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Raffi Hovasapian

Raffi Hovasapian

Example Problems I

Slide Duration:

Table of Contents

I. Classical Thermodynamics Preliminaries
The Ideal Gas Law

46m 5s

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

46m 2s

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

1h 6m 45s

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

1h 6m 33s

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

1h 2m 17s

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

1h 4m 39s

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

16m 50s

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

43m 40s

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

32m 23s

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

39m 15s

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

35m 52s

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

42m 40s

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

1h 23s

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

44m 34s

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

59m 7s

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

49m 16s

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

33m 59s

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

54m 37s

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

31m 18s

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

23m 6s

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

25m 42s

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

43m 39s

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

56m 44s

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

57m 6s

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

54m 35s

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

35m 5s

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

28m 42s

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

34m 38s

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

30m 50s

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

34m 6s

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

39m 18s

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

19m 40s

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

54m 16s

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

31m 17s

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

45m

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

58m 5s

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

25m 20s

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

34m 25s

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

59m 57s

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

42m 5s

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

30m 26s

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

21m 34s

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

56m 22s

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

45m 24s

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

48m 43s

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

46m 18s

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

39m 28s

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

35m 32s

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

29m 55s

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

54m 25s

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

46m 58s

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

44m 11s

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

35m 33s

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

43m 4s

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

26m 30s

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

41m 10s

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

30m 32s

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

35m 19s

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

33m 48s

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

46m 43s

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

40m

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

35m 58s

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

36m 18s

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

33m 55s

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

51m 53s

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

51m 53s

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

34m 29s

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

43m 49s

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

1h 1m 57s

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

48m 33s

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

48m 33s

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

48m 33s

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

59m 18s

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

34m 54s

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

38m 3s

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

57m 49s

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

1h 1m 20s

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

56m 34s

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

18m 24s

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

50m 2s

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

59m 47s

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

46m 22s

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

29m 24s

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

30m 53s

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

1h 1m 33s

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

33m 47s

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

1h 1m 5s

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

33m 31s

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

1h 1m 15s

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

47m 23s

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

54m 9s

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

48m 32s

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

57m 30s

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

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • 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

Transcription: Example Problems I

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

Today, we are going to start doing some example problems on the Schrӧdinger equation, particle in a box,0004

particle in a 2 dimensional box, 3 dimensional box, things like that.0012

We are going to be doing a lot of problems.0016

The only real way to wrap your mind around any of this is doing a ton of problems.0019

The only warning that I have is as you already figured out quantum mechanics can be notationally intensive.0023

The best advice I can give is keep calm, cool, and collect it, and work slowly.0030

Let us jump right on in.0035

Our first example problem is let ψ be a wave function for a particle in a 3 dimensional box.0039

Find del² of the ψ and show that the ψ is an Eigen function of Laplacian operator.0046

Let us see what we have got.0056

First of all, let us start off with our equations.0057

Again, I think is always a good idea to just write down the equations that you know simply0060

because repetition locks it in your mind, makes you feel more comfortable with it.0060

Let us go ahead and write down our equations.0070

Ψ for a 3 dimensional box, N sub X N sub Y N sub Z = 8/ ABC ^½ × sin of the N sub X π/ AX × sin of the N sub Y π/0073

B × Y × sin of N sub Z × π/ C × Z.0096

That is our wave function.0107

The Laplacian operator, del² was this thing.0110

The del² that was that D² / DX² + D² DY² + D² DZ².0116

What we have to do is we have to do is to apply the Laplace operator to this function.0133

I know what you are thinking, I'm thinking the same thing.0143

Our del² ψ = del² DX² + del² DY² + del² DZ² of ψ.0148

Again, operators they can distribute so we are going to do this, we are going to do this, and we are going to add.0166

Let us just jump right on in.0176

Let us go ahead and do the second partial derivative of the X with respect to X,0180

