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Raffi Hovasapian

Raffi Hovasapian

The Particle in a Box Part III

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
00:50
Convention for Heat
02:40
Convention for Work
04:24
Schematic Representation
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
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
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
02:16
N is the Number of Particles in the System
03:27
The Molecular Partition Function
05:06
Quantum States & Degeneracy
07:46
Thermo Property in Terms of ln Q
10:09
Example: Thermo Property in Terms of ln Q
13:23
Statistical Thermodynamics: The Various Partition Functions I

47m 23s

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

54m 9s

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

48m 32s

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

57m 30s

Intro
0:00
Example I: Make a Plot of the Fraction of CO Molecules in Various Rotational Levels
0:10
Example II: Calculate the Ratio of the Translational Partition Function for Cl₂ and Br₂ at Equal Volume & Temperature
8:05
Example III: Vibrational Degree of Freedom & Vibrational Molar Heat Capacity
11:59
Example IV: Calculate the Characteristic Vibrational & Rotational temperatures for Each DOF
45:03
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Lecture Comments (9)

2 answers

Last reply by: shashikanth sothuku
Thu Mar 26, 2015 12:07 AM

Post by shashikanth sothuku on March 25, 2015

hello prof,
why do the laplacian op. has an exponent 2 instead of a caret?

1 answer

Last reply by: Professor Hovasapian
Fri Jan 2, 2015 9:02 AM

Post by David Llewellyn on December 31, 2014

In the one dimensional case you had the wave function = 0 at x<=0 and x>=a but in the two and three dimensional cases you only mentioned boundary conditions when x and y and z <=0 but not when they were >=a, >=b and >=c respectively. Is there a reason for this?

0 answers

Post by Professor Hovasapian on September 27, 2014

Hello Everyone.

I hope the course is going well for all of you.

I wanted to mention a slight error in this lesson: for the Particle in a 2-dimensional box, I wrote h^2/2m for the coefficient of the energy. It should be h^2/8m. My apologies.

Raffi Hovasapian

2 answers

Last reply by: xlr z
Mon Sep 29, 2014 5:17 PM

Post by xlr z on September 27, 2014

I think at 22:13 equation for energy it suppose to be (-h bar/2m)

The Particle in a Box Part III

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
  • 2-Dimension 0:12
    • Dimension 2
    • Boundary Conditions
    • Partial Derivatives
  • Example I 6:08
  • The Particle in a Box, cont. 11:28
    • Operator Notation
    • Symbol for the Laplacian
    • The Equation Becomes…
    • Boundary Conditions
    • Separation of Variables
    • Solution to the 1-dimensional Case
    • Normalization Constant
  • 3-Dimension 28:30
    • Particle in a 3-dimensional Box
    • In Del Notation
    • The Solutions
    • Expressing the State of the System for a Particle in a 3D Box
    • Energy Level & Degeneracy

Transcription: The Particle in a Box Part III

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

Today, we are going to finish off our discussion of the particle in a box.0004

In the next lesson, we are going to start on the example problems.0008

Let us just jump right on in.0011

In the last lesson, we discussed the particle in a box, the Schrӧdinger equation, and we did for a 1 dimensional box.0015

Finding a particle on some interval.0021

We are going to do the 2 dimensional box which is just a plane region and the 3 dimensional box0023

which is really what you think about when you think about a box.0029

Let us start with dimension 2.0032

In dimension 2, the Schrӧdinger equation looks like this.0039

It is going to end up being –H ̅²/ 2 M.0044

I think it is always a good idea to keep writing these over and over and over again,0050

until you just feel comfortable writing them until they are just another part of your day.0054

