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Example Problems II

Slide Duration:

Section 1: Classical Thermodynamics Preliminaries
The Ideal Gas Law

46m 5s

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

46m 2s

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

1h 6m 45s

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

1h 6m 33s

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

1h 2m 17s

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

1h 4m 39s

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

16m 50s

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

43m 40s

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

32m 23s

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

39m 15s

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

35m 52s

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

42m 40s

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

1h 23s

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

44m 34s

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

59m 7s

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

49m 16s

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

33m 59s

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

54m 37s

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

31m 18s

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

23m 6s

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

25m 42s

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

43m 39s

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

56m 44s

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

57m 6s

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

54m 35s

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

35m 5s

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

28m 42s

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

34m 38s

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

30m 50s

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

34m 6s

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

39m 18s

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

19m 40s

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

54m 16s

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

31m 17s

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

45m

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

58m 5s

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

25m 20s

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

34m 25s

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

59m 57s

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

42m 5s

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

30m 26s

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

21m 34s

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

56m 22s

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

45m 24s

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

48m 43s

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

46m 18s

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

39m 28s

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

35m 32s

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

29m 55s

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

54m 25s

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

46m 58s

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

44m 11s

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

35m 33s

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

43m 4s

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

26m 30s

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

41m 10s

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

30m 32s

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

35m 19s

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

33m 48s

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

46m 43s

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

40m

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

35m 58s

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

36m 18s

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

33m 55s

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

51m 53s

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

51m 53s

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

34m 29s

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

43m 49s

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

1h 1m 57s

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

48m 33s

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

48m 33s

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

48m 33s

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

59m 18s

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

34m 54s

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

38m 3s

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

57m 49s

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

1h 1m 20s

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

56m 34s

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

18m 24s

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

50m 2s

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

59m 47s

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

46m 22s

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

29m 24s

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

30m 53s

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

1h 1m 33s

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

33m 47s

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

1h 1m 5s

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

33m 31s

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

1h 1m 15s

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

47m 23s

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

54m 9s

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

48m 32s

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

57m 30s

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

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### Example Problems II

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• 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

### Transcription: Example Problems II

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

Today, we are going to continue our example problems in molecular spectroscopy.0004

Let us jump right on in.0008

Our first example is the force constant for carbon monoxide is 1857 N/m.0010

The equilibrium bond length = 11.83 pm.0019

Using the rigid rotator harmonic oscillator approximation,0025

construct a table of energies for the first 5 rotational states for the vibrational levels R = 0 and R = 1.0029

The 5 rotational levels for R = 0, 5 rotational levels for R = 1.0039

Specify which transitions are allowed and calculate the frequencies of each transition,0043

associating each with its appropriate branch R or P.0049

Let us see what we can do.0054

Let me go ahead and work in blue today.0056

Under the rigid rotator harmonic oscillator approximation,0061

our energy equation is going to be energy RJ is equal to ν sub 0 R + ½ + this rotational constant × J × J + 1.0065

That is it, it is the vibrational energy + the rotational energy.0083

We have to find ν sub 0 that, and this, before we can actually start running J from 0 to 5.0088

Let us see what we can do.0107

Ν sub 0, the definition is 1/2 π C × the force constant divided by the reduced mass ^½.0109

We have to find the reduced mass.0124

The reduced mass is equal to the product of the weights divided by the sum of the weights.0128

12 × 16, 12 for carbon 16 for oxygen, divided by 12 + 16.0135

This is in atomic mass units, I’m going to multiply that by 1.661 × 10⁻²⁷ kg per atomic mass unit.0142

My atomic mass units cancel and I'm left with a reduced mass in kg.0153

When I do this calculation, I should get,0159

I hope that you are confirming my arithmetic, I am notorious for arithmetic mistakes.0161

The arithmetic is less important, the process is important.0167

The equations that you choose, that is what is important.0171

I may clear my throat a few more times than usual during these few lessons today.0176

× 10⁻²⁶ kg.0181

I want to keep writing these numbers over and over again.0189

When I go ahead and put this value of K into here, when I put this value ν into here and run this calculation,0192

