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Raffi Hovasapian

Raffi Hovasapian

Spin Quantum Number: Term Symbols I

Slide Duration:

Table of Contents

I. Classical Thermodynamics Preliminaries
The Ideal Gas Law

46m 5s

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

46m 2s

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

1h 6m 45s

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

1h 6m 33s

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

1h 2m 17s

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

1h 4m 39s

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

16m 50s

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

43m 40s

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

32m 23s

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

39m 15s

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

35m 52s

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

42m 40s

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

1h 23s

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

44m 34s

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

59m 7s

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

49m 16s

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

33m 59s

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

54m 37s

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

31m 18s

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

23m 6s

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

25m 42s

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

43m 39s

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

56m 44s

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

57m 6s

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

54m 35s

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

35m 5s

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

28m 42s

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

34m 38s

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

30m 50s

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

34m 6s

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

39m 18s

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

19m 40s

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

54m 16s

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

31m 17s

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

45m

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

58m 5s

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

25m 20s

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

34m 25s

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

59m 57s

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

42m 5s

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

30m 26s

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

21m 34s

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

56m 22s

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

45m 24s

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

48m 43s

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

46m 18s

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

39m 28s

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

35m 32s

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

29m 55s

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

54m 25s

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

46m 58s

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

44m 11s

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

35m 33s

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

43m 4s

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

26m 30s

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

41m 10s

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

30m 32s

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

35m 19s

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

33m 48s

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

46m 43s

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

40m

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

35m 58s

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

36m 18s

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

33m 55s

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

51m 53s

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

51m 53s

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

34m 29s

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

43m 49s

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

1h 1m 57s

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

48m 33s

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

48m 33s

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

48m 33s

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

59m 18s

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

34m 54s

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

38m 3s

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

57m 49s

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

1h 1m 20s

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

56m 34s

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

18m 24s

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

50m 2s

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

59m 47s

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

46m 22s

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

29m 24s

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

30m 53s

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

1h 1m 33s

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

33m 47s

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

1h 1m 5s

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

33m 31s

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

1h 1m 15s

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

47m 23s

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

54m 9s

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

48m 32s

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

57m 30s

Intro
0:00
Example I: Make a Plot of the Fraction of CO Molecules in Various Rotational Levels
0:10
Example II: Calculate the Ratio of the Translational Partition Function for Cl₂ and Br₂ at Equal Volume & Temperature
8:05
Example III: Vibrational Degree of Freedom & Vibrational Molar Heat Capacity
11:59
Example IV: Calculate the Characteristic Vibrational & Rotational temperatures for Each DOF
45:03
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Lecture Comments (5)

2 answers

Last reply by: Professor Hovasapian
Wed May 11, 2016 2:43 AM

Post by Tram T on April 23, 2016

Dear Prof. Hovasapian,

If Energy En of Hydrogen atom only depends on the Principal QN, then why 2p orbitals have higher Energy than 2s orbital? I thought they are all have E sub 2 energy.

Thank you!

1 answer

Last reply by: Professor Hovasapian
Fri Apr 10, 2015 12:02 AM

Post by dulari hewakuruppu on April 9, 2015

in the multi electron atoms section, where you considered 2P1 of Boron, if we know L we know about m sub L as well right? you mentioned that we dont know about Ml nor Ms so this kind of got me confused.. I dont know if I am wrong though.. could you kindly explain? :) thank you

Spin Quantum Number: Term Symbols I

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
  • Quantum Numbers Specify an Orbital 0:24
    • n
    • l
    • m
    • 4th Quantum Number: s
  • Spin Orbitals 7:03
    • Spin Orbitals
    • Multi-electron Atoms
  • Term Symbols 18:08
    • Russell-Saunders Coupling & The Atomic Term Symbol
  • Example: Configuration for C 27:50
    • Configuration for C: 1s²2s²2p²
    • Drawing Every Possible Arrangement
    • Term Symbols
    • Microstate