D² ψ of DX² + D² ψ/ TY² = E × ψ.0059

Where now X is going to be ≥ 0, ≤ A.0074

Y is going to be ≥ 0 and ≤ B.0078

The ψ is a function of two variables X and Y, not just some F of X.0089

It is going to be some ψ which is a function of X and Y.0097

When we do 3 dimensions, it is going to be XYZ.0107

We are looking for some function that has all 2 or 3 variables in it.0109

The boundary conditions are going to be as follows.0114

Do not worry, I’m going to draw this often just a second.0116

Our boundary conditions where the wave function goes to 0.0118

We are actually containing it in this little square region.0123

The wave does not exist outside of that.0126

Those conditions mathematically are represented as follows.0131

0 Y = 0 and ψ of X 0 = 0.0134

This is our mathematical problem that we have to solve.0143

This is their differential equation that we have to solve.0148

Here are the constraints on the variables.0150

Here are the constraints on the actual function itself on the boundaries namely on the outer edge.0152

Here is what we are looking at now.0161

I will do it over here.0164

Let me do this in blue.0166

We have our coordinate system.0172

This is Y and this is X, I’m going to go ahead and put A over here.0174

I will go ahead and put B over here.0179

In general, A and B are going to be different but they can be the same, square box.0182

We will go ahead and do this.0188

In this region, we are going to contain the particle in that region.0192

The particle is constrained to be in this 2 dimensional boxes.0204

2 dimensional region, that is where the particle is going to be.0220

These boundary conditions, all they are saying is that whenever the wave function for any one X0223

is always 0, for any value of Y wave function is going to be 0.0229

That is the same things here for all values of X is 0.0237

All it is saying is it is going to be 0 on that boundaries.0243

Let us go ahead and talk a little bit about this notation here, this partial derivative notation.0248

Let me go ahead, I will stick to blue.0254

If you are not familiar with this notation, this D ψ DX or D² ψ DX².0257

If you are not familiar with that, these are called partial derivatives.0267

Some of you may have seen them, for those of you have not, do not worry about it.0273

I’m just going to take a couple of minutes to explain it, do an example.0277

It is very simple, you actually know what to do.0279

This is just a notational difference.0282

They are called partial derivatives and what you are doing when you take a partial derivative of a function of 2 variables, 3 variables, or 4 variables,0284

you differentiate with respect to the bottom variable.0292

It is the same thing, it is like D ψ D.0308

It is the same thing, you are taking a derivative but now we use a different notation because we have another variable.0310

You differentiate with respect to the bottom variable treating the other variable as a constant.0316

You are doing the same exact thing you have always done for the past several years with derivatives.0323

Treating the other variable as constant.0327

That is all this notation means.0342

When we have ψ, we go ahead and we take the first derivative with respect to X.0344

We take these derivative of that what we got again with respect to X.0349

And then we take the derivative with respect to Y, the original function.0353

We take the second derivative with respect to Y and then we add them together.0356

That is what this means.0360

Let us go ahead and do an example of this partial derivative just to see how it works and then we will move on with the Schrӧdinger equation.0361

Our example is, let F of XY a function of 2 variables B X⁵ Y³.0371

This is a very simple function.0375

We want to find DF DX DF DY as well as D² of DX², D² of DY².0377

And then find D² F DY DX, DX² DF DY.0385

This one is just a derivative of this function with respect to X.0389

This is the derivative with respect to Y holding the X constant.0393

This one is taking the first derivative with respect to X and taking the second derivative0397

again with respect to X and this is the same thing with respect to Y.0402

This one here, it says take the derivative with respect to X and what you get then take the derivative with respect to Y, this time holding X constant.0406

And this is the reverse of that.0414

These are called mixed partial derivatives.0415

Let us go ahead and do this.0418

Let us do DF DX first.0421

We have DF DX, we are going to differentiate this with respect to X.0425

We are going to treat Y as a constant.0433

Y³ will stay as Y³.0435

This becomes 5X⁴ Y³.0437

That is it, you are just holding the other variable constant while you differentiate with respect to one variable.0444

Again, we can only do things one at a time.0449

This is the symbol for it.0451

Let us go ahead and do DF DY.0454

We will go ahead and differentiate with respect to Y except we are going to hold X constant.0456