I'm going to get a ν sub 0 equal to 214354.49.0201

The answer is actually going to be in inverse meters because this is N/ m.0212

When I convert that, I will just move that decimal over a couple of times,0218

I'm going to get 2143.54 inverse cm.0221

The problem is, the issues with spectroscopy are like many issues throughout quantum mechanics and thermodynamics.0228

They are just for conversion issues, just watch your units.0234

That is really all you have to watch out for.0239

We have that number.0242

Let us go ahead and find the rotational constant.0248

Let me go ahead and do that on the next page here.0251

Our rotational constant is equal to, the definition H/ 8 π² C × the rotational inertia.0255

We need I to put it into here.0267

I = that, that².0273

Therefore, I = R 1.139 × 10⁻²⁶ kg × RE is a 112.83 × 10⁻¹² m.0277

What we get is I = 1.45 × 10⁻⁴⁶ kg/ m², the unit of rotational inertia.0299

We put this value in here with all of the other values.0314

I will go ahead and write this one out, it is not a problem.0323

E = 6.626 × 10⁻³⁴ J-s divided by 8 π² × 2.9980328

× 10⁸ m/ s × 1.45 × 10⁻⁴⁶, that is the I.0344

When I do that, I will get 193.04 but this is going to be in inverse meters because a joule has meters.0355

That is going to give us, when I convert this, move the decimal over twice 1.9304 inverse cm.0367

Remember, inverse cm and inverse meters, it is the other way around.0376

It is going to be 100 inverse meters in 1 inverse cm, which is why this one goes to the left.0381

We have that and we have that.0390

We said that the energy was, let me go ahead and rewrite the equation.0392

The energy was E= sub RJ equal to ν sub 0 × R + ½.0397

This is the rigid rotator harmonic oscillator approximation + B ̃ J × J + 1.0406

Let me make my J a little more clear here.0417

For the R = 0 vibrational state, that is going to be E⁰ and J is going to run through these quantum numbers.0420

That is going to equal, when I put R and 0 into this equation, it becomes ½ ν sub 0 + B ̃ × J × J + 1.0435

And for the R = 1 vibrational state, I just put 1 into this equation and I get the E sub 1 J is equal to 3/2 ν sub 0 + B ̃ J × J + 1.0448

Let me double check that.0464

Therefore, our E0 J, when I actually put the ν 0 that I got and the B that I got in here,0469

I'm going to get equation that says 1071.77 + 1.9304, which is the rotational constant, × J × J + 1.0478

Now, I just set up a table of values.0490

I have J over here and I have my energy 0 J over here.0492

I’m going to take 0, 1, 2, 3, and 4.0498

I get 1071.8, I get 1075.6, I get 1083.4, I get 1094.9, I get 1110.4.0502

These are energies not frequencies of absorption or emission.0524

These are not spectral lines, these are not frequencies.0528

These are the energies of the actual rotational states in the R = 0 vibrational state.0530

Let us go ahead and do the energies for 1 vibrational state.0539

When I put the values in, I get 3, 2, 1, 5.0546

In other words 3/ 2, the ν sub 0.0549

3215.31 + 1.9304 which is the rotational constant, × J × J + 1.0551

Let us go ahead and set up my table of values again.0570

This is going to be E1 J.0573

I have 0, 1, 2, 3, and 4.0576

I have 3215.3, I have 3219.2, 3226.9, 3238.5, and 3253.9.0583

Let us go ahead and let me do it on one page here.0607

Let me see if I want to just go ahead and do this one in red.0614

I got J and I got E0 J.0617

Over here, I have J and I have E1 J.0623

I have 01234, 01234.0630

These are the first to 5 table of values.0636

I had 1071.8, 1075.6, 1083.4, 1094.9, and 1110.4.0639

Over here, I have 3215.3, 3219.2, 3226.9, 3238.5.0657

Again, I want them on the same page so I can see them together.0673

3253.9, the energies of the 0 vibrational state, the first 5 rotational for the 1 vibrational state.0677

The allowed transitions, δ J is + or -1.0690

The allowed transitions, the selection rule is δ J = + or -1.0699

Therefore, for the 0 state to the 1 state, the 0 vibrational state to the 1 vibarational state, I can go 0, 1, 2, 3, 4.0709