Transcription: Spin Quantum Number: Term Symbols I

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

Today, we are going to start talking about something called term symbols0004

and we are going to be spending 2 or 3 lessons on this.0009

Before I discuss terms symbols, I want to discuss a 4th quantum number, it is called the spin quantum number.0011

We will introduce that and then we will get into the topic of term symbols.0018

Let us go ahead and just get started.0023

I think I will go ahead and stick with blue today.0026

Three quantum numbers specify an orbital, a wave function.0034

A wave function represents an orbital for the hydrogen atom that we saw.0048

We can also call it a quantum state if we want.0055

Again, like I said we can just go ahead and call it a wave of function which is exactly what it is.0063

We have N which is the primary quantum number and it takes on the values 1, 2, 3, and so on.0070

When we have L, it is the angular momentum quantum number and it takes on the values of 0, 1, 2, all the way up to N -1.0081

L depends on N and we also have this one, we have the M.0093

Or I'm going to start occasionally putting a subscript of L on there.0098

M or M sub L is the magnetic quantum number and it is dependent on L, which is why we have this subscript L.0103

It is going to take on the values of 0, + or -1, + or -2, all the way up to + or – L.0111

There is a 4th quantum number.0124

I’m not going to spend a lot of time here or talking about where this 4th quantum number comes from.0126

I’m just going to introduce it, tack it on to the three quantum numbers that we have.0135

If we need to go a little bit deeper into what this 4th quantum number is, we will.0144

But I do not want to just unload a bunch of information that is unnecessary at this point.0150

For right now, just know that there is this 4th quantum number and its designated s and is called the spin.0155

In your books or, in any discussion you see online, they are going to talk about how the electron has this intrinsic property called spin.0175

If you want to correlate with why we chose the word spin, you can think of a electron as a ball and a ball is spinning.0186

A ball can spin in 2 directions if you are holding it like this and either spin to the right or it can spin to the left.0195

If you spin it to the right using the right hand rule that you remember from physics,0202

as it spins to the right, your thumb is pointing up.0207

It is up spin.0209

If it is spinning this way and if your fingers going to the direction of the spin, your thumb is pointing down.0211

It is down spin.0215

Now, the electronic is not actually spinning but you can think about it that way, if it helps you.0217

Spin is an intrinsic quality of the electron like its mass.0222

It is just its quality so spin is the 4th quantum number, it has only one value.0227

It has only one value for the electron and that is ½.0236

The actual quantum number that we are going to use and we often see are the M sub S.0244

Sort of like L and M sub L, we have S and we have M sub S.0268

This is the one that has the two values that you have probably seen before.0274

The + ½ and the -1/2, for up spin and down spin, respectively.0279

This is a spin quantum numbers.0286

An electron with a given set of N, L, and M sub L can have M sub S = + ½,0296

which we call up spin or M sub S = -1/2 which we call down spin.0323

I’m not going to hop on this whole idea of the spinning top or spinning electron, things like that.0333

It is not altogether important, it is an intrinsic quality of the electron.0338

When we have just N, L, and M sub L, we have an orbital.0344

We have a quantum state and a wave function.0366

When we add the 4th quantum number to it, when we add M sub S,0371

when we the 4th quantum number M sub S, I would call it something else.0378

Instead of an orbital, we would call it spin orbital.0393

We call these spin orbitals.0399

A spin orbital is just an orbital that actually includes the 4th quantum number.0410

Let us go ahead and change here.0424

Each orbital, each choice of N, L, and M sub L, has 2 spin orbits.0427

We know this already from general chemistry.0446

We know that each orbital that we come across is going to have 2 electrons in it in maximum.0448

One with up spin and one with down spin.0454

Each orbital has 2 spin orbitals.0456

There was 2 places for an electron to go.0461

Again, we already know that an orbital can hold at most 2 electrons of opposite spin.0473

It is not a problem to go ahead and use the things that we learned in general chemistry.0482

I know that that stuff actually comes from this, it comes from the quantum mechanics.0490

It is okay take to use this information, things that we already know to help us understand what is happening.0496

We already know that at least at this level, each orbital can hold at most 2 electrons of opposite spin.0504

