  Raffi Hovasapian

Example Problems I

Slide Duration:

Section 1: Classical Thermodynamics Preliminaries
The Ideal Gas Law

46m 5s

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

46m 2s

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

1h 6m 45s

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

1h 6m 33s

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

1h 2m 17s

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

1h 4m 39s

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

16m 50s

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

43m 40s

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

32m 23s

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

39m 15s

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

35m 52s

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

42m 40s

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

1h 23s

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

44m 34s

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

59m 7s

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

49m 16s

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

33m 59s

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

54m 37s

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

31m 18s

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

23m 6s

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

25m 42s

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

43m 39s

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

56m 44s

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

57m 6s

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

54m 35s

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

35m 5s

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

28m 42s

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

34m 38s

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

30m 50s

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

34m 6s

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

39m 18s

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

19m 40s

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

54m 16s

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

31m 17s

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

45m

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

58m 5s

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

25m 20s

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

34m 25s

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

59m 57s

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

42m 5s

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

30m 26s

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

21m 34s

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

56m 22s

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

45m 24s

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

48m 43s

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

46m 18s

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

39m 28s

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

35m 32s

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

29m 55s

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

54m 25s

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

46m 58s

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

44m 11s

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

35m 33s

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

43m 4s

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

26m 30s

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

41m 10s

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

30m 32s

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

35m 19s

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

33m 48s

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

46m 43s

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

40m

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

35m 58s

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

36m 18s

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

33m 55s

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

51m 53s

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

51m 53s

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

34m 29s

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

43m 49s

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

1h 1m 57s

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

48m 33s

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

48m 33s

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

48m 33s

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

59m 18s

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

34m 54s

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

38m 3s

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

57m 49s

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

1h 1m 20s

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

56m 34s

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

18m 24s

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

50m 2s

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

59m 47s

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

46m 22s

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

29m 24s

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

30m 53s

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

1h 1m 33s

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

33m 47s

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

1h 1m 5s

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

33m 31s

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

1h 1m 15s

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

47m 23s

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

54m 9s

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

48m 32s

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

57m 30s

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

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• 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

### Transcription: Example Problems I

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

Today, we are going to start our example problems for term symbols and atomic spectra.0004

Let us jump right on in.0009

What are the terms symbols for the NP1 configuration?0013

The 2P1, 3P1, 4P1, the N part does not matter.0019

It is actually just the electrons of the suborbital.0023

This is really just true for P1.0026

I'm going to solve this with a slightly different procedure than the one that I have used when I introduced terms symbols.0030

Back when I introduced term symbols a few lessons back,0038

we started by listing out all the microstates explicitly and then picking out those M sub L values and M sub S values.0041

Then, I introduced this shorter procedure where we are actually making a table of the ML and MS values.0049

Counting the largest ML, the largest MS, crossing them out until you come up with all of your term symbols.0056

What I'm going to do is essentially just the reverse of that.0064

I'm going to do this for this example problem and also the next several example problems.0067

When we do the MP2 configuration and the 3 configuration, the D2 configuration goes through quite a few of these.0074

Essentially, what we are doing is we are going to work with the ML and MS values directly0080

and then generate the microstates for these term symbols as we go long, as opposed to the other way around.0085

At some sense, it is the reverse.0093

You can decide which one you like better.0095

For this particular problem, I’m going to do both.0099

I’m going to do this new way and then go back into the table way, then you will see which one you like better.0101

They are essentially the same and at some point, you have to deal with the largest ML0107

and largest MS and you are going to have to deal with the microstates.0113

But this way with the numbers just tends to be a little bit faster.0116

Let us see what we can do.0120

Like everything else in science or anything for that matter, you have to do them a number of times in order to really understand.0123

Passive understanding just by looking and seeing that you can follow that,0133

that is one thing but being able to generate and it do it yourself, that is when you we only get your hands dirty,0137

that is when you really understand what is happening.0142

Let us see what we have got.0144

We always want to begin by finding the number of viable microstates.0146

Let me stick with black here.0151

Always begin by finding the number of viable microstates and that was, we have this great name for that,0153

it is the number of spin orbitals that we have available to occupy choosing the number of electrons.0176