The X⁵ stays the same.0464

This is going to be 3X⁵ Y².0466

Everything still applies, that you have learned from calculus.0472

The chain rule, everything else is just holding,0475

you just have to be a little extra careful holding something constant and remembering to actually carry it forward.0479

That is it, like anything else in calculus.0484

Just takes a little bit of extra vigilance and making sure because there is going to be a lot of things on the page.0486

Let us go ahead and do the second derivatives.0493

D² F DX², we have taken DF DX.0495

Now we are going to take the derivative of that with respect to X holding Y constant.0499

4 × 5 is 20 so we get 20 X³ Y³.0505

Now we get D² F DY.0512

Now, we are going to take the second derivative of the function with respect to Y so we have a DF DY.0514

We differentiate that so we hold the X constants.0521

2 × 3 =6, DX⁵ stays and the Y drops down to 1.0524

I will go ahead and put the 1 there, that is not a problem.0531

Let us go ahead and do the mix partials.0534

D² F DY DX, the order on the bottom does matter.0538

Here we are saying, we differentiate with respect to X first then differentiate with respect to Y.0543

What this means is do DDY of DF DX that is what this means.0548

This is why you get the D² F DY DX.0557

I’m going to take the derivative with respect to Y of my DF DX.0562

My DF DX is up here and this is 5 X⁴ Y³ but now I’m going to differentiate this function with respect to Y holding the X constant.0566

It is going to be 15 X⁴ Y².0574

Over here, we will go ahead and do D² F.0582

This time we will do DX DY which means we will take the derivative of DF DY.0586

We are going to do why first which we already did, now we are going to take the derivative of that with respect to X.0594

3 × 5 is 15 X⁴ Y².0600

Now notice something, DF DX DF DY is not the same.0608

D² F DX² DX² of DY², it is exhausting saying all this, they are not the same.0613

But when you take the derivative with respect to one variable then take the derivative with respect to the other.0619

You get this and you get this.0626

Notice that they are the same, this is not a coincidence.0629

Mixed partials are going to be the same, provided the function is well behaved.0631

You are always going to be dealing with well behaved functions.0636

You have already run across a function that is not well behaved.0639

In other words, continuous and continuous partial derivatives and things like that.0642

This will always be the case.0645

This is actually a very deep theorem in mathematics.0647

This is not a coincidence.0651

The equality of mixed partials is not a coincidence.0659

For those of you that actually went through the thermodynamics portion of this particular course, I have discussed this in great detail.0671

We actually used the property of the mixed partials being equal to simplify our equations.0677

Equality of mixed partials is not a coincidence.0683

Let us go ahead and get back to our particle in a box.0689

Let me go back to black here.0694

Once again, we have – H ̅²/ 2M × A² ψ DX² + D² ψ DY².0696

I can derive these things over and over again simply to help me become comfortable with the equation.0712

It is equal E × there we go.0720

In operator notation, an Eigen value notation.0724

Eigen function or Eigen value notation looks like this.0735

-H ̅²/ 2M, I'm going to pull out the ψ aside.0739

D² DX² + D²/ DY², the ψ out there = E × ψ.0745

Remember, this is an operator and an operator distributes.0759

It means do this, do this, do this to this function.0762

And I do this to this function, it distributes like a normal algebraic binomial or trinomial or whatever it is.0766

This thing right here is very important.0773

This is called the Laplace operator or just the Laplacian.0777

It is profoundly important in science and mathematics.0811

I promise it will be the last that you see it.0818

It is profoundly important in science.0821

We have a symbol for the Laplacian.0832

Let me go back to black here.0834

The symbol is called the Laplacian.0838

It is going to be this upside down Δ².0853

That is equal to this thing, this D²/ DX² + D²/ DY².0858

Later, we have 3 variables for the particle in a 3 dimensional box, it is going to be D² DZ².0865