For example, we have the 2P.0529

In this particular case, N = 2, L = 1, and M sub L = 1, 0.0536

I’m not going to use comma, if you do not mind.0546

It is 1, 0 and -1.0549

M sub L is, from –L to L passing through 0.0553

We already what this is, this is the P sub X, P sub Y, P sub Z.0557

This is the 2 P sub X that is N = 1.0563

The 2P sub Z is N = 0.0570

Notice, I’m putting the Z before the Y.0572

And because we decided to label it that way.0575

When N sub 0 and M or M sub L is equal to 0, it is actually the Z orbital.0578

0, and we have 2 P sub Y, this is M = -1.0586

1 orbital, 2 orbitals, 3 orbitals.0597

3 orbitals, each orbital can hold 2 electrons of opposite spin.0601

We have a total of 6 spin orbits.0609

Again, nothing strange here.0613

In other words, there are 6 places for electrons to go.0616

In the case of a single electron in the 2P orbital, there are 6 possible places for it to go.0628

For multi electron atoms, so we have dealt with hydrogen now0667

we are going to start dealing with helium, lithium, boron, and so forth.0671

For multi electron atoms, we simply add 1 electron at a time to successive orbitals,0678

within the confines of the exclusion principle.0710

An individual orbital can only hold 2 electrons maximum of opposite spin.0712

In general chemistry, what we did is we fill them up one at a time.0723

Like for oxygen, 1S2 S22 before, we ended up feeling 1 electron here up spin and 1 electron here parallel spin.0731

Another electron here parallel spin and then we went back and filled another one down spin0738

and work our way up until we close the shell.0745

For multilevel atoms simply add 1 electronic time to successive orbitals.0748

You know this from general chemistry.0756

The order of filling is, we have 1S2 and 2S2, 2P6, 3S2, 3P6, 4S2, 3D10, 4P6,0762

that just in a higher and higher energy levels.0785

We just throw electrons into it.0787

5S2, 4D10, 5P6, and so on.0791

I’m filling up the F orbitals.0798

Let us take a configuration like 1S2, 2S2, 2P1, this is boron.0800

Let us examine the 2P1 electron.0829

We have N which is right here and we have P, which means that we have L.0842

N is equal to 2, L is equal to 1.0854

We have information on N and L, but notice the 2P1 does not say anything about the M sub L.0858

It does not say anything about the M sub S.0874

In other words, this one electron has 6 possible places for it to go.0877

The M sub L, this can be either 1, 0, or -1.0880

In each of those cases, M sub S can be + or – ½.0886

M sub L = 0 can be ± ½ and the M sub L -1 can be ± ½.0893

This electron configuration that we know from general chemistry does not really give us a lot of information.0900

It does not tell us what primary level it is and gives us an angular quantum number0905

but it does not really give us much more information that we want to know where the electron are.0911

Are they up spin or the down spin, which orbitals they are in?0917

What suborbital they are in?0921

What is the N, what is M sub L, what is the M sub S?0922

I will repeat that by writing it down.0926

Notice that 2P1 says nothing about M sub L or M sub S.0928

We do not know if the electron is in 2 PX, 2 PY, or 2 PZ.0956

We do not know where this spin on this electronic is + ½ or -1/2.0981

This is important information.0995

There are 6 places for it to be.1000

We need a notation that gives us more information than standard electron configuration.1007

This 1S2, 2S2, 2P1, we need a notation that gives us more information than our standard electron configurations1033

about where an electron is exactly that come from the M sub L and what spin state it is in.1055

That comes from the M sub S.1081

Here is where we introduce term symbols.1085

The topic of term symbols.1090

The quantum numbers that you know the N, L, M sub L, and M sub S, notice that they are small letters.1101

Term symbols, we decide use capital letters, that is the difference.1108

When you see a capital letter, it refers to a term symbol.1112

How do we introduce term symbols?1120

That was the challenge, we will do our best here.1122

For a given configuration, when we talk about configuration, we are talking about a general state configuration.1125