In this particular case, we are looking for the NP1 configuration.0191

We have 1 electron and we are trying to distribute among the PX PY PZ.0195

We have 2, 4, 6, we have 6 spin orbitals and we have 1 electron.0202

It is going to be 6 choose 1, there are 6 microstates.0207

6 different ways to arrange 1 electron in the P orbital.0215

In this particular case, we already know which one they are.0219

It is going to be up here, up here, up here, or down here, down here, down here.0222

That takes care of it, but let us just go ahead and work it through.0226

This is the important thing, always begin with that.0228

Start by finding the number of viable microstates and as you find each term symbol,0231

knock out the number of microstates and it will tell you how many you have left.0235

We are also going to be finding the M sub L, that is equal to the m sub l1 + m sub l2 and so on.0240

In this case it is only 1 electron.0258

The M sub L, you remember that is the 1, 0, -1, in the case of the P orbital.0260

For the D, it was 2, 1, 0, -1, -2, because there are 5 suborbital in the D orbital.0266

We are going to be finding that number and we would be finding the M sub S.0275

That is equal to the M sub S1 + M sub S2, that is the spin of the electron, +½ or - ½.0281

We are going to add them all up for all of the electrons.0292

We begin by looking for the largest ML possible.0297

Here, because we are talking about 1 electron in the P.0314

Here, the largest ML is going to equal 1.0323

This particular thing, and the reason it is, is because we can put the 1 electron in the left most suborbital.0334

Let us do that.0363

Here, the ML value is 1 and ML value is 0, ML value is -1.0365

We want the largest ML possible, for 1 electron it is over here.0371

Here, ML = 1.0376

The largest MS possible, in this particular case it is 1 electron, it is up spin ½ as opposed to downspin - ½.0381

Given that ML is equal to 1, in this particular case, the largest MS possible is ½.0395

If ML is equal to 1 that implies that L = 1.0406

If MS is equal to ½ that implies that S is actually equal to ½, that implies that 2S + 1 is equal to 2 × ½ + 1 is equal to 2.0411

When L is equal to 1 and S is equal to 2, that gives me a doublet P configuration, a doublet P term symbol.0423

From here, we are going to calculate how many microstates are in this doublet P.0436

L = 1 remember, when L = 0 that is S.0442

When L = 1 that is P.0446

When L = 2 that is D.0448

When L = 3 that is F, and so on.0450

S is ½, 2S + 1 that is the left most top left number.0454

This is a doublet P configuration, we will worry about the J a little bit later.0462

We found our basic term symbol.0468

2L + 1 × 2S + 1, let us do it this way.0471

The number of ML × the number of MS that is equal to 2L + 1 × 2S + 1.0489

L is equal to 1, 2 × 1 + 1 is equal to 3.0501

2S + 1 is equal 2, 3 × to = 6.0505

There are 6 microstates in this level.0510

There are 6 microstates that belong to this particular term symbol.0517

These come from ML is equal to 1 or L is equal to 1, that means that ML is equal to 1, 0, -1.0521

MS is equal to ½, S is equal to ½, that means that MS is equal to ½, and ½ -1 is – ½.0537

There are 3, here is 2, 3 × 2 is 6.0547

The states that you are going to get, the microstates come from 1, ½, 1 - ½, 0 ½ , 0 - ½, -1 ½, --1 ½.0552

In this particular case, all 6 microstates are accounted for.0566

Let us go ahead and generate these.0574

The ML values takes on these values, MS takes on these values.0581

Let us actually generate the microstates.0586

Once again, doublet P we have ML = 1, 0, -1.0593

We have MS = ½ - ½.0607

We are going to have 1, ½, that is here.0612

ML is 1, spin is ½, 1, 0, ½.0625

ML is 0, spin is ½ - 1 ½.0632

ML is -1, spin is ½.0637

We will go 1 - ½ that is here, 0 - ½ -1 - ½, this accounts for all of the microstates in the doublet P.0640

