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Raffi Hovasapian

Raffi Hovasapian

Term Symbols & Atomic Spectra

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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Term Symbols & Atomic Spectra

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
  • Lyman Series 0:09
    • Spectroscopic Term Symbols
    • Lyman Series
  • Hydrogen Levels 8:21
    • Hydrogen Levels
  • Term Symbols & Atomic Spectra 14:17
    • Spin-Orbit Coupling
    • Selection Rules for Atomic Spectra
    • Selection Rules for Possible Transitions
    • Wave Numbers for The Transitions
  • Example I: Calculate the Frequencies of the Allowed Transitions from (4d) ²D →(2p) ²P 32:23
  • Helium Levels 49:50
    • Energy Levels for Helium
  • Transitions & Spin Multiplicity 52:27
    • Transitions & Spin Multiplicity

Transcription: Term Symbols & Atomic Spectra

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

Today, we are going to talk about term symbols and atomic spectra.0004

Let us jump right on in.0008

I'm sorry if my voice is a little scratchy today, I hope that I'm still well understood.0011

We talked about term symbols and let us see what the association is with term symbols with these atomic spectra.0017

Let us actually go to blue, I think.0024

Term symbols are also called spectroscopic term symbols or spectroscopic symbols0030

because the lines we see on atomic spectra,0055

like the one that you see below, which we will talk about in just a minute.0070

They represent transitions between electronic states.0073

Electronic states as we saw on the previous lessons, they are represented by term symbols.0097

We have seen a spectra like this before and we saw it earlier on in the course,0122

when we are talking about some of the early evidence for the quantum theory.0126

Let me go ahead and take an atom, we excite the energy to a higher level and then we allow0133

and it falls back down to, the excited electron fall back under to the ground state.0139

When we fall back down, they emit that particular excess energy as a photon of light.0146

They emit certain frequency that is what we see here.0152

The higher they go, they fall down, the higher the energy.0155

Of course at some point, it is some upper limit.0159

That is all these are, it absorbs light and energy at a certain frequency0162

and it spits it back out when the electron falls back down to the ground state.0167

This particular one is the Lyman series for atomic hydrogen.0172

Let me do red now.0182

This right here is the Lyman series for atomic hydrogen.0188

The Lyman series represents transitions from higher N values, the first quantum number N = 1, 2, 3, 4, 5.0213

Let me continue it down here.0237

These higher N values, you remember N is the primary quantum number, down to the ground state which is N = 1.0240

This represents transitions from level 2 to level 1, level 3 to level 1, level 4 to level 1, and so on.0262

Level 2 to level 1, that line.0272

Level 3 to level 1, that is that line.0278

Level 4 to level 1, that is that line.0279

We knock the electron, the electron of the 1S by the electron configuration 1S1.0280

We knock it up to our 2P and draws back down the 3P, drops back down we got the 4P, drops back down.0286

It is getting higher and higher energy but when it draws back down, it emits a photon of light.0296

Higher and higher energy, that is all that is going on here.0300

That is what spectra represent.0303

These names, do not worry about it.0305

Lyman α, Lyman β and Lyman gamma, that is just first level, second level, third level.0306

Or in this case, N = 2, N = 3, N = 4, that is unimportant.0311

Here, we have the wavelength given in angstroms.0316

Oftentimes, we will see spectra given in inverse cm.0320

For our purposes, we would be working mostly in inverse cm which is the wave number.0324

In other ways of converting between the two is not important.0328

What is important is the numerical value and the qualitative information that we get so far.0331

When we look at the first line, let us go ahead and take a look at this line.0338

When we actually look at that line under a higher resolution, in other words when we magnify it.0342

When we look at the first line and the second and the third actually, not just for the first,0354

when we look at these lines for the Lyman series, when we look at the first line under a higher resolution,0362

Let us actually spell things properly here.0371

It is getting a little ahead of myself.0374

Under a higher resolution, we end up seeing 2 separate lines.0376

This single line that appears in the spectra is actually made up of two individual lines that are close together.0384

We end up seeing two separate lines.0391

The question is, what is going on?0408

We excited from level 1 to level 2 and then it drops back down to the ground state.0411