For a given configuration, we will determine three things.1139

We will determine L, this will be the total angular momentum also called the total angular momentum quantum number.1150

Notice, small L is angular momentum quantum number.1168

Capital L is the total angular momentum quantum number.1171

Again, this is for more than 1 electron.1174

For hydrogen, it was just 1 electron.1176

L represents the angular momentum quantum number for the electron.1178

L represents adding up all the l for each electron in a multi electron atom.1182

Lithium for example, it has 3 electrons, so each one has an l value.1188

We are going to add those up and it is going to give us the L.1194

We will go into details just a little bit.1197

We are also going to calculate the total angular momentum.1199

S this is the total spin angular momentum.1204

Let me go back and talk about why it is called the spin.1222

An electron, by virtue of its orbital motion, imparts a magnetic field.1225

In other words, it is angular momentum that comes from the fact that the electron is spinning around the nucleus.1238

L represents the total orbital angular momentum.1245

The angular momentum that comes from the fact that these electrons are spinning around the nucleus.1256

This is the total orbital angular momentum.1260

An electron, by virtue of its existence, also has an angular momentum component1266

that it adds to the angular momentum of the total atom.1275

By virtue of its orbital motion, the electron has angular momentum.1283

And by virtue of its spin state, it has angular momentum.1286

The total orbital is L, the total spin angular momentum is S.1290

Of course we have something called J, that is the total angular momentum.1294

It is going to be the sum of the L + S.1298

J is going to be L + S, this is the total angular momentum.1301

The angular momentum that come through orbital motion.1309

The angular momentum that comes from spin state.1311

The scheme in adding up all the small l, adding up all the s, adding up the L + S is called Russel Saunders coupling.1324

In other words, we are going to couple the orbital angular momentum1348

with the spin angular momentum to come up with a total angular momentum for a particular multi electron atom.1351

The scheme is called Russell Saunders coupling and gives rise to the atomic terms symbol.1360

Atomic term symbol looks like this.1374

There is going to be and L value and on the left is a superscript is going to be 2 times S +1.1383

It is going to be a number.1391

On the right subscript, there is going to be a J.1393

Notice LSJ, LSJ, this is the atomic term symbol.1397

This is what is going to look like.1402

L represents the total, we call L the total orbital angular momentum quantum number.1405

We call S the total spin angular momentum quantum number.1429

We called J, let us make J a little bit better J here so it does look like something else.1440

We call J the total angular momentum quantum number.1445

The capital letters, they represent quantum numbers for multi electron atoms.1459

A hydrogen atom has N, L, M sub L, M sub S.1464

A multi electron atom has, because we are talking for each electron, it has an L,1471

it has a angular momentum quantum number just like the hydrogen atom did.1481

It has an S spin quantum number and it has a total angular momentum quantum number.1484

These are the quantum numbers that are used for multi electron atoms.1489

That is the correlation.1495

We have the quantum numbers that we have for the hydrogen atom.1497

We have quantum numbers for multi electron atoms.1499

That is the only difference.1501

When you see capital letters, we are talking about multi electron atoms.1503

Here is where it is interesting.1510

Couple of things I would like to say here but I will write them down.1513

The only way to really see what is happening is to manually workout1516

all of the possible arrangements for a particular electron configuration.1547

In other words, I'm going to pick a particular electron configuration and1570

I'm going to actually work out all of the possible ways that however many electrons we choose, going to which orbital.1573

You need to see every single possibility.1581

What we are going to do is we are going to group those into ones that have different energy levels.1583

Those groupings are going to be the term symbols.1589

It is going to give us a very detailed information about exactly where the electron is1592

and what configurations it can possibly take depending on its energy.1597

The only way to make sense of the term symbols, I can go ahead to wrap the process and1601

show you how to come up with a term symbol but you have to work out at least one manually just to see it.1606

It is the only way to do it, you only wrap your mind around.1613

The other way that is happening is to manually work out all the possible arrangement1616

for a particular electron configuration and see how these arrangements fall into groups1620

that is represented by the terms symbols.1643

Let us look to the configuration for carbon, 1S2, 2S2, 2P2.1665

We do not need to look, notice the 1S2, 2S2 are close shells.1700

We do not need to include those electrons in our scheme for coming up with a term symbol.1704