Let us go ahead and find the J values.0654

The doublet P, we find the J values by taking L + S.0656

In this particular case, L + S is going to be 1 + 0 = 1.0661

And we find the absolute value of L – S.0667

L + S is equal to L is 1, S is ½.0677

This is going to be 3/2 the absolute value of L - S is equal to absolute value of 1 - ½ = ½.0683

This is the upper value and we dropped by 1 until we get to that value.0695

J is equal to 3/2 and is equal to ½.0699

Therefore, our final term symbol we have the doublet P 3/2 and we have the doublet P ½.0705

That is a complete term symbol for the MP1 configuration.0717

2P1, 3P1, 4P1, it consists of 2 levels.0724

There is going to be doublet P 3/2 term symbol, there is a doublet P ½ term symbol.0729

Doublet P 3/2 energy state, doublet P ½ energy state.0735

We will talk about the degeneracy.0750

We know that the basic term symbol, the doublet P accounts for 6 microstates.0755

The doublet P actually consists of doublet P 3/2 and the doublet P ½.0760

The degeneracy, in other words the number of microstates in a given specific complete term symbol that is equal to 2 J + 1.0765

In the case of doublet P 3/2, J is equal to 3/2.0774

2 × 3/2 is 3 + 1 = 4.0781

4 of these microstates belong to the doublet P 3/2 energy level.0785

2 × ½ + 1 is 2, that means 2 of these microstates belong to the doublet P ½.0791

Final is, which is the ground state?0805

In other words, is it the doublet P 3/2 or the doublet P ½ the ground state?0814

Hund’s rules say, whichever has the largest S.0824

In this particular case, S2, S2.0828

This is ½, this is ½, S is the same.0831

Let us go ahead and put them out.0837

We take the largest S, over here, S is the same.0840

2, if S is the same, we take the largest L.0849

In this case, we have P and P, L is the same.0855

3, if that particular orbital is less than half filled which in this case it is less than half filled, it is only 1 electron in the P orbital.0862

We are going to take the smallest J value, smallest J.0874

This implies that we take the ½ which implies J = ½, which implies that the doublet P ½ is the particular ground state.0879

It is the lowest energy, the most stable.0898

Not necessarily the ground state, but of the two, that is the one that lower energy.0902

Let us go ahead and do this.0916

Recall that complimentary configurations have the same term symbols.0918

The complement of P1 is P5.0950

P5 also has a doublet P 3/2 and a doublet P ½ state.0952

Let us do this with an actual table.0969

The table version, this particular process that I just did.0975

I will make a lot more sense when we see it for the next few configurations that we do.0980

For the table, we have 1 electron, we are looking for the largest ML.0987

We know the largest ML value is going to be 1.0996

That means ML is equal to 1, 0, -1.1001

The largest MS for 1 electron is also 1.1007

MS is equal to 1, 0, -1.1013

MS is actually ½, we have ½ and we have - ½.1020

ML is going to be 1, 0, -1.1037

MS is the upper row, M sub L the column.1041

We have 1 electron, the sums of the ML values add up to these numbers.1047

The sums of the MS values add up to these numbers.1056

We get 1 + ½, 1 - ½, 0 + ½, 0 - ½.1059

We get -1 + ½ - 1 - ½.1069

We take the largest ML having a viable microstate in its row, 1.1076

This one.1085

We take the largest MS having a viable microstate in its column.1089

Between these two, it is this one ½.1093

We are essentially doing the same thing.1098

We are just doing it backwards.1102

The largest ML is equal to 1.1104

The largest MS having a viable microstate in its column is equal to ½.1110

ML = 1 implies that L = 1.1121

MS = ½ implies that S = ½.1132

It implies that 2 S + 1 is equal to 2.1137

Therefore, we have a doublet P.1145

We have ML is 1, 0, -1.1154

And MS = ½ - ½.1163

We are going to cross out 1 for each value of these.1167

1, ½, 1 - ½, and we go to 0 1/2, 0 - ½ and then - ½ -1 - ½.1170

We have accounted for all of them and from here, since we have the doublet P,1185