Which means we are going to hold this constant and this constant.0191

The only thing we are going to differentiate is this.0195

Let me actually do this in red.0198

This first term, we are just going to differentiate this function because we are holding all the other variables constant.0200

Let me go ahead and write this out.0210

When I take the first derivative of this, I’m going to write it this way.0215

N sub X π/ A × X, when I take the first derivative, just that D means take the derivatives.0219

I end up with the following.0227

I end up with NX π/ A × cos of NX π/ A × X that is the first derivative.0228

If I take the second derivative, if I differentiate again, I end up with N sub X² π²/ A² and the derivative of the cos is –sin,0238

I get the negative sin there and I get N sub X π/ A × X.0253

This is my second derivative.0262

Since, I’m holding everything else constant, this is the only thing that I need to concern myself with.0265

My D² DX² of ψ is going to equal to -8/ ABC ^½.0271

I will take this thing , -sin N sub X² π²/ A².0290

I have this × the rest of this.0301

I have included this, that is here.0305

The negative sin from here, this thing the N sub X² π²/ A² that is here.0308

Now, I have this × this and this.0314

What I get is a sin of N sub X π/ A × X × sin of N sub Y π/ A.0317

BY × sin of N sub Z × π/ C × Z.0329

This is the first thing that I want.0342

Let me go ahead and circle it, that is the first thing that I want.0346

That is my first distribution.0351

I have done 1/3 of my operation so far.0353

It is going to be same, these functions are the same.0358

It is the exact same thing except this time with respect to Y and with respect to Z.0360

When I operate on ψ with the second partial with respect to Y and operate with respect to Z,0368

I end up with the following.0375

I'm just working very carefully.0379

Let me go to red.0382

I get D² DY² that is going to equal the same thing as before.0385

Everything is the same except now the variable is different.0391

You get -H/ ABC ^½ and this time you have N sub Y² π²/ B² and everything else is the same.0395

Sin of N sub X π/ A × X, you will get sin of N sub Y π / B × Y × sin N sub Z π/ C × Z.0409

The last one, we get the del²/ del Z² it looks exactly the same.0425

8/ ABC¹/2 except now we differentiate it with respect to Z.0433

We have N sub Z², let me make this a little bit more clear here.0441

N sub Z² × π/ C² and everything else is the same.0452

Sin of N sub X π/ A × sin of N sub Y π/ B.0459

There is an X here, there is a Y here.0468

Sin of N sub Z π/ C × Z.0472

You can see that it is really easy for a problem in quantum mechanics to go south on U for no other reason than for arithmetic or missing some letter.0479

It is going to make you crazy. It is going to make you want pull your hair out but that is the nature of the game.0488

We have our 3 partial derivatives.0493

We have operated on the wave function.0495

We have this and this and the previous one, so now we just add them up.0499

That was the last part of the operator, we have to add them.0504

When we add them up, adding these 3 expressions together, notice the sins are all the same.0507

The -8/ ABC¹/2 that is the same, you have the π² that is the same.0529

When you put all the terms together, what you end up with is the following.0533

Let me write out the operator here.0541

Del² of ψ N sub E= - 8/ ABC ^½ × we have π² that is going to be × N sub X²/ A² + N sub Y²/0548

B² + N sub Z²/ C² × this thing.0574

I have the sin of N sub X π/ A × X.0581

I should have made a little bit more room here.0588

Sin of N sub Y π/ B × Y × sin of N sub Z π/ A × Z.0590

This is my final solution, the first part of the problem, that is it right there.0608

I wanted to apply the del operator ⁺to the wave function or particle in a 3 dimensional box that it is right there.0616

On the second part I have to show that it is actually an Eigen function of the operator.0625

For a function to be an Eigen function of an operator.0632

Let me write this down.0636

For a function F, speak in general terms, for a function F to be an Eigen function of an operator A, remember operators have a little hat on top of them.0639

The following must hold, this is the definition.0664

The following must hold applying the operator to F gives me some constant × F.0671