We will talk a little bit about why later on, but just real quickly, notice 1S2.1710

I will talk about that later but just understand that when you are doing terms symbols for a configuration,1718

you do not have to look at close shells.1723

You just have to look at everything that comes afterward, the orbit shells.1725

Let us just go ahead and write that down.1732

We do not need to look at closed shells.1734

In other words, you do not have to worry about these 4 electrons.1746

We only have to worry about these 2 close shells.1747

I will explain why when we actually done the process and wrap our minds around this.1753

2P2, the N itself does not matter.1760

In this particular case, they are all the same level.1767

They are all the same N value.1770

The 2P2, also N does not matter.1775

We are only concerned when we talk about a particular terms symbol is which suborbital they are in, S and P, and electrons and nodes.1784

Those are the one that we are concerned with.1795

In this particular case, we have just the P2 configuration.1797

We have 2 electrons that can be distributed among 6 spin orbitals.1803

2 electrons that can be distributed among 6 spin orbitals.1813

The P suborbital has 3 suborbital.1832

Each orbital has 2 spin orbitals, and 2 times 3 is 6.1839

2 electrons that can be distributed among 6 spin orbitals, because P implies that L is equal to 1.1842

M sub L is equal to 1, 0, and -1 or P sub X, P sub Z, and P sub Y.1855

In other words, I can actually draw out every single possible configuration.1872

Let us draw every possible distribution.1879

I will use the word distribution, let us use the word arrangement.1891

Every possible arrangement of the 2 electrons in 6 different spin orbitals.1895

Let us draw each one out.1904

I’m going to be calculating, once I do the drawings.1906

Next to the drawings I’m going to calculating some numbers.1916

I'm going to be calculating the sum of the individual M sub L and that is going to be M sub L.1919

I'm going to be adding up the individual M sub S.1929

That is going to equal M sub S and I'm going to end up also writing M sub L + M sub S is going to equal M sub J.1936

You see this pattern that is developing.1956

There is an L, there is an S, and there is a J.1959

There is M sub L, M sub S, and M sub J, just like we have the L and M sub L, this is the correlation that we are developing.1964

We are trying to keep things parallel.1976

These three numbers. M sub L is just adding up the individual M sub L for that particular arrangement.1982

Let us go ahead and do that.1989

It is going to take a little bit while here.1990

I'm hoping I can actually get to it solve on one page.1994

I'm going to try my best to.1996

I have got 12345678910.1999

Let us do 123, 123, 123 123, that is 4, 5.2005

123, 123, 123, 123 and 6 123456789 and 10.2018

Make this one a little bit better.2036

Let me go over here and do another set.2039

123, 123, 123,23, 123, that is 15.2041

123, 123, 123, 123, 123, 12345678910.2052

And we need one more, let me write that over here actually.2067

It is not a problem, let go over here.2072

This page, as we move forward, you do not have to keep referring to this page.2079

I, myself, can actually go back but you, yourself can by just rewinding and looking back on this page2086

when you see how is that we are going to take the arrangements.2094

Anyway, let us go ahead and draw out all the possible ways that 2 electrons can be distributed among 6 different spin orbitals.2097

This is PX PZ PY, these are P orbitals.2104

How many different ways can I actually put 2 electrons into this?2108

You just you have to do this, that is one possibility.2114

You have to take your time account for every single possibility.2122

It is going to turn out that they are 21 different ways of doing this.2125

That is another possibility, do not worry about whether it violates the exclusion principle or not yet.2133

We just want to throw all the possibilities out there and we will cross off those that we can use for one reason or another.2141

That is another possibility, 2 electrons can go there, you can have down spin.2149