we can go ahead and generate J value like we did before.1190

That is all we are doing when we are doing the table.1193

We are finding the largest ML and the largest MS, and then L is equal to 1.1196

ML is equal to 1, 0, -1.1206

S is equal to ½, MS is equal to ½ - ½.1209

And we just combine these to knockout one for each.1213

We are leftover with, we just continue on the same way.1217

We pick the next largest ML, the next largest MS.1221

I think an example will be a lot better and a lot more clear.1227

What are the term symbols for the MP 2 configuration?1242

We have 2 electrons and we still have the P orbitals.1248

We have 6 spin orbitals but we have 2 electrons.1252

MP2 means 6 choose 2, that is 6!/ 4! =2!.1258

What we end up with is 15 viable microstates.1272

We always want to begin with a number of viable microstates so that we know how many we are dealing with.1280

We are looking for the microstate that is going to give the largest value of M sub L.1287

The microstate giving the largest value of M sub L is going to be this one.1299

We are going to put both of the electrons with 2 electrons.1316

We are going to put both of the electrons in the m sub l into one.1318

We have ML, M sub L for the first electron.1326

This is getting confusing here.1330

The first electron m sub l is equal to 1.1334

The second electron m sub l is equal to 1.1338

For the largest value that we can actually get is that putting them both into that m sub l to get a value of 2.1341

Here, M sub L which is the sum of the m sub l is equal to 2.1350

In this particular case, because I get the largest M sub L equal to 2,1359

that I have to put them both in the same suborbital, they have to have opposite spin.1366

In this particular case, the largest MS that I can generate from this configuration,1373

this 1 configuration is going to be 0 because it is going to be m sub s = + ½.1378

It is going to be m sub s = - ½.1387

When I add these two together to get the M sub S, it is going to be 0.1390

For this microstate, the largest M sub S is going to equal 0 because the 2 electrons must have opposite spin.1399

We have ML is equal to 2 which implies the L is equal to 2.1424

We have M sub S is equal to 0 which implies that S is equal to 0, which implies that 2 S + 1 is equal to 1.1434

L is equal to 2 gives me D.1444

2 S + 1 is 1 so I have a singlet D state.1446

I need to find out how many microstates are in this particular energy level.1450

In other words, the degeneracy of this basic term symbol.1458

I'm going to do 2L + 1 × 2S + 1.1462

2L + 1 = 5, 2S + 1 is equal to 1.1470

5 × 1, there are 5 microstates in this level.1475

Let us list the microstates, when you get the basic term symbol that is when you want to list those microstates.1485

Let us list the microstates for the singlet D.1494

Recall, we have L is equal to 2.1505

L is equal to 2 which means the ML goes to 1, 0, -1, 2.1510

MS is equal to 0 that means MS is equal to 0.1517

We are going to have a microstate that is going to have the sum of the ML is 2, the sum of the MS is 0.1522

2 0, 1 0, 0 0, -1 0, 2 0, those microstates are as follows.1529

Up down, up down, up down, up down, there is a certain symmetry to this.1545

I generate the largest ML with one configuration.1563

The M sub L runs through all the values 2, 1, 0, -1, 2.1568

MS =0.1573

This is 2 0, this is 1 0, because ML is 1 and ML is 0, that give me a M of 1.1576

Spin is +½, spin is -½ gives me a total spin of 0.1587

Over here, m sub l is 1 and m sub l is -1.1593

1 -1 is 0 so M sub L is 0.1599

Up spin down spin, the total spin is 0.1604

That make sense.1609

These 5 microstates represent this term symbol.1611

We will worry about that J values later.1617

The next largest M sub L possible is ML is equal to 1.1630

And this is achievable as I can do that.1655

I can be do 2 up spins, the M sub L is equal to 1 because it is 1 + 0.1676

M sub l is 1 and m sub l is 0.1683

The total M sub L is 1, that is one possible way of getting ML is equal to 1 or I can do this and this.1686