In other words, when I apply an operator to a function, what I end up getting is just some constant × that function.0682

If that holds, then the function is an Eigen function of the operator.0690

In other words, an F is an Eigen function of the operator A and L happens to be the Eigen value associated with that particular Eigen function.0697

That is what is happening.0707

Notice what we have, our wave function.0710

I will do this in black here.0714

Our original wave function ψ, in other words, what we want is this.0719

We want Del² ψ to equal sum constant.0723

I will just use λ again.0730

To be some λ × ψ.0732

I want to get ψ back so I operated on it with the del² operator and I got this.0734

My function, my original ψ is this one right here.0743

It includes this, that, and this.0745

That right there is ψ.0749

I operated on it, I got this thing to actually to equal.0763

I can write it this way, that is my function.0768

I end up with del² of ψ = I'm left with - π² × N sub X²/ A² + N sub Y²/ B² + N sub Z²/ C² × ψ.0772

Thing is just the actual function ψ.0804

Sure enough, when I operated on it, I got this thing.0808

This thing is nothing more than a constant which is that × the original function.0812

Therefore, ψ my wave function for a particle in a 3 dimensional box is an Eigen function of the Laplacian operator.0819

This thing happens to be the associated Eigen value for the associated function.0830

Let me write that down.0837

Do I have an extra page here?0838

I do, so let me go ahead and use it.0841

Our ψ N sub X N sub Y N sub Z is an Eigen function of the Laplacian operator.0846

The corresponding Eigen value is - π² × N sub X²/ A² + N sub Y²/ B² + N sub Z²/ C².0867

That is it so again, I have an operator and I operate on some function F.0897

If what I end up getting when I do that is just the constant × the function back, that means that F is an Eigen function of this operator.0902

That means it is very intimately connected and λ is the corresponding Eigen value for this particular Eigen function.0913

There are going to be many Eigen functions for a given operator.0920

It one of those Eigen functions has an Eigen value.0923

What you are doing is this, it is saying when you take the derivative, when you operate on a function you are just getting that function back.0926

All you are doing is multiplying it by some constant.0933

That is very extraordinary and we will say more about it as we go on in the course.0936

Let us go on to another problem here.0945

Example 2, what are the most likely positions of a particle in a 1 dimensional box of length A when it is in the N = 2 state?0950

What is the most likely position of a particle?0959

In other words, what position in a 1 dimensional box?0962

This is 0 and this is A, where along this interval is the highest probability that0967

I’m going to find the particle when the wave function is in the 2 state?0979

That is all the question is asking.0986

Let us go ahead and write down our equations.0989

We know that the equation for a 1 dimensional box ψ sub N = 2/ A.0991

I hope you do not mind.1002

I’m going to make one slight change here.1004

It is a capital here, I am accustomed to using the small a so I’m going to change this, of length A.1007

I’m just accustomed to writing the small a.1015

We have 2/ A ^½ × sin of N π/ A × X.1018

This are actual wave function for a particle in a 1 dimensional box.1029

We said that it is in the N = 2 state.1034

Ψ of 2 = 2/ A ^½ × sin of 2 π/ AX.1037

We just actually pick the value.1047

We are talking about probabilities here.1051

We said that ψ conjugate × ψ or in the case of a real function just ψ² is the probability density or in other words, the probability.1055

When we speak of probability density, we can go ahead and say probability.1081

We said that when you multiply the conjugate of ψ × ψ or in the case of a real function just ψ² is a probability density.1085

This is the function that we have to maximize.1095

This is the function we have to maximize.1098

Let me write a little bit better.1104

Ψ² is the function we must maximize because we wanted the most likely positions.1107

Once we have the function which we do, ψ² for the probability, we want to maximize the probability.1124

That is the function we have to maximize.1130

You remember from calculus, maximize means that the DDX of this ψ² function = 0.1131