I have 0, 1, 2, 3, 4.0724

I can go from 0 to 1 that is + 1.0727

1 to 2, 2 to 3, and 3 to 4.0731

Let me do this one in black.0736

I can go from the lower 0, 1 to 0, to the 1, 3 to 2, 4 to 3.0738

Those are my allowed transitions.0746

The 0 state to 1 state, I can go 0 to 1, 1 to 2, 2 to 3, 3 to 4.0749

Or the 0 to 1 vibrational state, I can go rotational 1 to 0, 2 to 1, 3 to 2, and 4 to 3.0753

Those are my allowed transitions, the frequencies of those particular transitions.0762

The frequencies of these transitions, in other words the frequencies of the spectral lines of the R and P branches,0773

we have ν sub R branch is equal to ν sub 0 + 2B × J + 1, where J takes on the values 0, 1, 2, and so forth.0817

The P branch is going to be ν sub 0 -2B J.0835

Here, J takes on the values 1, 2, 3, and so on.0844

My ν sub R is going to be 2143.54 + 2 × 1.9304 × J + 1.0852

My ν sub P, those branches are going to be 2143.54 -2 × 1.9304 × J.0876

The R is 0 branch that represents the 0 to 1 transition, that was going to take place at 2147.4.0894

I will just put the J values in here.0904

J = 0, J = 1, J = 2, J = 3, J = 4.0905

That is the R0, let me be label it with the particular J value.0912

We put the R branch for J value 0 and it is going from 0 to 1.0919

When we do the P branch, it is actually going to be a P1 because it is going to be going from 1 to 0.0924

The R1, that represents the 1 to 2 transition.0932

The R2 represents the 2 to 3 transition.0937

The R3 represents the 3 to 4 transition.0941

These are the frequencies that we have.0946

We have 2151.3, we have 2155.1, and we have 2159.0948

We have the P1 transition, the P1 line, the P2 line, the P3 line, and the P4 line.0962

This one represents the 1 to 0 transition.0971

This one represents the 2 to 1 transition.0973

These are subscripts or the beginning energy level.0976

The energy level of the lower vibrational state.0979

For the lower rotational state, on the lower vibrational state.0984

In other words, the departure state not the arrival state.0991

I like that better, departure and arrival.0995

I think the mathematical terms are actually a lot better.0997

This one represents the 3 to 2 transition and this one represents the 4 to 3 transition.1003

These are 21392139.7, 2135.8.1012

These are going down, these are going up.1020

R branch increases, P branch decreases.1022

2132 and 2128.1027

There you have it, we found the energies, we found the allowed transitions.1035

These are the frequencies of the actual transitions that are allowed.1040

These are the labels for them representing this particular transition.1043

Let us go back to red here.1048

Be very careful to distinguish, especially on test.1054

On a test, you are going to be stressed out and your mind is going to be racing.1061

You need to really slow down when dealing with spectroscopy because they can be asking about energy,1065

Those are not the same.1072

Be careful to distinguish between the energy of a given level and the energy of transition,1074

the energy of transition between two energy levels.1113

The transition between two energy levels that is what we see in spectra are the transition energies from one level to another.1129

One final thing to notice, we found the energy levels for the E sub 0 J and E sub 1 J.1161

We used equations, the ν sub R, ν sub P to actually find those frequencies.1173

What you could have just done is take the difference between energies in the table.1179

Notice that the frequencies of transition,1184

instead of using equations for ν R branch and ν of the P branch,1203

we could have just taken δ E between the two vibrational states1235

for the appropriate J values.1262

For example, if I wanted the R1 line,1271

The R0 line that represents the transition from 0 to 1.1280

We have the energies already in the table.1289

We could have just taken the energy of 1 1 - the energy of 0 0.1290

This is the vibrational number, that is the vibration number.1304

Upper lower, the transition going from the 0 to 1 transition.1308

Upper lower, we could have just done that, just subtract the two values.1316

I will do that to the problem a little bit later on in this lesson.1319

Let us see what is next.1325

Using the table of parameters below and equations which correct for anharmonicity and vibration rotation interaction,1330

construct a table of energies for the first 5 rotational states of hydrogen iodide, for vibrational levels R = 0 and R = 1.1344

Specify which transitions are allowed and calculate the frequencies of these transitions,1353

associating each to its appropriate peak in the R and P branches.1356

The same sort of thing except now we need to make adjustment to the rigid rotator harmonic oscillator1361

and we are correcting for anharmonicity and vibration rotation interaction.1367

Notice, we are not correcting for centrifugal distortion.1371

We can always correct for all three, for two of those things, or for one of those things.1374

