 π ^½ ×, - and – is +.2518

Negative and negative this becomes positive.2554

We get sin φ cos θ E ⁺I φ + I cos θ cos φ E ⁺I φ.2558

We have to multiply this just LX of S11.2581

We have to take S11 conjugate multiply on the left by S11 conjugate L sub X of S11.2586

That is going to equal 3/8 π ^½ sin θ E ⁻I φ because it is S11 conjugate × this thing which is,2602

This goes away, we have actually taken care of a positive.2623

I H ̅ 3/8 π ^½.2627

This is crazy, this is actually crazy.2634

Sin φ cos θ E ⁺I φ + I cos φ cos of θ E ⁺I φ.2638

This is really something.2662

Let us see what we have got when we multiply everything out.2664

We have got the 3, I , H, = 3 I H ̅/ 8 π × sin θ cos θ sin φ E ⁺I φ.2669

E ⁻I φ and E ⁺I φ that goes away, + I × sin θ cos θ cos φ.2704

This is our final, the integral of S11 conjugate L sub X S11 is equal,2723

Let me go ahead and write it over here.2737

It is going to be I H ̅ × 3/ 8 π × the integral 0 to π.2740

0 to π of everything that we have just wrote, which is going to be sin θ cos θ sin of φ × sin θ cos θ cos φ × sin θ D θ D φ.2751

Let the software do this for you.2785

It all = 3 I H ̅/ 8 π × 0 to 2 π.2788

We are going to separate this φ and θ.2806

Sin φ D φ × the integral from 0 to π of sin² θ cos θ D θ + I × the integral from 0 to 2 π.2809

This is a cos φ D φ, make sure that you understand how was I separated this.2830

There is this one and there is the I part.2837

I separated two integrals, the integral of this and the double integral of that.2842

Within E I have separated out the φ and the θ, that is what I have done.2850

The integral from 0 to π of sin² θ cos θ D θ.2856

Once again, the integral of 0 to π of sin of φ, this is equal to 0, this is equal to 0.2872

Our integral is equal to 0.2881

Once again, the average value for S11 which is this integral, this whole integral = 0.2884

For S1 -1, I’m not going to go through the process.2895

S1 -1 you actually end up with the same thing that you get for S11.2899

For S1 -1, the integral ends up the same as for the S11.2904

The average value of S sub X for S1 -1 also = 0.2924

It is true in general.2938

We just had checked 3 of the spherical harmonics.2947

It is true in general that for all L, the average value of the X component of the angular momentum is equal to 0.2949

The average value of the Y component of angular momentum is equal to 0.2967

Let us not forget what the average value is.2973

If I take a particular measurement, I'm going to get whatever I happen to measure for the angular momentum.2975

On average, if I take 100 measurement, a 1000 measurement, a 1,000,000 measurements,2981

on average I’m going to get many different values that they are all going to average to 0.2986

That means for every one that I get to the left, I will get the same to the right.2991

For every one that I get going up, I get when going down.2997

For every one that I get going this way, I get when going that way.2999

On average, you can end up canceling, that is what is going on here.3003

What this means is that, remember we know what L² is.3008

We know the L², it is equal to H ̅² L × L + 1.3026

We know the magnitude of the angular momentum.3038

We also know this one, we also know the Z component of the angular momentum.3043

In other words, the projection of the angular momentum vector on the Z access is specified by M.3049

The magnitude is actually specified by L.3054

What we do not know is X and Y.3057

We do not know them specifically.3060

This is an application of the Heisenberg uncertainty principle.3062

This operator, they commute but they do not commute with the X and Y.3067

X and Y are uncertain which is why on average, we are going to end up being equal to 0.3074

What this means is that we can specify L² and LZ.3080

That we can do but we cannot specify the X and Y component of the angular momentum.3096

We know how long the angular momentum is.3121

We also know the X component of it, the problem is we do not know what direction3123

the angular momentum vector is at a given moment.3129

We can specify the X and Y, we can only tell you its projection along the Z axis.3133

We know how long it is but we do not know which direction it is pointing.3139

It can be pointing anywhere but we do know how long it is and how long the Z value is.3142

We will do a little bit more just in a moment.3150

This problem is actually sufficiently important to further discussion.3155

We did discuss in a previous lesson but I actually did go through it again here.3160

This problem is sufficiently important.3169

In fact, very important.3182

Angular momentum is everything in quantum mechanics.3184

This problem is sufficiently important to discuss it in context.3187

We know the magnitude of the angular momentum vector is equal to H ̅ × √ L × L + 1.3203

When I have L, I can tell you how long the angular momentum vector is.3214

It is just a function of L.3220

I also know the Z component of the angular momentum.3222

That is H ̅ M, that is specified by M.3226

I know what N is, because M is just -L to + L.3233

The quantum numbers L and M K.3238

The magnitude of the angular momentum.3249

The magnitude of L can never equal LZ.3254

This is a function of L and this is a function of M.3264

They are never going to equal each other because they never equal each other, they do not lie on the same direction.3266