The equation becomes –H ̅²/ 2N L² ψ = energy ψ.0873

That is our equation.0892

Once again, let us go ahead and do the boundary conditions.0894

Our boundary conditions are ψ of 0 Y = 0 and ψ of X 0 = 0.0906

Of course, we have X ≥ 0, ≤ A and Y ≥ 0, ≤ B.0918

A and B are the limits here.0926

When we solve this equation, we use the method called the separation of variables.0930

I’m not going to go through it here, you can go ahead and see if you like, either in your book or in one of the appendices that I have here for this course.0938

Separation of variables, this particular technique it requires us to assume that our function that we are looking for,0946

this ψ of XY is actually equal to some of function of X × some function of Y and that is a product.0965

It is not some mixed function.0977

It is actually a separate function of X alone and a separate function of Y alone that are multiplied together.0979

This particular technique requires that this be the case.0985

I want to throw that in there.0988

In other words, what we do is we find F of X and then we find G of Y.0991

And we multiply them together that is it.1007

Then multiply them together to form our ψ, our wave function which is now a function of X and Y.1011

What is interesting is the following.1030

The F of X and G of Y, they turn out to be solutions to the one dimensional case.1032

The 1 dimensional particle in a box along the X axis or the 1 dimensional particle in a box along Y axis.1041

We just happen to put them together to create the solution to the 2 dimensional particle in a box.1046

It is extraordinary that we can do that.1053

And I will talk about that a little bit more towards the end of this lesson.1056

We find F of X, we find G of Y with this technique of separation of variables.1059

We multiply them together to form our ψ of XY.1063

F of X and G of Y turn out to be solutions to the 1 dimensional case.1068

In other words, F of X is, from the last lesson we get some D sub X some constant × the sin of N sub X π A.1091

Remember, these were the solution for the 1 dimensional particle in a box.1105

I'm putting these little subscripts for the N, I can just call it N because there is going to be N for the X and there is going to be N for the Y.1108

We are going to have two quantum numbers.1115

It ends up being, if you remember 2/ A¹/2 × the sin of N sub X π/ A × X.1119

F of Y ends up being some constant B sub Y sin of N sub Y × π/ BY.1131

The constant is the same and the normalization constant, except it is going to be 2/ B.1143

We are doing from 0 to B ^½, the sin of N sub Y π/ B × Y.1149

Personally, I think the hardest part in quantum mechanics is not the quantum mechanics, it is not the concepts.1162

It is not the math, it is writing the math down.1166

There are so many things to write.1170

Many indices, this star, that star, this subscript, that superscript.1172

Keeping it all straight can be really difficult.1177

We have the following.1182

Our ψ which is now we have the subscript N sub X N sub Y is going to equal,1184

You are multiplying F of X and F of Y.1196

It is going to be B sub X B sub Y.1198

Let us here.1202

The sin of N sub X π/ A × X × the sin of N sub Y π/ B × Y.1209

This B sub X and B sub Y, those are right here.1227

That is the B sub X and that is the B sub Y.1231

If you want you just put those in.1233

It is just going to be 2/ A × B.1235

The 2¹/2 and 2 ^½ they multiply it to form 2.1238

The A ^½ and B ^½ ends up being AB ^½.1241

I will write it over here.1248

This BX BY, the coefficient, the normalization constant is just going to be 2/ AB ^½.1251

I just left it in generic form right there.1261

Now the energy, this is a solution to the particle in a box, this whole equation.1264

And notice, it is a function of X and Y.1271

It just happens to be two separate functions that we have multiplied together.1275

The energy N sub X N sub Y = A²/ 2M × N sub X²/ A² + N sub Y²/ B².1280

These are your solutions right here.1301

This is the solution to the Schrӧdinger equivalent for the problem for a particle in a 2 dimensional box.1304

This is the wave equation, this being the normalization constant, and this is the energy for a different values of N.1310

Let me go ahead and actually say N sub X takes on the values 1, 2, 3, and so on.1317