It emits energy that is what we see here.0419

Given it is the photon of a certain frequency and that frequency in this particular wavelength happens to be that.0422

It jumped up from 1 state ground state to a higher state and it drop back down.0430

We should only see one line, why are we seeing two lines?0435

When we see two lines, each line represents the energy level.0438

Basically, what is happening is its jumping up and it is dropping back down but it seems like they are two different levels.0442

Either it is jumping up to two different levels and each of those level is dropping back down.0453

If this is the ground state, it is either jumping up to here and jumping up to there.0457

When it fallback down to the ground state, you are getting two lines0462

because there are two different energy levels very closely spaced,0467

that ends up looking like one line under a lower resolution or we have a double level jumping up to a single like that.0470

But these are going to end up being the same energy.0478

More than likely, what is happening is that you have the ground state and you have two higher energies0482

but what is it that is actually going on?0489

Let us take a look, we ask ourselves what is going on.0492

This is what is going on.0502

This is an energy level table for atomic hydrogen, this is pretty much what you are going to see0505

when you look at the standard table.0519

There is other information that could have been put in here but I have blocked out all that other information.0521

I just want you to concentrate on the information that absolutely is important.0526

Let us go through this very carefully and see what is that we have on this table.0530

This first column is the electron configuration 1S, 2P, 2S, 23P.0534

When you do not see a number here that just means it is 1 electron there.0540

This is actually the 1S1 configuration.0544

The term symbol, remember this is electron configuration.0548

In the previous lessons, we said that electron configuration is not enough.0551

We need more information than just the primary quantum number and the angular momentum quantum number.0554

We want to know what the spin states are.0561

We want to know as much information as we can.0564

We came up with this thing called term symbol.0567

For the 1S configuration 1S1, the ground state, the term symbol is a doublet S.0571

Its J value is ½, the term symbol for this is doublet S ½.0577

The energy level is 000, that is the ground state.0583

All ground states are listed as 0 energy.0587

Let us go to the second level N = 2.0593

Very interesting thing here.0596

The level 2 has a 2P and 2S.0600

And the 2P itself actually consists of, it has a term symbol.0608

It is a doublet P ½ and also a doublet P 3/2.0612

The 2P state, in other words, if I keep this 1 electron to the 2P1 configuration,0619

there are two possible energies that it can represent.0625

It can be the doublet P ½ which is this energy and it could be the doublet P 3/2 which is this energy.0630

The N = 2 there is also 2S.0637

That is the doublet S ½, that has this energy.0640

Notice that level 2 actually consists of 3 closely spaced energy levels.0646

We have the doublet P ½ which is this one right here, doublet P ½ which is 82258.9191 and this is in inverse cm.0677

Another one has the doublet P 3/2 that is energy 82259.2850.0691

And then we have the doublet S ½ which is the ground state.0699

Let me make this ½ a little bit clearer, that is 8258.9543.0704

Interesting is not it?0716

You got this 2589543, notice that I have arranged this table in term order not in energy order.0719

On the web site that I got this, the National Institute of Science and Technology0731

has this database of spectral lines and spectral energy tables, and things like that.0734

The way I arrange this particular table, I want it to be in term order.0741

I did not necessarily say energy order.0746

Notice that this energy 258.95 is actually higher than the 258.919, which is the 2P.0748

Of course, the highest is the 2P but it is the doublet P3/2 which is the highest energy, that is the 259.2850.0756

The level 2 actually consists of 3 closely spaced energy levels.0766

I will talk about this in just a second.0770

Let us spend a little bit more time on this energy table and just sort of getting accustomed to what it is that is going on here.0777

What you want to concentrate on what this is, once you have a particular configuration,0784

notice 2P 2S, this gives the full breakdown.0791

If you just want to consider the level 2 on its own without worrying about the individual breakdown into energy levels,0795

you can just go ahead and take this number right here.0801

That is all that number means.0804

This two is just sort of a combination, if you will, of all of these.0806

We are going to be concerned about is the breakdown.0811

The 2P level has a doublet P ½ and doublet P 3/2.0816

The 3P level, notice the 3P is the same thing.0821

The 3P this is 3P1, this is 2S1 2P1.0823

Again, if there is no number there it just means that 1 electron.0828

If there is an electron in the 3P1 energy level, it is been kicked up from the 1S1 and the 3P1.0832