It just depends on what the problem is asking for.1378

We have our table of parameters, our rotational constant, our Α sub E,1381

our fundamental frequency and our correction for anharmonicity.1389

We said that the energy for the rigid rotator harmonic oscillator approximation1399

that was E sub RJ = ν sub 0 × R + ½ + B × J × J + 1.1405

This is the rigid rotator harmonic oscillator approximation.1417

It does not really matter.1422

The dependence of B on R, B sub R is equal to B sub E - Α sub E × R + ½.1424

This thing goes in where B is, that is the adjustment for vibration rotation interaction.1438

The adjustment for harmonicity, the energy is ν sub E × R + ½ - this X sub E ν sub E × R + ½².1446

This is the adjustment for anharmonicity.1465

Now when we put this together, when we put both of these into this,1469

we get a new equation for the energy which is RJ is equal to,1474

It gets a little complicated, a little long, no doubt about that.1482

R + ½ - X sub E ν sub E × R + ½² and the rotational term which is going to be B sub E - Α sub E × R + ½ × J × J + 1.1487

This is the ν for the two corrections that we have to make.1510

Notice again, this problem does not ask us to account for centrifugal distortion.1516

If it did, we have one more – that D thing, D × J² J + 1², but that does not matter here.1520

For R = 0, we get E of 0 J is equal to ½ ν sub E -1/4 X sub E ν sub E + this B sub E - Α sub E.1530

Again, these are just a bunch of parameters that we just have to account for.1551

It is not terribly a big deal.1554

R is 0 here so we can just go ahead and put the -1/2 Α sub E × J × J + 1.1557

This ends up being E sub 0 J is equal to 1154.1569

I will just put the numbers in.1577

In other words, I will put the ν sub E in, we put the X sub ν E in, the BE, the Α E, we just put it in.1579

That is it, nothing strange.1586

9.911 + 6.4275 × J × J + 1.1588

For the R = 1 vibrational state ,we get E of 1 J that is going to equal 3/2 ν sub E - 9/41601

X sub E ν sub E + B sub E – 3/2 Α sub E × J × J + 1.1614

And we end up as far as numerically is concerned, we end up with 3463.521 - 8189.199.1629

This is nothing more than just a bunch of tedious arithmetic, that really is L comes down to.1645

+ 6.2585, which is why I personally love theory as opposed to the tedium.1650

That is why we work with symbols, you do not want to do those arithmetic.1662

× J × J + 1.1665

This equation right here, let me go to blue, we use this equation and we use this equation.1669

We create a table of values.1676

We have J, J is going to take on the values 0, 1, 2, 3, 4, that is the first 5 states.1680

E sub 0 J, all the values are in inverse cm.1689

E1 J we have 1144.6, 1157.5, we have 1183.2, we have 1221.7,1695

we have 1273.1, we have 3374.3, we have 3386.8, we have 3411.9, 3449.1713

I completely understand, this is not something that you actually get used to.1735

After a long time doing this, you lose your mind from what are all these numbers.1739

3449.4 and this is going to be 3499.5.1751

The allowed transitions are δ J = + or -1.1760

We are going to have the 0 to 1, 1 to 2, 2 to 3, 3 to 4.1767

Or we are going to have the 1 to 0, 2 to 1.1773

Basically, all you have to do when you need to find the actual frequencies of the absorption,1777

the frequencies of the spectral lines, you are just going to take for example the 0 to 1 transition,1783

you are just going to take this number - this number.1788

The 1 to 2 transition, this number - this number.1793

You do not actually need the equations, which is why the energy equation is really the most important.1796

If you have that, you can get everything else.1801

The δ J = + or -1, when we do that, let us go ahead and create a new table of values here.1805

Once again, let us say δ J = + or -1.1815

The R0, R1, R2, R3, that represents the 0 to 1 transition, this is the 1 to 2 transition,1823

this is the 2 to 3 transition, this is the 3 to 4 transition.1834

This is going to be at 2242.2.1838

It is just the second number the second column - the first number the first column.1842