We are going to form ψ², we are going to differentiate it with respect to X.1149

We are going to set it equal to 0 and we are going to find the X values that actually maximize it.1152

What you are going to get are some of that maximize and minimize it.1159

We are going to have to pick the ones that maximize it.1161

That is what we are doing.1162

This is a straight maximum problem from single variable calculus.1164

Let us go ahead and write down,1169

The ψ * × ψ which is the same as ψ² because we are dealing with a real function.1171

That is just going to equal this function × itself.1178

We get 2/ A × the sin² of 2 π/ A × X.1182

This is our ψ².1196

We need to go ahead and differentiate that function.1198

When we do DDX of our ψ², I will leave the 2/ A.1203

I’m going to differentiate this, this is going to be × 2 × sin of 2 π/ A × X.1213

Chain rule so I differentiate this, I get the cos of 2 π/ A × X and I differentiate what is inside.1228

It is going to be the 2 π/ A.1238

That is the derivative with respect to X of this function.1242

When I simplify it, let me go ahead and put some things together here.1246

I end up with 4 π/ A² × 2 × sin of 2 π/ A × X × cos of 2 π/ A × X.1253

I’m going to use the identity, 2 sin θ cos θ = sin of 2 θ cos θ.1270

I'm going to rewrite this using that identity and I end up with the following.1284

I end up with the DDX of our ψ² is going to equal 4 π/ A² × sin of 4 π/ A × X.1291

That is our derivative.1308

That is the derivative that we now need to set equal to 0.1310

That means that sin of these 4 π/ A × X = 0 because this is just a constant.1315

Now sin of angle = 0 whenever this 4 π/ X = 0 π 2 π 3 π 4 π 5 π 6 π 7 π.1325

Therefore, this is true.1342

Whenever 4 π/ A × X = some integer × π, where K = 0, 1, 2 and so one.1344

That is what makes this possible.1359

Rearrange for X, go ahead and cancel the π, and I get X = K × A/ 4.1360

For K= 0, 1, 2, 3, 4.1372

Try different values of K.1379

Let me go ahead and rewrite my function ψ² just so I have it.1385

My ψ² which we have actually maximized.1391

You have to go back to the original function not the derivative.1396

Ψ² = 2/ A × the sin² of 2 π/ A × X this is the function that we have maximized.1399

We differentiated it, set it equal to 0, these are the values.1411

When K =0, what you will end up with is.1416

Let me put K and 0 into here, and we put this value of X that we get into here.1424

What you will end up with is the following.1434

You end up with 2/ A × the sin² of 2 π/ A × K 0.1436

This is 0 that means X is 0 × 0 = 0.1447

Let us try K = 1.1454

When I do K= 2 in here, I get 2A = 4, X = 2A/ 4.1458

I put that into this X so I end up with 2/ A × the sin² of 2 π/ A × 2A/ 4.1470

The sin of π is 0, so again this = 0.1492

Let us try K=4.1497

4 is as far as we can go, I will tell you why in just a minute.1509

When we use K =4, we get 4A/ 4 that is X.1512

We put that into this and we end up with,1518

Again, we are doing the original function sin² of 2 π/ A × 4A / 4, that cancels.1521

A and A what you get is a sin of 2 π², sin² of 2 π.1534

The sin of 2 π is 0 so we end up with 0.1540

These are the minimum values.1544

4 is as far as we go for K and here is why.1549

When K = 4, 4/ 4 will give us A.1552

We got as far as we are going to go.1556

Again, we are going from 0 to A that is our box.1558

We cannot go farther than that.1564

We cannot go for example 2, 6A/ 4 which is 3/ 2.1566

That is out here.1571

We are sticking here is the case, from 0 to 4.1573

If K is equal 0, 2 and 4 give us minimum values that means that the 1 and the 3, when K = 1 and K = 3,1576

that is going to give us the maximum values.1584

When K = 1 and K = 3 give us the Max values of this function.1592

With K = 1, X = ¼ and when K = 3 X = 1 A/ 4.1614

It is A/ 4.1632

This is going to be 3A/ 4.1633

Therefore, let us do an extra page.1640

A/ 4 and 3A/ 4 are the most likely positions of finding a particle in a 1 dimensional box for the N = 2 state.1646