The term symbol for the P1 configuration is a doublet P.0839

The doublet P has 2 energy levels, 2 states, doubled P ½ and doublet P 3/2.0844

The S and S1 have a doublet S ½.0850

When we solved the Schrodinger equation for hydrogen, we found that the energy of the electron dependent only on N,0858

which is the primary quantum number.0900

Again, level 2 and level 3 and level 4 like what we saw in the energy table.0907

Y then is split into 3 energy levels.0917

It should not be, it should just be 1 energy level based on N2, N3, and N4.0931

That is it, single energy level but it is not.0936

The answer is spin orbit coupling.0939

I will mention it, I will define it, and then we will not worry about it anymore.0943

Spin orbit coupling is the interaction of the magnetic moment caused by an electron’s intrinsic spin with the magnetic field,0960

induced by the electric current caused by the electron’s own orbital motion.1000

In other words, we have orbital angular momentum and we have spin angular momentum.1035

The fact it is orbiting, the fact that this charged particle is orbiting, it is moving around, it is causing a current.1040

That current creates a magnetic field.1050

Do you remember from general physics?1055

The particle itself, the spin of the electron itself also has a magnetic moment.1057

The interaction of those two, the magnetic moment and the magnetic field that is what causes this energy splitting.1063

You do not need to know any more about this spin orbit coupling.1071

Just know that this spin orbit coupling is actually going to take a certain level1074

and split that level into closely spaced energy levels.1080

It is not going to be just 1 energy level, when you jump up a primary quantum number.1084

It is actually going to be several, that is all that is going on here.1089

As we saw level 2 is split into 3 energy levels.1096

I hope you keep referring back to that table, 3 energy levels.1117

Those 3 energy levels were doublet P 3/2, a doublet P ½ and a doublet S ½.1121

The Lyman series under higher resolution shows only 2 lines.1131

The Lyman series shows only 2 lines, why does not it show 3 lines representing the 3 following jumps?1144

Why does it not show 3 lines for the following transitions?1163

We said that the Lyman Series, we are going to take the first line.1184

That is going to be from N = 2 to N = 1.1188

The transition from N = 2 to N = 1.1191

We saw that level 2 was actually split up into 3 energy levels.1194

Basically, we are wondering that means that the 1S1 electron should go to 2S1.1197

They can go to 2P1 here, you can go to 2P1 that term symbol, or it can go to the 2S1 that term symbol.1207

We should see 3 lines but we do not.1220

We only see 2 lines, why is that?1223

Again, technically it should show because we have 3 energy levels, should be the following.1225

It should be the doublet P ½ down to the doublet S ½.1234

The doublet P 3/2 down to the doublet S ½ and the doublet S ½ to the doublet S ½.1244

These represent the transition from the 2P1 configuration to the 1S1 configuration.1259

This one is the 2S1 configuration down to the 1S1 configuration.1263

And again, configuration S1 S1, notice the term symbols are the same.1268

However, they are different primary levels, level 2 to level 1.1276

We should see 3 lines, one for this transition, one for this transition, one representing this transition.1281

But we only see 2, why is that?1286

Here is the answer.1289

The answer is selection rules.1291

They are selection rules that stipulate the allowed transitions1299

that an electron can make from one state to another.1332

These rules are expressed in terms of the changes in the quantum numbers.1349

The selection rules are as follows.1365

For atomic spectra, the selection rules are δ L = + or -1, + 1 for absorption spectra, -1 for emission spectra.1367

We are mostly going to be concerned with emission spectra.1385

We are mostly going to be concerned with higher levels to lower levels.1387

It is the same, except in reverse.1392

You absorb energy of a certain wavelength, you release the energy of a certain wave length.1393

We just had to speak more of emission spectra.1398

Δ S = 0, δ J = 0, or + or -1, and the J = 0.1406

If J = 0 and there is a transition, J = 0.1416

The J = 0, this transition is not allowed.1422

In this particular case, δ J = 0 but if you are going from 0 to a 0 that transition is not allowed.1426