The energy of the 1 - the energy of the 0.1848

The energy of the 2 - the energy of the 1.1852

For the two vibrational states.1854

We do not necessarily need equations for Ν sub ρ and ν sub P.1858

2254.4, 2266.2, 2277.8, and we have the P branch.1863

This is going to be P1, P2, P3, P4.1875

This represents the 1 to 0 transition.1880

This represents the 2 to 1 transition.1883

This represents the 3 to 2 transition.1886

This represents the 4 to 3 transition.1888

This is going to be 2216.8.1893

This is going to be 2203.6.1897

This is going to be 2198.2.1901

And this is going to be 2176.3.1905

I really like the black.1918

If we wanted specific equations for ν sub ρ and ν sub P, basically the observed frequencies.1920

These numbers, we have to do some algebra, that is the problem.1934

We would have to do a lot of algebra actually.1945

We have to do a lot of algebra.1952

Essentially, what you have been doing is the following.1954

For ν sub R, you take the energy of the upper states and subtract the energy of the lower state.1959

That is what you are doing.1963

You are going to end up with, it is going to be the energy E R + 1 J + 1 - ERJ.1964

You have to do all that and come up with some final equation.1975

For the P branch, that is going to end up being E R + 1 J - 1 ERJ.1978

Based on the equation above, that long equation that we had for the energies,1990

you have to take one for the upper state then subtract that long equation, one for the lower state.1994

Go through all the algebra and come up with some equations, which is essentially what you see in your books.1999

Most of what you see in your books, the equations, the derivations of the spectroscopy section are just that.2004

They are just taking one energy and subtracting other energy.2010

It tends to look complicated but it is just a lot of algebra.2013

In order to come up with equations for the spectroscopic lines that we see,2016

based on whatever correction we are making it.2021

It is either going to be the harmonic oscillator rigid rotator approximation for the energy.2023

You can either make a correction for vibration rotation interaction.2029

You can make a correction for anharmonicity.2033

You are going to make a correction for centrifugal distortion.2035

Either 1, 2, or all 3 of those.2038

The equation is longer for every correction that you make.2041

You remember all this from the first lesson, the initial lesson that we did in spectroscopy2046

where all these things are coming from.2050

That it is not this ocean of equations that you are dealing with.2053

It is just the energies of the upper state - the energies of lower state that give you the frequency of the spectral line.2057

Let us do some more examples here.2068

I think we are on example number 3 now.2070

Wait, I think we are missing some data here.2078

We do not have this data but that is not a problem.2084

I actually have it right here with me, I will go ahead and write it.2086

For some odd reason, it did not end up on this page, my apologies.2088

Calculate B sub 0, B sub 1 of the equilibrium bond length for the R = 0 vibrational state and2091

the equilibrium bond length for R = 1 vibrational state, the B sub E and Α sub E for carbon monoxide.2100

I apologize, it did not show up on this slide.2118

Ν observed, we observed an R 0 line of 2173.81.2122

We observed R1 line at 2177.58.2132

We observed a P1 line at 2166.15.2140

And we observed a P2 line at 2162.27.2146

Given this data for two R lines and the first two lines of the P branch, how can I calculate all of these?2153

We see B0 and B1, that equations I want to deal with are going to only account for the vibration rotation interaction.2162

I do not see anything else here.2184

I do not see any X sub E, ν sub E.2185

I do not have to worry about the anharmonicity.2187

I do not see a D here so I do not have to worry about centrifugal distortion.2190

I see B sub 0 B sub 1, the equations that I’m going to be looking at,2193

I'm going to be looking at the rigid rotator harmonic oscillator corrected for vibration rotation interaction.2197

That is how you choose your equation.2203

You take a look at what constants are at your disposal, what they want, and I will tell you what to choose.2205

We see B0 and B1, we consider only the vibration rotation correction.2211

I always like to begin with the basic equation and then add to that.2234

It is just a nice way to keep sort of refreshing yourself and see that the equation over and over and over again.2238

Our harmonic oscillator rigid rotator equation E sub RJ, the energy is ν R + ½ + B × J × J + 1.2245

I do not need that parenthesis at J × J + 1.2263

The vibration rotation correction, that one says that this B, the rotational constant depends on R.2267