What is it that we have done?1681

We have the wave function, let me go back to black.1683

We have the wave function ψ.1690

We know the probability or the probability density.1692

In this particular case, because it is a real function, under normal circumstances is this one.1694

This, because it is a real function it is just ψ².1700

We form the ψ² and we differentiated it.1704

We took the derivative of it.1708

We set the derivative equal to 0.1709

We found the values of X that make that true and we took those values of X for different values of K 0, 1, 2, 3, 4.1714

We plug them back in the original equation to see which one give us a minimum value,1725

which in this case was 0 and which one give us a maximum value which was in this case, you end up with a sin =1.1729

That is what we did.1738

We found out that when K = 1 and K = 3, that gives us the maximum values.1739

Therefore, when we put that back into the X = KA/ 4 which gives us our max and min values,1746

we end up with A/ 4 and 3/ 4 as the most likely positions to find a particle.1755

I hope that makes sense.1762

Let us try our next example here.1766

For an electron in a 3 dimensional box which sides are 1 nm × 2 nm × 4 nm and X, Y, Z respectively,1772

X is 1, Y is 2, Z is 4, calculate the frequency of the radiation required to stimulate a transition from the state 123 to the state 234.1781

In other words, N sub X is 1, N sub Y is 2, N sub Z is 3.1793

For a particle in a box, we have 3 quantum numbers.1798

In this case, 123 state.1800

I want to stimulate it to the 234 state.1802

How much energy do we need?1808

Or what is the frequency of the radiation that I need?1810

Let us go ahead and see if we can work this one out.1813

Let us start off with our equations.1818

I know the energy N sub X, N sub Y, N sub Z, is equal to planks constant/ 8M × N sub X²/ A² + N sub Y²/ B² + N sub Z²/ C².1820

The change in energy from the state is going to be the energy of the 234 state - the energy of the 123 state.1845

Energy of the 123 is the difference between them.1857

Let us find the energy of the 234 and the energy of the 123.1865

The energy of the 234 state, we just plug the values in.1868

That is fine, let us go ahead and do this way.1874

I will write down what these values are.1886

A²/ 8 × the mass of the electron ×,1888

Well, NX² NX is 2 so this is going to be 2²/ A is 1 nm.1894

1 × 10⁻⁹².1902

You want to work in meters, you want to work in kg, you have to watch the units.1907

Time is going to be in seconds, mass is going to be in kg.1912

Length is going to be in meters as energy is going to be in Joules.1916

+ N sub Y,1922

N sub Y is 3²/ B².1924

B is going to be 2 × 10⁻⁹² + 4²/ 4 × 10⁻⁹² because that is 4 nm long.1928

Planks constant H = 6.626 × 10⁻³⁴ J/ s and the mass of the electron, the rest mass of the electron is 9.109 × 10⁻³¹ kg.1944

When I put these values in to here and run the calculation on my calculator, I get an energy for state 234 = 4.37 × 10⁻¹⁹ J.1963

Pretty standard range.1979

Let us go ahead and calculate the energy of 123.1982

The energy of the 123 state is again, the same thing A²/ A × the mass of the electron.1985

This time, we have 1²/ 1 × 10⁻⁹² + 2²/ 2 × 10⁻⁹² + 3²/ 4 × 10⁻⁹².1993

And when I go ahead and calculate this, I end up with 1.54 × 10⁻¹⁹.2014

Again, I hope that you are going to be checking my arithmetic.2021

I am notorious for arithmetic mistakes.2024

Therefore, the change in energy is equal to, like we said the energy of the 234 state –2029

the energy of the 123 state is going to end up equaling 2.83 × 10⁻¹⁹ J.2036

We know that the change in energy = planks constant × the frequency of the radiation2047

which implies that the frequency of the radiation = the change in energy ÷ planks constant.2053