M sub l is 1, m sub l is 0.1695

However, in this particular case, the two ways of actually achieving the largest M sub L is equal to 1 which one of these1698

do I actually choose as my basis to decide what my M sub S is going to be.1705

This one is accounted for already.1710

If I look at the previous page where I listed the microstates,1714

this microstate is already there where we have the first suborbital up spin, the second one suborbital with a downspin.1717

Since this is accounted for, the one I pick is this one.1724

For this particular one, the largest MS is going to be 1, both of them up spin.1728

Here, M sub L is equal to 1 and M sub S is equal to 1, that implies that L = 1, that implies that S is equal to 1,1737

that implies that 2S + 1 is equal to 3.1746

L = to 1 is a P, 2S + 1 = 3, we end up with a triplet state.1749

Let us go ahead and find the 2L + 1 × 2S + 1, 2L + 1, 2 × 1 + 1 is equal to 3.1756

2S + 1 that is equal to 3, there are 9 microstates.1771

There are 9 microstates in the triplet P basic term.1776

Let us list what these microstates are.1782

We have ML is equal to 1, 0, -1.1786

MS is equal to 1, 0, -1.1793

The microstates are going to represented such that we have 1 1, 1 0, 1 -1, 0 1, 0 0, 0 -1, -1 1, -1 0, -1 -1.1796

There would be 9 of them.1809

Here is what it would look like 123, 123, 123, 123, 123, 123, 123, 123, 123.1811

This is the one that we chose, our basic so we start with that one.1827

That is the 1 1, 0 1, -1 1, we are going to go to 0 1.1843

That is what we just did the 1 1, 0 1, -1 1.1863

We are going to do 1 0, 0 0, -1 0.1868

The last 3 are going to be 1 -1, 0 -1, -1 -1.1874

We are running through all of them.1880

This first one, in order to pick the largest ML and the largest MS as possible given.1882

The largest ML possible here was 1, that was achievable 2 ways.1891

I can even go up spin up spin, down spin down spin.1895

Which one do I choose to decide what my MS is going to be, what by largest MS?1898

This was already accounted for in the D1 configuration, what we did in the previous page.1902

Therefore, I choose this one.1907

Here I get down up, down up, down up.1910

I have down down, down down, down down, down down.1921

These microstates all belong to the triplet P basic term symbol state.1930

You basically just run through each combination, 1 1, 1 0, 1 -1.1939

This is the 1 1 state, this is the 1 0 state, this is the 1 -1 state.1947

M sub l is 1, m sub l is + 0.1957

1 + 0 is 1 - ½ - ½ -1, I have to run for each possibility, that is all I'm doing.1960

We have 9 microstates and we have 5 microstates previous.1975

We have account for 14 of the microstates and we only have 1 left.1979

We have accounted for 14 microstates that mean there is 1 left.1987

The next possible largest ML is equal to 0.1998

Lead me see, how can we actually get ML equal to 0?2018

This is achievable as I can do that, that is one way or I can do that is one way, that is another way,2027

here the MS is equal to 0.2059

Here the MS is equal to 0, here the MS is equal to 0.2062

I can go up up, here the MS is equal to 1.2068

Here, I can go down down, MS is equal to 1.2073

I'm looking for the next largest ML.2081

Everything is already accounted for, the next largest M sub L is going to be 0.2087

And the only way to get that is this one particular configuration or I can do this.2092

Each of these configurations gives me an ML equal to 0.2096

The problem is these of all that accounted for in the previous microstates, the previous 14.2100

We have only one left with is this one.2114

In this particular case, I have ML is equal to 0.2118

I have MS is equal to 0 that means that L is equal to 0, that means that S is equal to 0,2124

that means that 2S + 1 is equal to 1.2131

L = 0 is S, 2S + 1 is 1, I have a singlet S state.2135

In this particular case, 2L + 1 × to S + 1 is equal to 1 × 1, there is 1 microstate.2143

We have accounted for all 15 microstates.2152

In this particular case, this microstate ML is equal to 0, MS is equal to 0.2155

The only microstate left is that one.2165

There you go, we have singlet D state, we have a triplet P state, and we have a singlet S state.2171