That does not work, that is not allowed.1432

For the 3 possible transitions that we just listed for the hydrogen atom from level 2 to level 1 and1436

the first line of the Lyman series, we have a doublet P ½ going down to a doublet S ½.1456

The doublet P 3/2 transitioning down to a doublet S ½ ground state.1467

A doublet S ½ that is the 2S1 electronic, it is the 2S orbital following down to the ground state which is also a doublet S ½.1474

Here, your δ L from P to S, = -1.1489

Your δ S22, in this case S is 0, S is 0 δ S = 0.1496

Here our δ J is ½ to ½, = 0.1502

That qualifies, this transition is allowed.1509

Because that transition is allowed, we see that line.1513

Let me put a little separation here.1518

Here, our δ L from P to S, remember S is the top left, L represents the middle, and J is down here.1522

2S + 1 S is 0, because these top numbers are the same, δ S is the same.1532

P to S, we are concerned with δ L.1539

P to S that is -1 δ S = 0, 3/2 to ½ δ J = -1.1541

This qualifies, this is also allowed.1552

Δ L -1, δ S= 0, these are the allowed transitions.1558

For this final one, SS here.1562

Δ L = 0, δ S = 0, and δ J = 0, this is not allowed.1568

This transition is not allowed because δ L = 0.1577

Let us go to red.1587

This one right here, this transition is not allowed.1589

Therefore, we do not see this line.1591

Therefore, 1 electron is jumping up from when we energize the electron.1594

The 1S1 electron of hydrogen.1605

It will jump up to the 2P.1607

It will jump up to this one and it will jump up to this one because there are two energy levels for the 2P.1612

However, the 1S1 will not jump up to the 2S1.1618

That transition is not allowed.1622

Therefore, when the electrons fall back to the ground state, they are going to fall back from the 2P levels.1624

The doublet P ½ to doublet ground state, the double P 3/2 down to the ground state.1629

We would to see 2 lines.1634

2 lines because they are 2 slightly different energies.1636

We do not see a third line because this is a not allowed transition.1638

That is it, that is all that is going on.1643

Let us go ahead and write that out.1648

Only 2 lines appear, the lines for the allowed transitions.1650

We have a doublet P ½ to the doublet S ½ and we have the doublet P 3/2 to the doublet S ½.1666

Do not worry, the numbers will drive you crazy because these are numbers and symbols all over the place.1676

The wave numbers for these transitions are calculated straight from a table.1688

That is it, you just take the higher level - the lower level.1699

The wave numbers for these transitions are calculated straight from a table.1703

These tables are great.1707

They are calculated straight from the table.1709

The doublet P ½, this transition doublet P ½, the wave number frequency1721

for that particular transition is we just look at the table.1740

I find the entry for the doublet P ½ level 2, and I subtract the doublet S ½ the ground state which is 0.1746

It is going to be 82258.9191 -0.00, that transition takes place at 82258.9191 inverse cm.1755

I see one of those double lines at that frequency.1775

The other one, the doublet P 3/2 to the ground state doublet S ½, that frequency = 82259.2850 -0.00.1781

That = 82259.2850 inverse cm.1800

Clearly, these are very close together.1808

It is easy to see how under lower resolution, it is just go look like one line but it is not one line, it is 2 lines.1812

The 1S1 electron of hydrogen jumps up to that 2 level, it is not allowed to jump up to 2S level but it will jump up to 2P orbital.1819

However, the 2P orbital is split into 2 energy levels.1831

This one, the doublet P ½ and the doublet P 3/2, one that drops back down it releases photons.1836

That frequency is going to release a photon of that frequency, of that wave number.1843

We will see these as individual lines, these lines are closely spaced together.1852

It is the first line of the Lyman series.1856

You are going to see doublets for all the lines in the Lyman series because the 3P, 4P, 5P, 6P, 7P, and so on.1858

They all consist of the same doublet P ½, doublet P 3/2.1865

When we fall back down that is what happens.1869

Let us see, the appearance of the greater spectral complexity, in other words the multiple lines does not have to be double.1876