We get 2.83 × 10⁻¹⁹ J ÷ 6.626 × 10⁻³⁴ J/ s.2061

Joules cancel and we end up with a frequency of 4.27 × 10¹⁴ inverse seconds which is as you now is a Hertz.2074

There you go, nice and simple.2088

You just plug the values in, that is all you got to do.2091

Let us see, what do we got next?2097

For a 1 dimensional box of length A, calculate the probability of finding a particle between A/ 2 and 3A/ 4.2106

Let us do this one in blue.2117

We have 0 and we have A, they are saying calculate the probability of finding a particle between A/ 2 and 3A/ 4.2120

This is 3/4 of A.2134

What is the probability?2136

What are the chances of me actually finding a particle right there?2139

Let us do it.2143

This is a 1 dimensional box so let us start off with our 1 dimensional wave equation2144

which is equal to 2/ A ^½ power × sin of N π/ A × X, that is our equation.2150

We also know that the probability of finding a particle between, when X is between some left and right point,2160

that is our L, this is our R, the probability of that is equal to the integral from L to R of ψ × ψ DX.2173

This is the probability density when we integrate over the entire interval that we are interested in,2185

we end up getting the probability of finding a particle there.2192

This is the definition of probability.2196

You just take the complex conjugate × the function, or in this case it is a real function.2199

Ψ² and you integrate it, that is all you are doing.2203

We know what ψ² is, ψ² is just equal to 2/ A × sin² of N π/ A × X.2208

It is just this thing squared.2219

Our probability is actually equal to, in this particular case it is going to be the integral from A/ 2 to 3A/ 4 of 2/ A sin² N π/ A × X DX.2223

Again, you can use mathematical software for this.2245

You can use a table of integrals for this.2247

It just depends on what it is that I happen to be doing.2250

In this particular case, I’m going to go ahead and use a table of integrals to look up this integral,2252

to see what it is, put the values in so I did this manually.2257

There are going to be other times when you just want some good straight numerical value and you just plug it into your software.2260

It does not really matter.2264

From a table of integrals I found the following.2268

Let us go ahead and do this in red.2271

The integral of sin² B X DX = X/ 2 - sin of 2 BX/ 4B.2276

In this particular case, our B is this N π/ A.2293

I'm going to just plug in N π/ A whenever and wherever I have a B and I’m going to evaluate this.2301

We end up with the following.2309

We end up with this integral, this constant comes out of course.2311

I have the integral of sin² N π/ X DX and again this is going to be our B here.2321

This is what I’m going to use.2329

I end up with the following.2331

I end up with the probability of being equal to 2/ A, that came out.2333

And when I integrated this thing, it is going to be X/ 2 - sin 2 N π/ A × X ÷ 4 N π/A.2338

We are evaluating it from A/ 2 to 3 A/ 4.2360

When I put these values in, here is what I get.2368

I’m going to write all this out.2371

2/ A × 3A/ 4 /2.2374

I get 3A/ 8 - sin of 2 N π/ A × X which is 3 A/ 4 all of that / 4 N π/ A -, now we will do A/ 2 in here.2380

I get A/ 4 - sin of 2 N π/ A × A/ 2 all over 4 N π/ A.2410

I just put this in here and this in here.2431

I’m just evaluating the integral when I start combining things 3A/ 8 -2A/ 8, I get A/ 8.2433

Here I get some cancellations.2444

The A and A cancel, the 2 and 4 cancel here, the 2 and 2 cancel, the A and A cancel.2446

I end up with the following.2455

Let us see here.2457

This one, I end up with sin of N π.2458

The sin of N π, regardless of when N is going to be 0, this term actually goes to 0.2463