B sub R is equal to B sub E - Α sub E × R + ½.2279

This we put into here and get a new equation.2289

The corrected equation for the vibration rotation interaction is equal to,2297

Sometimes, I will put a tilde over the E, sometimes not.2302

It is in wave numbers, do not worry about it.2305

That is equal to, it is going to be ν R + ½2308

+ this thing B sub E - Α sub E R + ½ × J × J + 1.2323

Unless specifically stated otherwise,2338

always take the R = 0 to the R = 1 vibration transition.2357

That is the one that you want to take.2365

At normal temperatures, room temperatures, most molecules are actually going to be in their ground vibrational state.2368

They are also going to be in their ground electronic state.2374

However, at normal temperatures, it is the rotational states that molecules tend to be the higher states,2378

they cannot be the ground state.2384

J = 0, they can be anywhere from like J = 2 or 3, above that.2386

At normal temperatures, most molecules are in the vibrational ground state and electronic ground state.2393

It is only the rotational state that there are in excited states at normal temperatures.2401

We will talk more about that when we do statistical thermodynamics next.2405

When we look at this, our new observe, what we are looking for is this.2419

We need an equation that is going to be the energy of the upper state - the energy of lower state.2423

The energy being the equation that we just had.2433

Our R branch is going to be E of 1 J + 1 - E of 0 J.2435

Our ν sub P is going to be E of 1 J-1 E0 J.2448

When I do the algebra for these and I put the energy equation that I just had in the previous page,2459

one for the upper state and one for the lower state.2464

When I work out that algebra, here is what I get.2466

When we work out these differences, we get a rather daunting looking equation actually.2475

Ν sub 0 + B1 - B0 J² + 3 B1 - B0 J + 2 B1.2491

The J takes on the value starting from 0, 1, 2, and so on.2512

For the P branch, I get ν sub 0 + this same term B1 - B0 J².2518

The third term is different, this is going to be B1 - B0 × J.2529

Here, the J takes on the values of 1, 2, 3.2540

And again, for the R branch 0, so on.2543

The P branch is 1 and so on.2545

Let me go to blue here.2561

The R is 0 line that represents the J = 0, that is going to be ν sub 0 + 2 B1.2565

In other words, I put J = 0 into this equation and I see what I get.2577

I get ν sub 0 + 2 B1.2584

The data gave us the frequency of absorption, the ν sub 0, the R sub 0 line is at 2173.81.2588

The R1 line that is where J is equal to 1, that is ν sub 0 + 6 B1 – 2 B0 is equal to 2177.58.2602

The P sub 1 line that represents that J = 1.2622

I’m going to use this equation, the one for the ν sub P.2627

I get ν sub 0 -2 B0 = 2166.15 inverse cm.2631

For the P2 line that is the J = 2, I get ν sub 0.2642

I get + 2 B1 -6 B0.2647

And that one was observed at 2162.27.2654

These are the equations that I’m going to work with now.2660

I’m going to take the R1 equation - the P1 equation.2668

When I do that, I get 6 B1 = 11.43.2674

Therefore, B1 is equal to 1.905 inverse cm.2684

I found B1.2694

Now, I take the R0 line and I subtract from it the P2 line.2696

When I do that, I get 6 B0 = 11.54.2701

Therefore, B0 is equal to 1.923 inverse cm.2711

We found B1, we found B0.2722

BR, this B sub R that we just found is equal to H/ 8 π² C × I.2726

I is the reduced mass × this, whatever this is².2739

Now, we need to find the equilibrium bond length for each of these.2744

When I put the values in, I will get the following.2751

When I rearranged this for RE², R sub E vibrational state 1² is equal to 6.626 × 10⁻³⁴.2754

I’m not going to put the units in and I’m just going to put the numbers in.2771

8 π² 2.998 × 10¹⁰.2774

I went ahead and use cm directly.2782

1.139 × 10⁻²⁶ that was the reduced mass and then the 1.905 that is this number right here.2786

This goes down here, this comes up here.2797

When I solve for this, I get that it is equal to 113.6 pm.2800

The same for R0, when I do the same for R0, R sub E for the 0 state, I end up with a value of 113 pm.2812