It can be triple, quadruple, whatever, depending on how many levels there are.1908

Multiple lines are called fine structure.1913

It is called the spectral fine structure, I will just say it is called a fine structure of the spectral line.1918

Let me write this a little bit better, me and my fast writing.1929

Let us take a look at an example.1938

Using the energy table for atomic hydrogen below, calculate the frequencies and1944

wave numbers of the allowed transitions from the 4D level to the 2P level.1949

The transition is going from the 4D to the 2P.1957

This is a notation that you often see.1959

More often than not, I tend to put parentheses around the electron configuration1961

that we remember from general chemistry 3D, 4D, 5D, 3S, 3P, 1S, things like that.1965

I tend to put parentheses around that and then the actual term symbol.1975

The 4D electron configuration, the term symbol for that is double t D.1981

It is going down to the 2P level, that transition.1988

The 2P consists of a doublet P.1993

Let us take a look at those and let us see what transitions are actually taking place.1996

Calculate the frequencies of the allowed transitions.1999

Let us go ahead and go to 4D, that is going to be right here.2002

It is a doublet D, notice the doublet D, the 4D energy and the 4D level also consists of 2 energy levels.2014

We have a doublet D 3/2 and we have a doublet D 5/2.2025

This could be one of the things that we are looking at.2029

This is not the ground state.2033

The state is going to fall down to, it is going to be the 2P and the 2P we see that it has 2.2036

We have 2 coming down to 2.2044

There are going to be 4 possible transitions.2047

Out of those 4, we want to see which ones are actually allowed.2049

It is going to be the doublet D 3/2 down to the doublet P 1/2.2053

The doublet D 3/2 down to the doublet D 3/2.2058

The doublet D 5/2 to the doublet P 1/2.2062

The doublet D 5/2 to the doublet P 3/2.2066

Those are 4 possible transitions, there may be up to 4 lines.2070

4 lines is the maximum that we see for the spectra.2074

We want to calculate the frequencies for those and we want to see which ones are allowed first.2076

These are the values that we are going to take a look at.2081

Let us go ahead and move on to the next page here.2085

The 4D doublet D, that consists of a doublet D 3/2,that is at 1028230.8943inverse cm.2092

And it also consists of a doublet D 5/2 which is at 102823.9095.2110

The 2P configuration which is a doublet P, that is also consists of 2, that is a doublet P ½,2121

that is going to be 82258.9191.2131

And there is a doublet P 3/2 which is going to be 82259.2850.2139

The available transitions are to be, this to this, this to this, this to this, this to this.2150

Possible transitions are therefore, we would have the doublet D 3/2 down to the doublet P ½.2161

For this one, our δ L = -1, our δ S = 0, our δ J 3/2 + 1/2 = -1.2190

This transition is allowed.2203

We will see a line, we will calculate the frequency in just a minute.2204

We will calculate that and we are just going to see which ones are allowed and which ones are not allowed.2214

The next possible transition is going to be the doublet D 3/2 down to the doublet P 3/2.2217

Here, we have a δ L, the D to the P that = -1, that is good.2227

We have a δ S 22 that = 0.2233

3/2 δ J = 0, this one is also allowed, not a problem.2236

The next possible transition we have is the doublet D 5/2 down to the doublet P ½.2249

Here, our δ L D to P = -1, so far so good.2261

Our δ S value 2 2 = 0 and our δ J = -2.2268

This transition is not allowed.2280

This transition is not allowed because our δ J = -2.2283

Δ J can only be + or -1.2294

This line, this transition will not happen.2297

An electron will not go from the 4D configuration whose state is represented by doublet D 5/2 to the 2P configuration.2301

The state is represented by doublet P ½, it is not going to happen.2313

The selection rules would not allow it.2317

The final transition is going to be a doublet D 5/2 all the way down to a doublet P 3/2.2320

Here we have δ L = -1, we have δ S = 0, and we have δ J = -1.2329

This one is allowed.2337

In this particular case, we have 3 allowed transitions.2339

The fine structure is going to show 3 lines.2342

The 4 possible transitions only 3 are allowed.2353

You would be doing this the same way.2365

Always find the possible transitions, check the δ L, δ S, δ J values and see which ones are allowed.2367