You can have that, you can have that, you could have that.2155

2 electrons can go there, 2 electrons can go there.2163

You can have one up spin, one down spin.2167

You can have one down spin, one up spin.2170

You can have up down here, you can have down.2174

You can have up, you can have up.2178

You can have down, you can have down.2180

You can have up, you can have done up and down the same or done the same.2183

These are the 21 different ways that you can have 2 electrons and distribute them in among 6 spin orbitals.2195

Let us go ahead and calculate what we said the M sub L and M sub S and the M sub J.2208

This is M sub L1, M sub L0, M sub L-1.2221

We are going to add up the M sub L.2228

In this case, it is 1 to 0.2230

M sub L could be the first number.2233

That is going to equal 1 + 0, it is going to equal 1.2236

M sub S, I'm just going to add up the spins.2241

This is up spin and up spin, ½ + ½, ½ + ½ S= 1.2244

The sum of those 2 M sub J is equal to 2.2252

I’m collecting some numbers here.2260

M sub L M sub L 1, 0, -1.2264

In this particular case, M sub L = 1, + and -1 that is equal to 0.2268

The M sub S is still ½ + ½ because it is up spin and up spin, so it is equal to 1.2277

The sum of these 2, the M sub J is equal to 0 + 1 is 1.2285

Here, I'm going to go ahead and drop the M sub L and M sub S.2298

The first I’m going to calculate is going to be the M sub L and the second numbers M sub S.2303

And the sum is going to be M sub L sub J.2307

In this particular case, I have a 0 and -1.2310

0 + -1 = -1, they are both up spin.2315

½ + ½ = 1.2321

In this particular case, the M sub J is equal to 0.2326

Here we have 2 electrons, they are both in the same orbital and they both have up spin.2331

This is a violation of the exclusion principle.2335

These we can just throw out.2340

The M sub L is 1, 0, -1.2350

1 + 0, the M sub L is equal to 1.2353

The spins, they are both down spin.2359

-1/2 + -1/2 = -1.2362

M sub J is equal to 0.2369

Here 1 + -1 = 0 down spin -1/2 + -1/2 = -1.2371

Here M sub J = -1, 0 -1, I’m just going to start doing these quickly.2388

0 -1 = -1, - ½ - ½ = -1.2395

M sub J = -1 and -1 = -2.2415

Here we have a violation of the exclusion principle.2421

We have a violation, we have 2 electrons of the same orbital of parallel spin.2428

Here we have a violation.2434

Here, we have 1 + 0, the M sub L, 1 + 0 is equal to 1.2437

And here we have up spin and down spin, ½ and ½ at 0.2446

Here MJ is equal to 1.2452

Over here we have the same thing.2456

It is going to be 1 + 0 = 1, this time it is - ½ + ½ that is equal to 0.2460

Again, we have MJ is equal to 1.2472

1 + -1, let me write -1.2477

1 -1 = 0 so that is M sub L.2484

½ - ½ = 0, that is M sub S.2488

Therefore, our M sub J is equal to 0.2494

Same thing here, here we have 1 -1 is equal to 0.2497

Here we have - ½ + ½ is equal to 0.2502

Again, we have an M sub J equal to 0.2507

We have 0 and -1, 0 -1 = -1, that is our M sub L.2510

We have ½ - ½ = 0.2521

Our M sub J = -1.2526

Here we have 0 and -1, 0 -1 = -1.2530

We have - ½ spin + ½ spin = 0.2536

We are left with M sub J = -1.2540

Over here, we have 1 + 1 is equal to 2.2546

½ - ½ is equal to 0.2552

Our M sub J is equal to 2.2556

Here we have, 0 + 0 are both in the Z orbital.2560

0 + 0 = 0, that is our M sub L.2566

Up spin down spin ½ - ½ = 0, that is our M and S.2570

Therefore, our M and J, the sum of those two is equal to 0.2576

Over here, we have -1 -1 which is -2 and then we have + ½ - ½ = 0.2581

Therefore, our M sub J = -2.2602

I will go ahead and erase this.2606

This is it, these are the total number of ways that 2 electrons can be distributed among 6 spin orbital.2609