Let us go ahead and find the J values for this.2186

For the singlet D state L + S is equal to 2 + 0 is equal to 2,2191

the absolute value of L - S is equal to the absolute value of 2 -0 is equal to 2.2199

J is equal to 2, we have a singlet D2 microstate.2207

There we go, the degeneracy as we said before is 2J + 1.2214

J is equal to 2, 2 × 2 is 4, 4 + 1 is 5.2222

I will go ahead and put that in parentheses underneath.2227

There are 5 microstates in that singlet D2.2231

We have a triplet P.2238

Our triplet P, our L + S is equal to 1 + 1 that is equal to 2.2241

Our absolute value of L - S is equal to 1 -1 is equal to 0.2250

J is equal to this, all the way down to this in increments of 1.2257

2, 1, 0, we have a triplet P2, we have a triple a P1, we had a triple a P0.2261

The degeneracy is 2J + 1.2271

Of the 9 microstates belong to the triplet P state.2274

All of those 9, 5 belong to the triplet P2 state, 3 belonged to the triplet P1 state and 1 belongs to the triplet P0 state.2280

We have broken down even further.2294

There are symbols all over the place.2297

Singlet S, L + S = 0 + 0 = 0.2304

The absolute value of L - S is equal to 0 -0 = 0, means that J is equal to 0.2310

We have a singlet S0 and its degeneracy is going to be 2J + 1.2319

It is going to be 1, 1 + 1 + 3 + 5 + 5 is equal to 15.2325

All 15 microstates are accounted for.2331

There is a singlet D level that has 5 microstates.2334

There is a triplet P level that has 9 microstates.2337

There is a singlet S level that has 1 microstate.2340

The singlet D is actually a singlet D2 that contains all 5 microstates.2343

The triplet P is broken up into 3 different energy levels, a triplet P2, triplet P1, triplet P0.2348

Each having 5, 3, 1 microstate respectively.2356

The final was here which is the ground state.2360

When we say ground state, we mean which is the one with the lowest energy.2371

We have 1, the first rule says largest S.2374

The largest S is S = 1 which means that triplet P state.2388

2, largest L, of these PPP, L is the same.2399

3, less than half filled.2413

It is less than half filled, we only have 2 electrons in there.2417

Less than half filled means we are going to look for the smallest J, where the smallest J is the 0 value.2420

Therefore, the triplet P0 is the ground state of all of these.2430

Let us do another.2442

What are the term symbols for the MP3 configuration?2449

We have 3 electrons, we have 6 spin orbitals, 3 electrons, 6 choose 3.2453

I will do that over here, I think.2464

We have 6 choose 3 that is equal to 6!/ 3! is 3!.2468

I will write it out, 6 × 5 × 4/ 3 × 2 × 1.2476

One of these 3!s cancels the 3 × 2 × 1 up above.2486

When you do that 6 × 5 × 4 divided by 6, 2 × 3 you are going to get 20 viable microstates.2489

There are 20 possible ways of distributing 3 electrons in the spin orbitals,2500

in the 3 suborbital of the P which has 6 spin suborbital.2511

6 spin orbitals, there are 3 electrons, there are 20 different ways of actually arranging them.2518

The largest ML possible is achievable as, if I put one there, one there, and one there.2525

In this particular case, I have ML is 1, m sub l is 1, m sub l is 0.2550

1 + 1 + 0 is equal to 2.2560

The largest M sub L is equal to 2.2568

In this particular arrangement, this and this spin they cancel out.2573

Here, the largest M sub S is equal to ½.2577

We have ML is equal to 2, we have M sub S equal to ½, that means that L is equal to 2,2592

that means the S is equal to ½, that means that 2S + 1 is equal to 2.2601

L = 2, that is a D, this is 2.2607

We have a doublet D state.2611

2L + 1 × 2S + 1 is going to be 5 × 2 is equal to 10.2618

There are 10 microstates in this doublet D level.2629

Let us list them.2635

Let us list the 10 microstates, the one with the next largest ML.2642

When we have a choice, we want to see which one of these microstates is accounted for.2653