It drops out.2468

When I take 3 A/ 8 – 2 A/ 8, I get A/ 8.2470

What I’m left with here, 2 and 3 is 6.2475

Here is what I end up getting.2491

I end up with the following.2495

I end up with 2/ A × A/ 8 - A × sin of 3 N π/ 2 all over 4 N π.2497

When I multiply, when I distribute this I end up with the following.2515

I end up with the probability equaling.2519

The A cancels so I end up with ¼ - 1/ 2 N π × sin of 3 N π/ 2.2526

This is my probability.2542

Whatever N happens to be, I will put it in and this is going to be a probability that we are seeking.2544

This is the probability that I will find a particle between A/ 2 and 3 A/ 4.2551

Notice something really interesting here.2563

As N goes to infinity, 1, 2, 3, 10, 20, 30, 40, 50, 100,2566

The term 1/2 N π goes to 0.2574

The probability ends up equaling ¼.2589

Here is how you calculate the probability.2601

It is going to be ¼ - something, ¼ + something depends.2603

But as you take N higher and higher and higher, as quantum numbers become larger and larger and larger,2607

the probability just comes down to ¼.2614

In other words, this particular area accounts for ¼ of the interval.2621

Therefore, my chances of actually finding the particle there is ¼ or 25%.2627

This is classical behavior and here is the correspondence principle in action.2635

The probability is going to change based on what N is.2641

It is not going to be ¼.2644

For N even it is going to be 1/4 but for N odd it is going to be different values ¼ - something, ¼ + something.2646

But as N gets higher and higher and higher, this term goes to 0 because this term goes to 0,2652

which means that the probability is going to be fixed at ¼.2659

If I do a million experiments, I'm going to find 1/4 of the time that the particle is going to be2665

in this particular area because it accounts for 1/4 of the interval.2670

Again, the correspondence principle says that as quantum numbers become larger and larger and larger,2674

quantum mechanical results which is this thing, they end up approaching classical mechanical results which is just the ¼ in general.2679

Let us see our last problem here.2693

Let A = B = 2C for the sides of a 3 dimensional box.2702

What are the degeneracy of the first five energy levels of a particle in this box?2707

The first five energy levels are the lowest, the next highest, next highest, the next highest, different numbers.2713

Let A = B = 2C.2719

When we talk about degeneracy, we set A = B =C, a perfect cubed.2722

In this case, two of the sides are equal but one of the sides is not.2726

What is going to happen?2729

Are there going to be degeneracy?2730

Are there not going to be degeneracy?2732

Let us find out.2733

Let us go ahead and do this in blue.2736

Here we have A= B and we have A = 2C.2738

Therefore, C = A/ 2.2744

When I take C² I'm going to get A²/ 4.2750

The energy in a 3 dimensional box, the energy of N sub X, N sub Y, N sub Z = planks constant²/ 8 M ×2755

N sub X²/ A² + N sub Y² / B² + N sub Z²/ C².2768

But in this particular case, A = B and C happens to equal A/ 2.2780

I can plug in, I can fix this so the energy actually ends up becoming H²/ 8M × N sub X²/ A² +2786

N sub Y²/ A² because B = A + 4 N sub Z²/ A²,2804

Because C² = A²/ 4.2817

When I flip it, the 4 comes on top and I’m left with A².2821

Now A² is everywhere in the denominator, I can pull it out as a constant.2824

I'm left with H²/ 8 M A² × N sub X² + N sub Y² + N sub Z².2829

This right here, this is a constant.2843

This is what is going to change.2846

N is going to change, it is going to vary independently.2848

111, 121, 131, 657 whatever.2850

This is what we are going to analyze right here.2854

This term is what we will analyze and which will give us the different values of the energy, what we will analyze.2859