N sub Y takes on the values 1, 2, 3, and so on.1325

This N sub X and N sub Y, the N can vary independently.1331

It is not just 11, 22, 33.1334

It could be 111, 12, 13,14, 7, 8, 14, 16.1337

They vary independently.1341

There are all kinds of different energy levels.1344

In a particle in a box, it actually have.1346

The normalization constant, we found it just by multiplying the normalization constant for F of X and G of Y, that is fine.1355

The normalization constant is found normally the same way as we did before.1367

B sub X B sub Y is found the same way as before.1373

Which was you have to take the integral over the entire region of C* × C.1384

I will go ahead and put DA here and set it equal to 1.1402

Again, that is all we did.1404

In order to satisfy the normalization condition, this is the probability.1407

When add up all the probabilities, the probability that the particles can be somewhere in the box has to equal 1.1413

That is our normalization condition, it always = 1.1420

We use this to solve for the BX BY.1422

This is the equation that we actually have to solve in order to find our normalization constant.1427

In order to normalize the wave function.1433

You just take the function, multiply it by its conjugate, and you integrate.1436

In this case, the conjugate happens to be the function itself because the functions are real functions, they are not complex.1441

This is the process that you always go through.1446

Here it is DA.1450

I wrote DA because we are integrating over an area.1455

We are going to integrate over X, then we are going to integrate over Y.1459

We are integrating over an area.1462

This is actually a double integral.1464

You have to watch what dimension you are working in.1465

Because we are integrating over an area, in other words we got this region here.1471

We integrated in the X direction, and then we integrate in the Y direction, the double integral.1492

Let me go back to black here.1498

This integral is symbolic.1501

This one right here, it is symbolic.1502

It actually means this.1505

In this particular case, a 2 dimensional case ψ* ψ DA= the integral from 0 to A of the integral from 0 to B of,1507

when we take the function and multiply itself,1519

we are going to end up with B sub X² B sub Y² ψ² N sub X π/ A × X and D sin² of N sub Y π/ B × Y DY DX.1522

DY DX is the DA, it is a differential of area element.1545

DX DY when you multiply them together you will get DA, the differential of area element.1554

That is what this means.1560

We have to solve this particular integral and do not worry about it, we will be doing these in the example problem.1561

We will see.1567

You can do it this way to solve the integral or you can just go ahead and take the coefficients for the 1 dimensional case and just multiply them together,1569

which is exactly what we did.1579

We had B sub X = 2/ A ^½ and we have the B sub Y = 2/ B ^½1581

which means that the normalization constant BX BY is such as this × that.1595

It is equal to 2/ AB ^½.1600

That is it, nice and simple.1605

Let us go ahead and say a little bit more here.1611

This ψ N sub X N sub Y which is a function of X and Y is the product of ψ, for the 1 dimensional case sub X and ψ NY.1620

For the 1 dimensional case in the Y variable.1651

Let me write it out.1667

Our wave function for 2 dimensions is the product of the wave functions for 1 dimensional case.1672

The energy for the 2 dimensions is the sum, here we have the product,1679

the energy is the sum of the energies for the individual cases, for the 1 dimensional cases.1684

N sub X N sub Y.1693

This is actually very important result and I will be talking about it again a little bit more formally towards the end.1698

Let us go ahead and move on to the 3 dimensional case.1707

I will go ahead and go back to black here.1712

We have dimension 3.1714

A particle in a 3 dimensional box, now our equation is going to be –H ̅²/ 2 M × the Laplacian operator for 3 variables.1723

DX² D²/ DY² + the second derivative with respect to ψ of our wave function that we are looking for,1746