What we are doing this procedure is finding the basic term symbol and list the microstates for that term symbol.2658

We are going to refer back to it, if we need to.2662

Let us list the term well.2666

ML, we said L is equal to 2 that means the ML is going to equal to 1, 0, -1, 2.2668

M sub S or S is equal to ½ which means the M sub S is equal to ½ and -1/2.2677

The microstates are going to be such that the total angular momentum is going to add up to 2.2682

The total spin angular momentum is going to be ½ all the way down the line.2694

We are going to have one with 2 ½, 1 ½, 0 ½ , -1 ½, 2 ½, and 2 – ½, 1 - ½, and so on.2701

For total of 2 × 5, 10 microstates.2709

Here is what they look like.2712

We have 123, 123, 123, 123, 123, 123, 123, 123, 123.2719

I will put the 10th one right down here, 123.2733

Up down, up down, up up, up down, that is down electron.2740

I just moved it once over here and once over here.2757

We are just writing out the microstates that allow you to achieve a total angular of 2, total angular of half.2762

Total angular 1 total angular half, total angular 0 total spin ½.2774

Total angular -1 total spin ½, total angular -2 total spin ½.2778

And then, total angular 2 total spin - ½, total angular 1 total spin - ½.2785

Total angular 0 total spin - 1/2, total angular -1 total spin - ½, total angular -2 total spin - ½.2791

You are just arranging them.2800

It is going to be that, that and that.2803

We have this, this and this.2808

That that down, they have down down up, down up down, down down up, down down up.2815

These are the 10 microstates that are in the doublet D level.2838

Let us look for the next largest ML.2848

The next largest ML is equal to 1 and achievable as follows.2856

I can do this, this this.2866

I could do this, this this, or I could do up down up.2881

These are different ways that I can have a M sub L, a total angular momentum number of 1.2890

This was has been account for in the previous set.2899

That is why I listed them.2907

I have listed list the microstates so that when I move to the next level, in this particular case the largest M sub L2910

and I see the different ways that I can actually achieve that largest M sub L,2919

I ignore the ones that were already accounted for.2923

This one has not been accounted for, this is the one that I choose.2927

In this particular case, the MS value for this one is ½.2932

I have an ML equal to 1 and I have an MS = ½, that implies the L = 1, that implies that S = ½,2938

that implies that to 2S + 1 is equal to 2.2950

L = 1 is a P, this is a 2 so I have a doublet P state.2955

2L + 1 × 2S + 1, 2L + 1 that is equal to 3 × 2S + 1 2, there are 6 microstates in the doublet P level.2963

Let us go ahead and list those.2976

L = 1, therefore M sub L = 1, 0, -1.2989

M sub S, S is equal to ½, M sub S is equal to ½ - ½.2995

2 × 3, there are 6.3001

The one that we started with first, boom boom boom move is over one, leave out the same.3003

Bring this here, I just have to make sure what is that I am writing, has noted that account for previously.3014

That is all, that is all I’m doing them.3023

I have that, that that, that, and this.3026

This is why you want to have a list of the microstates that have been accounted for.3043

If you are listing these microstates to satisfy these relations, if you list the one that are account for, just list another one.3048

Find another way of doing it, there will always be another way of doing it.3057

There has to be because that is 6 choose 3, there have to be 20 microstates available for you to do what you do,3061

that is what all these numbers mean.3069

If you end up listing something, as you are going through it here,3072

listing the microstates to satisfy these conditions 1 ½, 0 ½, -1 ½, 1 - ½, 0 – ½, -1 – ½.3075

If you list them and satisfy them up spin downspin, just make sure you have listed one that already been accounted for.3084

If have, just list another one.3093

Let me write that down.3095

If you list a microstate that has been accounted for, just find another that satisfies the values of M sub L and M sub S.3099

Find something that satisfied, just find another one that satisfies those values that is not been accounted for.3145

There will always be one.3163

Let me take a couple minutes here. And when you doing these problems, you are not probably going to be asked do more than a couple of these problems.3175