We found RE 1, RE, we found B0, we found B1.2832

Let us go ahead and find for B sub E and Α sub E.2836

We said that b sub R is equal to B sub E – α sub E × R + ½.2850

We know what B0 is.2862

B0 is equal to B sub E – ½ Α sub E.2866

We know what B1 is, let us put that 0 into here, 1 into here for R.2874

We get B sub E - 3/2 Α sub E.2881

We know what B0 is, we already found it 1.923.2888

1.923 = B sub E – ½ Α sub E.2892

And we know that 1.905 which is B1, that = B sub E - 3/2 Α sub E.2903

Two equations and 2 unknowns.2914

Let us go ahead and take, this is equation 1 and this equation 2.2916

Let us go ahead and take equation 1 - equation 2.2921

When you do that, you end up with Α sub E is equal to 0.018 inverse cm.2928

Put this into one of these other equations and you end up with B sub E is equal to 1.932 inverse cm.2944

There you go, I hope that make sense.2957

Let us see what is next.2962

The IR spectrum of ML shows a fundamental line at 1877.62 inverse cm and the first overtone at 3728.66 inverse cm.2970

Find the values of ν sub E and X sub E ν sub E for ML.2983

Let us go ahead and work in black here.2993

The frequencies of the overtones for the anharmonic oscillator2998

are given by G of R – G of 0, just looking at vibrational transitions right now.3027

G of R to G of 0, the anharmonic oscillator.3037

Under normal temperatures, most molecules are in the ground vibrational states.3043

They are all going to start at the 0 vibrational state.3046

But δ R is no longer + or -1.3050

It can be + or -1, + or -2, + or -3.3053

The + or -1 is the fundamental.3056

The + or -2, those are the first overtone.3059

The first overtone, second overtone, third overtone.3062

The G of R equation that is equal to ν sub E × R + ½.3066

The anharmonic oscillator – X sub E ν sub E R + ½².3076

When we take the G sub R – G of 0, when we do the algebra,3078

what we get is the observe line equal to ν sub E × R – X sub E ν sub E × R × R + 1.3093

R1 runs from 1, 2, 3, and so on.3109

The fundamental transition is the transition from 0 to 1, that is the one that we see.3117

That is the brightest one that we see.3131

That is equal to, I put into this equation, fundamental 0 to 1.3135

I put R = 1, it is going to be ν sub E × 1 - X sub E Ν sub E × 1 × 1 + 1.3142

We end up with ν sub E - 2 X sub E ν sub E.3157

It tells us what that is already.3178

The fundamental is 1877.622.3179

1877.622 that is one of the equations that we are going to use.3184

The first overtone represents the transition from 0 to 2.3191

It is a weaker line.3198

That is going to be ν sub E × 2.3202

Just using this equation right here, taking care of the R = 1.3208

Now the first overtone R = 2 - X sub E ν sub E × 2 × 2 + 1.3214

What we end up with is 2 ν sub E – 6 X sub E ν sub E, it tells us what the first overtone is 3728.3227

3728.66 that is the second equation that we are going to use.3240

When you solve simultaneously, I will not go through the process.3247

I will let you go ahead and do that.3249

This equation and this equation, when you solve simultaneously,3250

you are going to get a value of ν sub E is equal to 1904.20 inverse cm and X sub E ν sub E is equal to 13.29 inverse cm.3253

That is it, nice and easy.3274

I know it is not nice and easy, believe me I do.3279

I have been there, the stuff still gives me grief too.3282

Let us take a look at our final example.3289

I’m actually wondering whether I should leave this example off because the first two lessons took care of it.3291

Of course, we had an example from the previous lesson that did this but we will see.3297

In examples 6 of the previous lesson, we asked you to calculate the frequencies of the first two lines of the R and P branches3304

for the vibration rotation spectrum of HBR under the harmonic oscillator rigid rotator approximation.3312

Here we ask you to repeat the problem.3320

But instead of the harmonic oscillator rigid rotator approximation,3323

we asked you to do so by accounting for all three corrections we made to the HIR equations.3326

We accounted for vibration rotation interaction for centrifugal distortion and for harmonicity.3335

Use the following spectroscopic parameters as necessary.3340

This is a great example because we get a chance to put everything together,3343

the HRR approximation and all of the corrections that we made.3350

I’m not going to write all the equations here.3367

I can do so in the last couple of lessons, basically what we are going to be doing is,3368

in this one you are going to take the harmonic oscillator approximation.3372

You are going to make the corrections for all three of these things.3378

The vibration rotation interaction, the centrifugal distortion, and the anharmonicity.3381