6 or more violations 123456, 123456.2617

Yes, that leaves 15 possible, 15 viable ways that the 2 electrons that the 1S2, 2P2, that 2P2 configuration, there are 15 possible places for those electrons to be.2624

What we are going to do is we are going to come up with terms symbols to tell us which configurations they can be and2638

what the energies of those configurations are and then all of those terms symbols that2648

we are going to find one that is in ground state, that is where we are interested in.2653

You are going to be referring back to this, I myself once I move forward, I cannot move back.2659

But you, yourself can move back.2664

When I make a statement, just come back here and take a look.2666

The important numbers are going to be M sub L and M sub S.2669

For right now, those are the one that we concern ourselves with in the next set of steps.2674

This 1 and 1, this 0 and 1, this -1 and 1, all of these numbers.2679

The first one is the M sub L, the second one is M sub S.2686

Let us go ahead and see what we can do.2693

We looked on this list and we are going to look for the largest value M sub L.2698

The largest value M sub L is 2.2704

That is what we do, first step.2723

The largest M sub L is equal to 2.2725

This implies that L is equal to 2.2734

The actual values of the M sub L, they are represented, are going to be 2, 1, 0, -1, and 2.2748

Remember, just like we had the L and the M sub L for the hydrogen atom number,2757

if L was 3 then M sub L had the values of 3, 2, 1, 0, -1, -2, -3.2764

This is the same thing, there is a correlation here.2773

There is a L and there is the small values M sub L takes on all of these values.2775

We chose the largest one to account for everything from the largest to smallest.2782

Let me get the largest M sub L that implies L =2.2787

Therefore, all the possible values M sub L are 2, 1, 0, -1, and 2.2794

I hope that makes sense.2799

We have L is equal to 2, we found our L.2805

For this particular ML is equal to 2, the largest MS value for the ML = 2.2812

The largest MS value is equal to 0.2823

This implies, if the MS is equal to 0 that means S is equal to 0, which means that the only value that MS actually has, M sub S has is 0.2837

In other words, the S are 1, we have 1, 0, -1.2850

If S were 2, we have 2, 1, 0, -1, -2.2854

If S were 3, we have M sub S = 3, 2, 1, 0, -1, -2, -3.2858

There is a correlation.2866

L, all the values of M sub L from + L to –L.2868

S, the M sub S, all the values from +S to –S, passing through 0.2875

Let us go, M sub S is equal to 0.2885

S is equal to 0.2888

I’m going to go ahead and before I actually write the term, you remember the L values.2894

Let me draw a little bit of something here.2903

Remember, we set the terms symbol is L 2S + 1 and J, we found that L is 2.2910

We found that S is 0.2918

2 times 0 + 1 is 1.2919

However, instead of writing the number 2 here, we would actually use a letter.2922

The correlation is as follows.2927

If you remember L, the small L from the hydrogen atom 0, 1, 2, 3, 4, we call this S, P, D, F, G, and so on.2928

It was the same correlation.2946

When L is equal to 0, 1, 2, 3, 4, we call this the S term, P term, the D term, the F term, the G term.2948

In this particular case, we found that L is equal to 2 that correspond to D.2965

S is equal to 0, there is that correspondence that we set up because we prefer to use letters.2970

L is equal to 2 means D term.2979

S equal to 0 means 2 times 0 + 1, 2S + 1 is equal to 1, D1 term symbol.2988

We found our first term symbol.3011

There are some arrangements that fall into the D1 level.3014

It is a term that represents the particular arrangements that have the same energy.3018

Everything is D1 level, all those arrangements they have the same energy.3026

How many are there?3030

Since ML is equal to 2, 1, 0, -1, -2, and M sub S is equal to 0, this comes from L = 2, this comes from S = 0.3034

There are 5 microstates.3057

In other words, arrangements in the D1 level.3067

The number of M sub L times the number of M sub S.3084

This is 5 times 1 = 5.3091

There are five of those arrangements that we drew on the previous page.3097

Five of those arrangements belong to the D1 energy state along the D1 term.3100

We have grouped five of them.3109

Of the 15 viable candidates, they all have the same energy.3111

In other words, the electrons, they are 5 different arrangements of them but all of them have the same energy.3117