There is a table in your book that lists with different term symbols are for each configuration,3183

whether it is P1, P2, P3, whether it is D1, D2, D3, D4, D5.3188

You are not going to be asked this is particular process.3197

It is reasonably straightforward, you just have a lot of things to account for is have a lot of things on the paper.3205

A lot of symbols floating around, a lot of things to check and doublet check again.3212

You have clearly figured out by now, that is the nature of quantum mechanics.3217

As we go deeper in science, things become more complex.3223

Beautiful things emerge, symmetries start to emerge.3227

Larger picture start to emerge but things become more complicated.3232

This is the nature of the game and it is part of the excitement to be able to know that we can do something like this3236

and achieve this magnificent, intellectual thing and know that we are right.3243

Enough of that, let us go on here.3250

We have taken care of ML is 1, ML is 2.3255

The next largest ML, the next largest M sub L is equal to 0 and that is achievable as which is not been accounted for.3262

We are good.3299

ML = 0, here MS up, the largest MS is 3/2.3305

This implies that L is equal 0, this implies that S = 3/2, this implies that 2S + 1 is equal to 4.3313

L = 0 gives me an S term symbol.3324

2S + 1 give me a spin multiplicity of 4.3327

We have a quartet S or quadruplet S.3330

2L + 1 × 2S + 1, 2L + 1 = 1 × 4 = 4 microstates.3338

We have 10 microstates, 6 microstates, 4 microstates.3349

We have accounted for all of them.3353

ML is equal to 0 and M sub S is equal to 3/2, ½ - ½ -3/2.3363

The total MS have to add the 0, total MS is ½ add to 3/2, and then add to ½ to - ½ to -3/2.3381

123, 123, 123, 123, there are 4 microstates.3389

The one that we chose as our basic one was that.3396

I can go ahead and do this, I can go ahead and do this, I can go ahead and do this.3402

From this one that has been accounted for, there are 3, I just switch them around3412

until it is one that has not been accounted for.3418

For the full symbols, we have a doublet D, we have a doublet P, and we have quartet S.3421

For the doublet D, our L + S is equal to 2 + 0 is equal to 2.3444

This is not right, 2S + 1 = 2, that means S = ½.3459

L + S = 2 + ½ that =, 2 ½ that is equal to 5 ½.3467

The absolute value of L - S is equal to 2 - ½, that is equal to 3/2.3478

Therefore, J is equal to 5/2 down to 3/2.3486

Therefore, we have a doublet D 5/2 and we have a doublet D D 3/2.3493

The degeneracy of 2J + 1, 2 × 5/2 + 1, the degeneracy of this one is 6 and the degeneracy of this one is 4.3502

Let us look at our doublet P state.3517

Here L + S is equal to 1 + ½ which is equal to 3/2 and the absolute value of L - S is equal to 1 – ½ = ½.3520

Therefore, J is equal to 3/2 and ½.3545

We have a doublet P 3/2, we have a doublet P ½.3550

The degeneracy is going to be 4 and 2.3557

The doublet P have 6 microstates.3564

Of all the 6 microstates, 4 of them belong to the doublet P 3/2 state, 2 of them belong to doublet P ½ state.3567

The quadruplet S, the quadruplet S L + S is equal to 0 + 3/2 is equal to 3/2, L - S sequel to absolute value 0 -3/2 = 3/2.3574

J is only equal to 3/2 which gives us a quadruplet S 3/2.3598

The degeneracy of which is going to equal 4 2J + 1.3606

I will go ahead and put that over here parentheses.3612

The ground state, the largest S is this one, equal to 3/2.3618

Our quadruplet S 3/2, that is the only one, that is the only term symbol for the largest S.3636

That is the ground state, I do not have to look at the L, I do not have to look at the J.3651

There we go.3655

I know that it is tedious and I apologize for that.3658

I hope that at least the procedure made a little bit of sense.3661

If not, there still a couple of more in the next lesson of example problems and3665

I'm going to be using the same procedure.3670

Hopefully, just going and seeing it over and over will make a difference.3672

Thank you so much for joining us here at www.educator.com.3677

We will see you next time, bye.3679

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