You got to find an equation for the total energy.3385

You are going to take the upper level - the lower level.3388

In other words, you are going to find δ E.3391

That is going to give you your equation.3395

You do not necessary have to go through the algebra, the equations are actually have been done for you.3398

They are in your books, I think we actually did in our lessons too.3402

When you do it, the equations that you come up with are the following.3404

Ν sub R, that is going to be ν sub 0 × 1 – 1.3411

2648, we are going to have ν sub E × 1 - 2 XE + 2 × BE × J + 1 - Α sub E × J + 1 × J + 3 - 4 D × J + 1³.3434

The J values are going to run from 0, 1, 2, and so on.3465

This is the equation that you get when you take the total energy,3471

the correction for all three things and the energy of the upper state – the energy of the lower state.3475

It is a lot of algebra but this is the equation you come up with.3482

This is the frequency of the spectral line that we should see.3484

The ν sub P branch that is going to be ν sub E × 1 - 2X sub E -2 B sub E × J - Α sub E × J × J - 2 + 4D J³.3492

Here J = 1, 2, and so on.3525

These are the two equations that we actually get.3529

Notice the equation for the spectral line changes depending on what corrections we are making.3532

In this case, we corrected for all three.3537

They give us ν sub E, we have that one.3542

We are going to need that parameter.3544

We are going to need this parameter.3546

We are going to need this parameter.3549

We do not need those others.3554

We also need the X sub E but they gave us N sub E ν sub E.3555

Actually, you have to take this.3559

We would need that.3561

We would have to take this parameter divided by that, in order to get just the X sub E.3561

Let us go ahead and do that first.3567

The data gives ν sub E and X sub E ν sub E.3570

Let us find X sub E alone.3586

The X sub E is equal to X sub E ν sub E divided by ν sub E.3591

We end up with 45.217 divided by 2648.975 and we end up with X sub E = 0.01707 inverse cm.3599

We put all these parameters into the equations.3617

What we end up with, ν of the R sub 0 line.3621

That is going to equal 2648.975 × 1 - 2 × 0.01707 + 23636

× 8. 465 × 0 + 1 - 0.2333 × 0 + 1 × 0 + 3 - 4 × 3.457 × 10⁻⁴ × 0 + 1³.3653

When I calculate that, I end up with 2574.768 inverse cm.3683

This is my R0 line.3693

I should see it there.3696

Notice, we expected something lower than what we get with the rigid rotator harmonic oscillator approximation.3699

Exactly what we got.3706

Now, when I do the same for the R1 line, I will write all this out.3708

What I end up with is 2590.522 inverse cm.3714

Notice the spacing between the lines.3724

The spacing between the spectral lines is this line - this line.3728

This line - this line, the absolute value there.3729

The spacing is ν of R1 - ν of R0, that is equal to 15.755 inverse cm.3735

2B E = 2 × 8.465 that is equal to 16.93 inverse cm.3750

You see that the spacing, the corrections that we have made.3763

On the actual spectrum, the spacing is less than 2B.3766

2B is 16.93.3769

The corrections that we made give us the spacing which is about less than 2B, which is exactly what we expect.3771

For the R branch, the spacing is smaller.3780

For the P branch, the spacing will actually be bigger than 2B.3783

The 2B is the rigid rotator harmonic oscillator approximation.3787

Let us go ahead and do the P branch.3792

I think I have one more page, If I’m not mistaken, yes I do.3794

I got ν of P1 and when I put into the equation for the P1, I end up with 2541.8.3801

Let me write this one, at least.3822

It is going to be 2648.975 × 1 - 2 × 0.01707 - 2 × 8.4653824

× 1 - 0.2333 × 1 × 1 - 2 + 4 × 3.457 × 10⁻⁴ × 1³.3844

This will give me 2541.843 inverse cm.3863

And when I do the same for the P2 line, I end up with 2524.690.3870

Now, the difference here, the δ ν.3885

In other words, one of them - the other, that is going to equal 17.153, which is greater than 2B,3891

which is exactly what we expect for the P branch.3903

That is it, thank you so much for joining us here at www.educator.com.3910

We will see you next time.3913

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