That energy is represented by this term symbol D1.3121

In other words, the total angular momentum for those things is going to be 2.3127

The total spin angular momentum for the state is going to be 0.3132

That is what this means.3139

D tells you that L = 2, that is the total angular momentum.3141

2S + 1 is equal to 1.3144

When you solve that equation 2S + 1 = 1, we get S = 0.3147

That means that the spin angular momentum for those states is going to be 0.3152

We will worry about J a little bit later.3157

I hope this last part makes sense.3161

There are five of those arrangements that fall into this category.3166

Let us go ahead and see.3172

Let me just finish up what I want to say here.3176

For each combination of M sub L and M sub S, we choose a microstate.3177

In other words, what you are going to end up actually doing,3205

you are going to go back to the page where we drew out all those individual arrangements.3207

What we call them microstates, the possible arrangements for 2 electrons and 6 spin orbitals3210

and you are going to choose M sub L, 2M sub S0.3215

You are going to choose that microstate, that is going to belong to D1 group.3220

You are going to choose 1, 0.3225

You are going to choose 0, 0.3228

You are going to choose the one with ML =-1 and MS =0.3229

Then, you are going to choose the one with ML = -2 and MS =0.3233

That is what we are doing here.3237

Those all belong to the D1 level.3239

For each combination of ML and M sub S, we choose a microstate.3242

If there is a choice of more than one, in other words if you have 2 microstates that have 1,0 and 1, 0, just choose one of them.3247

It does not matter which one you choose.3266

If there is a choice of more than one for each M sub L and M sub S, it does not matter which you choose.3268

The reason is because that is the same energy.3292

Either what you choose but pick only one.3295

When go back to the H where we draw all those microstates, for every value of M sub L 2, 1, 0, -1, 2, and this equal to 0.3306

2, 0, 1, 0, 0, 0, -1, 0, 2, 0, those that have those numbers, I’m going to choose those microstates.3318

Here is the one that I actually pick out.3327

They have 123, 123, 123, 123, and 123.3330

There is this one, there is this one, there is that one, there is that one, and there is that one.3339

In this particular case, here the ML is equal to 2.3357

Here, the MS is equal to 0.3365

We have taken care of the two 0.3367

In this particular case, M sub L is -2, M sub S =0.3369

Here, M sub L =1, M sub S was 0.3376

Here, M sub L was -1 and M sub S = 0.3383

This one was M sub L = 0, M sub S = 0.3388

These 5 possible arrangements, these all along the D1 energy states.3395

These microstates belong to the D1 level, the D1 term.3401

It is a notation that describes the energy of these 5 states, the D1 level.3421

They have the same energy.3429

We will take care of the J.3441

Now, we have the D1, we still have to figure out what this thing is right here.3455

We still do not know what the J value is.3460

We will take care of the J after we have found the primary terms symbols for the groups.3462

Then, I will go back and fill in the J values after we have grouped them.3473

All those 15 viable microstates, the arrangements, we are able to find 5 of them that fall into a certain category.3488

That category has the term symbol D1.3496

This D1 tells you about what is happening here.3499

The D tells you that the L is equal to 2.3504

The total orbital angular momentum for the state is 2.3507

The 2S + 1 = 1.3510

This one number here tells you that S is equal to 0.3513

It tells you that the total spin angular momentum of these is 0.3516

Up spin down spin 0, Up spin down spin 0, Up spin down spin 0, Up spin down spin 0, Up spin down spin 0.3521

These terms give you information on what is happening here electronically3529

in terms of angular momentum, orbital angular momentum, and spin angular momentum.3534

I’m going to go ahead and stop this lesson here.3540

The next lesson is going to be the continuation of these examples.3542

Just consider one long lesson.3546

I do not want to make it one huge long lesson, I want to break it up.3547

The next lesson is going to be just a continuation of this particular example.3552

Thank you so much for joining us here at www.educator.com.3555

We will see you next times, bye.3557

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