Let us go ahead and start.2871

I think I would actually do this on the next page.2873

Let me rewrite the expression.2878

Our energy = H²/ 8 × M × A² × N sub X² + N sub Y² + 4 N sub Z².2881

We will go ahead and take the 111 state.2900

For 111, I’m just going to plug the 1 here, the 1 here, the 1 here, I end up with,2903

1² + 1² + 4 × 1² that is going to equal 1 + 1 + 4 that is going to equal 6.2913

6 × H²/ 8 MA² that is the energy of the 111 state.2924

I hope that makes sense.2932

I’m going to write it this way.2937

I will put E of 111 is equal to, this is what we are analyzing.2943

It is going to be 1 + 1 + 4 = 6 that is level 1.2950

We are going to try different values in here.2959

We are going to try 121, 211, 212, and you end up with different numbers.2960

You can try it at any random order but you are going to arrange the energy levels increasingly.2966

Here is what you come up with when you do this.2971

The energy of the 211 state, that is going to be 4 + 1 + 4 that is going to equal to 9.2974

For this particular state, the energy is going to be 9 H²/ 8 MA².2989

This is the constant so I’m just working out the numbers in the parentheses.2998

Now the energy of the 121 state that equals, if I put the 121 into this expression, I end up with 1 + 4 + 4 = 9.3002

Sure enough, it ends up having the same energy.3018

Let us try 131, energy of 131.3022

It is going to be 1 + 9 + 4 is going to equal 14.3025

Let us try the energy of the 311 state.3035

That is going to be 9 + 1 + 4.3040

It is going to equal 14 so we see this.3044

When we do the energy of the 321 state, I will put the 321 into here and I get 3² is 9, 2² is 4 + 4, that is going to equal 17.3053

The energy of the 231 state is going to be 4 + 9 + 4 so I end up with 17.3073

Let us try the 112 state, energy of 112 = 1 + 1 + 16 = 18.3087

This is level 1, it has 1 degeneracy.3104

This is level 2, it is level 2 because it is the next highest energy based on whatever N can be.3109

It is the next highest but it is the lowest of the bunch.3116

I can put in any N I want, I’m going to get a bunch of numbers.3120

I'm going to end up with a bunch of energies.3124

It is going to be 6 × this thing, 9 × this thing, 9 × this thing, 14 × this thing.3127

What I'm looking for are the number of degeneracy for that particular energy level.3131

For the energy level of 9 H²/ 8 MA², 2 states have that same energy.3136

The 211 state, the 121 state.3143

This particular energy level is 2 fold degenerate.3146

It has a degeneracy of order 2.3150

That is what I'm doing.3151

The question asked, find the degeneracy of the first five energy levels for a particle in a 3 dimensional box.3153

Here is level 1, here is level 2, here is level 3, I see it has 2 degeneracy.3163

Here is level 4, it has 2 degeneracies.3171

Here is level 5, it has no degeneracy.3174

It is just one thing.3178

That is what we have done.3180

The degeneracy of order 1, its next level up has degeneracy of order 2, the third level is 2 fold degenerate,3182

the fourth level is 2 fold degenerate, and the fifth level is not degenerate at all.3190

It has 1 energy level.3197

It might be, I continue on to see, that is the whole point.3200

I have to try each different state.3203

I hope that actually makes sense.3207

We calculated the formula for the energy and we just tried different values of N by varying the NX, NY, NZ separately3210

to end up with some increasing energies of the particles of different states.3219

These are 2 different states, 2 entirely different wave functions that they have the same energy 9 H²/ 8 MA².3225

For this particular setup, where A = B = 2C, the 131 and the 311 state also happen to have the same energy.3233

The 321 and 231 state happen to have the same energy.3241

That is all we are doing here, that is all we have done.3245

I hope that makes sense.3248

Do not worry about it, this is just the first set of example problems.3251

We are going to be doing a lot of example problems here.3254

I’m going to be doing them in the next several lessons.3257

In the meantime, thank you so much for joining us here at www.educator.com.3260

We will see you next time, bye.3264

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