It is going to equal that × ψ.1756

This time we have X ≥ 0, ≤ A, Y ≥ 0 and ≤ B, Z ≥ 0 and ≤ C.1760

We are constraining at a box which is A × B × C.1769

A long, B wide, C high.1777

However, you want to look at it.1778

The ψ is a function of 3 variables.1781

Ψ is now a function of 3 variables X, Y, and Z.1788

Ψ is ψ of X, ψ of Y, ψ of Z.1800

It is going to be exactly what you think it is.1804

We are just going to end up forming the product of the 3 functions.1807

It looks like this1811

Let us go ahead and draw this one out.1813

The coordinates is in here.1816

This time what we have is the following.1821

This is going to be our X coordinate, and this is going to be our Y coordinate, this is going to be our Z coordinate.1825

This is the standard 2 dimensional representation of a 3 dimensional object.1829

The right hand coordinate system.1835

The X is this way, the Y is this way, and the Z is this way.1837

A is over here and B is over here and let us go ahead and put C like right there.1841

Here is what we have.1852

Here is our box, let me go ahead and do the box in blue.1854

I’m going to go across and I will go down.1879

Our particle in a 3 dimensional box.1887

The particle is somewhere there.1890

That is all we are doing.1892

We have our equation which is this one.1895

We have the constraints on the variables.1898

Our boundary conditions are going to be as follows.1901

Our boundary conditions are ψ of 0 YZ = 0 ψ of X 0 Z = 0 and the ψ of X Y 0 = 0.1908

All these boundary condition say is that once you actually hit the walls of this box, the wave function goes to 0.1927

The wave function is 0 at that point and beyond.1936

That is all we are doing.1941

In Del notation, it is nice to see it.1944

-H ̅²/ 2N Del² ψ = energy × ψ.1952

Again the Del² is just this Laplacian operator, that is all it is.1965

Our ψ again, we do it with the method of separation of variables.1983

Our wave function of XYZ is going to equal the product of F of X × G of Y × H of Z.1987

The F of X = 2/ A ^½ that is just the wave function for the particle in a 1 dimensional box in the X direction.2000

A × the sin of N sub X π/ A × X.2015

If we have G of Y = 2/ B ^½ × the sin of N sub Y × π not A.2023

Be really careful, I always do that.2039

It is like I get stuck on one value A and I write A for everything.2045

This is N sub Y × π/ B.2048

The variable is Y and of course our H of Z is going to = 2/ Z ^½ × the sin of N sub Z × π/ A × Z.2053

Again, N sub X and N sub Y and N sub Z, they all range from 1, 2, 3, and so forth.2069

They are very independently.2084

You have 3 quantum numbers for a particle in a 3 dimensional box.2086

Now, our wave function N sub X N sub Y N sub Z, it ends up very independently,2106

= 8/ ABC ^½.2114

N sub X π/ A × X × sin of N sub Y × π/ B × Y × sin of N sub Z × π/ ψ × Z.2129

This is crazy, these functions get really big and yes we are going to be integrating with them.2148

We are going to be differentiating them.2154

It does tend to get notational intensive and again personally I think it is the most difficult part of quantum mechanics.2156

It is just keeping all of the notations straight.2163

The energy of a particle in a box, first state N sub X N sub Y N sub Z.2168

This is going to equal planks constant²/ 8M.2177

N sub X²/ A² + N sub Y²/ B² + N sub Z²/ C².2184

The energies are just the some of the energies in the individual directions.2203

A particle in a 1 dimensional box in the X, 1 dimensional box in the Z, 1 dimensional box in the Z.2208

Put them together, you form the product of the functions, you get the overall wave equation for a particle in a 3 dimensional box.2213

You add up the energies to get the energy of the particle in a 3 dimensional box.2219

Once again, the N sub I they vary independently.2225

Do not get stuck on the idea that it all has to be 111, 222, 333.2232

It does not.2236

Let us talk about something interesting that happens.2240

Something happens when the sides of the box are equal.2244

A can be anything, B can be anything, C can be anything.2248

When they are equal, here is what happens.2250

The energy N sub X, let us specify what we mean by this sides being equal.2254

We mean when A = B = C, when you have a cubed.2262

Here is what happens to the energy.2270

The energy of N sub X, N sub Y, N sub Z, that is equal to H²/ 8M.2271

Now, it is going to be N sub X²/ A² + N sub Y²/ A²,2279

Because B = A so I can just put in A².2287

They are all the same length, + the N sub Z²/ A².2292

The A² is a constant so you can just pull it out.2299

The energy = H²/ 8 M A² × N sub X² + N sub Y² + N sub Z².2302

This is just a constant.2316

What is important is this thing.2318

The N depending on what N is, the energy of that level is going to be different.2321