Therefore, the actual angular momentum vector itself can never lie along the Z axis.3274

Let us see what we have got.3292

Let me actually go over here.3296

The average value of L sub X is equal to 0.3303

We just demonstrate it with the first few spherical harmonics but we know that is true in general.3307

The average value of L sub Y is equal to 0.3311

In other words, the X and Y components on average for the angular momentum are equal to 0.3317

These two equaling 0 mean that when we measure L sub X and L sub Y,3325

which are the components of the angular momentum along the X axis and along the Y axis.3340

Respectively, the values will average to 0.3345

We will get a bunch of values but the average is going to be 0.3358

This means that each and every value is represented.3365

For every positive value we get in the X direction, you have at least one negative in the X direction3369

because they have to cancel out to become 0.3375

Let us to take a look at what this actually means graphically and I think it will make sense.3380

Graphically, we mean this.3384

Let us see here.3393

Here is what is going on.3400

We have a coordinate system here, the Z axis is vertical.3400

In this particular picture, they have the Y axis lying this way.3404

They have the X axis going this way.3409

This is a 3 dimensional version.3411

We can specify the length of the angular momentum vector.3413

In other words, we can specify this vector right here.3417

That is the angular momentum vector, we know how long it is.3423

We also know its projection on the Z axis.3426

In other words, if I shine a light this way towards the vector from this side, this is going to be a shadow.3431

That is what we call the projection.3439

The projection of a vector on the Z axis is the Z component of that vector.3440

Again, we are talking about a vector in 3 space.3445

L it has an X component in the I direction, it has a Y component in the J direction, Y direction,3448

and it has a Z component in the Z direction which is specified by the unit vector K.3459

If we use this engineering notation I, J, K.3464

We know this and we know the magnitude of this.3467

That is this, we know how long it is.3472

We also know how long this is.3474

We know that value.3477

The problem is we cannot specify simultaneously all of them.3482

In other words, we do not know its projection along the Y axis.3487

We actually do not know this length and we do not know its projection along the X axis,3490

we do not know that it is a length.3496

The Heisenberg uncertainty principle, the L² operator and LZ operator they commute3499

but that does not commute with all of simultaneously.3506

The LZ and the LY do not commute.3510

The LZ and LX do not commute.3512

This commutes with one of them, one at a time.3514

We can specify how long the angular momentum vector is and we can tell you the projection3518

along the Z axis but we cannot tell you where, how far along.3524

This vector, we know that it is pointing this way.3529

But notice the projection down on to Y or down on to the X axis, we do not know how long these are.3532

Therefore, they can be any value at all.3541

This one could be one of this, this can be 5 but what the average value tells us is that for every one.3544

In other words, if I project this down onto a circle, it can be this way, this way, this way.3551

It can be in any direction along the circle in the XY plane, which is why you have this angular momentum vector.3564

I know the length, I know along Z but it is could actually sweep out at a cone because3570

I do not know where it is a long the X and Y directions.3575

I do not know, it can be anywhere in the X and Y directions which is why it sweeps out a circle in the XY plane.3579

And depending on where it is, you are going to get different values.3586

For L1, it is going to be this one.3589

For L2, it is going to be this one.3591

For L3, it is going to be this one.3593

It is going to sweep out a cone, that is what these columns are.3596

Recall the angular momentum cones because the vector itself, we do not know which direction is pointing.3599

We only know how long it is, that is all we can say.3606

We know its projection along the Z axis but we do not know in which direction it is going, so it can be in any direction.3609

Another picture that may or may not help, I do not know.3617

It is going to be the following.3619

This is a physical interpretation of the orbital angular momentum in quantum mechanics.3621

This is a way of thinking about it.3625

We know that in angular momentum vector, it happens when something spins.3626

Something spins this way, it has an angular momentum in that direction.3630

If it spins this way, its angular momentum is in this direction.3634

We can think of electrons spinning like that in orbit.3637

And it is going to give rise to this orbital angular momentum vector.3642

That is this vector right here.3647

We know how long it is, we also know its projection along the Z axis.3649

That is specified by M.3653

However, we do not know what its projection is along the Y or the X.3656

It can be anywhere along those.3661

It can be anywhere, which is why it actually sweeps out a cone.3663

That is what is happening.3669

This angular momentum vector sweeps out that way and because it can be this way or this way,3670

when we take measurements, for every measurement we get in one direction,3678

we are going to find a measurement of the opposite direction.3682

If we get a measurement this way, we are going to get a measurement this way.3685

When I add all of these up, they average to 0, that is what this means.3688

That is with this cone means.3692

Sometimes the angular momentum in the Y direction is going to be here, sometimes here, sometimes there.3695

That is what is going on, I hope this makes sense.3702

In any case, let us go ahead and leave it that.3707

Thank you so much for joining us here at www.educator.com.3711

We will see you next time for a continuation of more examples.3713

Take care, bye.3717

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