That is it when I calculate frequencies.2373

This transition which in this particular case is 4D to 2P, but it could be any D down to 2P,2375

it belongs to the Ballmer series of atomic hydrogen spectra.2390

We have a line in series which was from any N level down to the ground state.2396

This one is any D level down to the 2P level, that is it.2401

It belongs to the Ballmer series.2408

There are a whole bunch of series for each atom.2413

The hydrogen goes on, the posh and series and things like that.2416

Belongs to the Ballmer series for hydrogen.2421

The line representing this transition, the single line representing this transition result of the 3 lines of the fine structure.2435

Results into 3 lines under higher resolution.2453

Let us go ahead and calculate the frequencies here.2468

We have the doublet D 3/2 transition to the doublet P ½.2471

For this particular case, our frequency is going to be the upper – lower.2482

It is going to be 102823.8943 - 82258.9191.2490

Our frequency is going to equal 20564.9752.2506

We will see a line at that wave number.2514

The next one, I will do on the next page.2525

The next transition we had was the doublet P doublet D 3/2 down to the doublet P 3/2.2533

Our wave number is 102823.8943 - 82259.2580.2544

Our frequency of that particular transition is going to be at 2564.6363.2562

We will see a line at that particular frequency, that particular wave number.2572

Our file transition, we have a doublet D 5/2 to a doublet P 3/2.2577

This one is going to be 102823.9095 - 82259.25802.2585

It is the process that matters.2606

20564.6515.2612

There we go, we have 3 lines.2620

Let us take a look at what is looks like graphically.2622

I think I have to put here, put it here, just put that there.2627

This is the 4D configuration, our doublet D term symbol, our doublet D state.2653

This is the 2P configuration, our doublet P.2661

It consists of 2 lines, this one is the 4D doublet D 5/2.2665

This is the 4D doublet D 3/2.2678

Here we have 2P which is the doublet P 3/2 and we have the 2P doublet P ½.2683

The doublet D is consists of 2 energy levels.2697

The doublet P is consists of 2 energy levels.2702

Here is the breakdown in energy, here are the transitions that we have.2704

The basic transition, when you see a single line it is just the transition from the 4D to the 2P.2709

The doublet D to doublet P.2725

Single line, that is what we see when we see a single line.2727

When we look at the spectra under lower resolution.2730

On higher resolution, what is happening is the following.2732

You have this transition, you have this transition, you have this transition.2735

Notice this transition does not exist when we look at the spectra.2744

We are going to see a line here, we are going to see a line here.2757

We are going to see a line here.2764

This one is the 20504.9572.2771

This one right here is the 20564.6515.2783

That one right there is the 20564.6363.2795

Single line, higher resolution, you are going to see 3 lines for this transition.2807

All of these under lower resolution is just one line.2812

That is it, you just magnify and look at what is going on.2816

Multiple energy levels, multiple transitions.2819

4 possible transitions and 3 are actually allowed so we see 3 lines.2822

The tables are used in these lessons are available from the National Institute of Science and Technology,2833

the comic spectra database.2882

If you have ever taken a look at the National Institute of Science and Technology,2887

anything to do with science and technology is right there.2891

Wonderful databases like a gold mine of scientific information, absolutely wonderful.2895

In any case, if you are interested I will go ahead and put down .2901

It is www.NIST.gov/pmi/data/atomic spectra database asd.cfm.2905

When you go there, you do not need to, all the tables that you need are all going to be provided for you in the book, in the back of a book.2927

On your exams, you do not have to have to go there.2935

But if for any reason you are interested, you want to be around little bit and see what is available to you.2938

You want to play around and see all the different ways you can actually have the output come out.2941

Remember, I said I did it in terms of term order.2946

You can do in terms of energy order, all the different of columns.2948

There are other columns you can have on there.2952

There are fewer columns, just depends on what is it that you want.2955

Check it out, when you go to the spectra database just click help and that will lead you through2957

how to actually choose the information in order to recover a particular spectral data.2964

If you want to do that, you will just go ahead and get in touch with me here at www.educator.com.2971

Send me a question on line and I will be more than happy to lead you through it.2975

Let us go ahead and finish this discussion here.2980

Let us go ahead and take a look at helium, the energy table for helium.2984

This is the energy table for helium, the energy levels of helium.2994

Let us take a close look at this.2997

We have 2 electrons 1S2 in the ground state that is here.3001

Let us go ahead and do this in red.3005

0 energy ground state, 1S2 is the configuration.3007

The term symbol for that is singlet S0.3011

The J value was put in a separate column.3015

This just means singlet S0.3017

We can kick up, we can excite one of these electrons to a higher energy level.3020

Let us say we kick one of the electrons up to the 2P level.3026

We have 1S1 2P1.3031

Something interesting happens, 1S1, 2P1, that is a triplet P state.3034

That triplet P state, I have actually 3 individual states.3041

It is going to be triplet P2, triplet P1, triplet P0, and these are the energies.3045

That all electron can actually go to any one of these.3050

However, notice we not only have a triple in the state but the 1S1 2P1 configuration also has a singlet P state.3053

And that one has just 1, it is a singlet P1.3064

The question becomes which one does it actually jumped to?3069

And if you just take a look at the rest, you will see it is often like that.3072

You will see like over here, we have the 1S 3D 1S 3D.3076

We have a triplet D state, we have a singlet D state.3081

We have the 1S 2S, the 2S there is a triplet state and there is a singlet state.3083

The 3S there is a triplet state and there is a singlet state.3092

For helium, you have triplet states and singlet states.3095

Here it demonstrates helium and that is not the only one that does, other atoms do as well.3101

Helium demonstrates more than 1 spin multiplicity.3109

This number on the top left that is the spin multiplicity.3117

More than 1 spin multiplicity for a given level.3122

For example, we solve the 1S1 2P1, that has a triplet state and has a singlet state.3132

You will know the other levels, they are triplet and singlet.3145

In the transition from 1S to the ground state to the 1S1 2P1 excite state, the possible transitions are 1S2.3153

We can go from a singlet S to a triplet P.3190

I have left of the J values, I’m just speaking using basic terms symbols.3199

Or I can go from the 1S2 state which is a singlet S.3202

I can go to the 1S1, 2P1, I can go to singlet P.3207

The first is not allowed.3213

The first is not allowed because δ S is not equal 0.3227

Let me go to black here.3233

This is 1, this is 3, δ S has to be 0 in order for the transition to take place.3234

Here and here is what is allowed.3247

When this transition takes place, the electron in the S orbital goes up to the single P state.3249

It does not go up to the triplet P state.3251

This means the transitions and we see this for many atoms.3256

This means that transitions are only allowed between states with the same spin multiplicity.3270

We knew that already because δ S have to be 0 but we thought it would be nice to actually to say it again.3297

The spin multiplicity is the 2S + 1, it is the number on the top left of the term symbol.3304

Enclosing the selection rules that we have presented are appropriate3312

and are valid for atoms with relatively small atomic numbers.3337

Say less than 30 or 35, I’m just throwing out a number right there.3360

At higher atomic numbers, as atoms get bigger and bigger,3367

all the selection rules that we threw out this δ S equal 0, these rules have to break down.3370

There are larger atoms where states between singlet or transition do actually occur between singlet and triplet states.3377

With a spin multiplicity, when δ S does not have to be 0.3386

These set of selection rules that we threw in out and at this level is absolutely appropriate.3390

This is sort of where we would be staying.3394

We are not going to be worry about spectra for higher and heavier atoms.3397

This is appropriate mostly for relatively low atomic number atoms.3400

We just thought you should know that these selection rules are not set in stone.3406

It is not like this across the board.3411

As atoms get bigger, the selection rules break down.3413

We just want to let you know that.3415

At higher atomic numbers, these rules start to break down and all kinds of transactions take place.3418

It is very interesting, very complex.3449

I will go ahead and leave it that.3459

Thank you so much for joining us here at www.educator.com.3460

We will see you next time to work on some example problems for term symbols and atomic spectra.3462

Take care, bye.3468

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