If N is 1, if N sub X is 1, N sub Y is 1, N sub Z is 1.2326

You are going to end up with 1 + 1 + 1 it is going to be 3 A²/ 8 MA² that is the energy of the 111 level.2331

Now, for a particle in a 3 dimensional box.2341

The state of the system is expressed by the N sub X N sub Y N sub Z.2345

Let me say that again.2351

For a particle in a 3 dimensional box, the state of the system is expressed by these quantum numbers.2358

It is expressed by N sub X, N sub Y, and N sub Z.2376

Let us go ahead and actually work out some energy for some different states.2382

When we have N sub X = 1, N sub Y = 1, and N sub Z = 1, this is going to be the energy of the 111 state,2388

That is going to equal H² / 8 MA² 1² + 1² + 1².2399

I’m just putting it into the equation.2409

I end up with 3 H²/ 8 MA², that is the energy of the 111 state of a particle in a 3 dimensional box.2412

If I said I have a particle in this 3 dimensional box, the sides are length A.2425

What is the energy of the 111 state?2433

That particular wave function, what is the energy of that particle?2441

There it is, planks constant², multiply by 3, divide by 8, divide by the mass of whatever the particle is,2444

electron, proton, whatever and then divide by the square of the sum of the side length.2450

Let us do some more.2459

This time let us do N sub X = 2 and N sub Y = 1 and N sub Z = 1.2461

This is the energy of the 211 state.2470

That is going to equal A²/ 8 M, A² this time it is going to be 2² + 1² + 1².2473

N sub X, N sub Y, N sub Z.2486

You will end up with 6 H²/ 8 MA².2489

Let us do another level.2501

For N sub X = 1, this time N sub Y = 2, N sub Z = 1.2503

This is not the same state 211, this is 121, this is an entirely different wave function.2511

An entirely different wave, an entirely different set of probabilities, an entirely different of state of the system.2517

We know the energy, the energy of the 121 state = H²/ 8 MA².2524

This time it is going to be 1² + 2² + 1².2533

Again, we end up with 6 H²/ 8 MA².2538

It ends up being the same energy as the 211 state.2543

You can imagine what we are going to do next.2549

This time we are going to do X = 1 and Y = 1 and this time we are going to take the N sub Z = 2.2551

This is going to be the 112 state.2557

Again, a completely different state 112.2562

Sorry to elaborate the points here, I think it is actually good to go through it like this.2570

+1² + 2²,2575

Once again we end up with 6 H²/ 8 MA².2578

For 3 different states, the 211, 121, and 112, the energies are the same.2585

For 3 different states, the energies are the same, this is what we call degeneracy.2599

Let me go ahead and do this in red.2615

We say this particular energy level, this energy level in this particular case, the 6 H², this particular energy level.2622

Let us say the 6 H²/ 8 MA², whatever the energy happens to be.2637

That level is 3 fold degenerate or has degeneracy of order 3.2642

In other words, there are 3 states that have that energy.2651

That should be the case.2657

A particle in one state is given by a particular wave function should have a different energy.2660

All of a sudden, simply by making the side of the box equal, whether it is a perfect square or a perfect cubed,2665

all of a sudden you have these particles that are completely different states, completely different wave functions,2673

if they have the same energy that is what we call degeneracy.2680

We say that this energy level is 3 fold degenerate.2683

That is the level has 3 states that have that particular energy.2693

Let me go ahead and go back to blue here.2726

I will just write in general, degeneracy is the number of different states having the same energy.2729

That is it that all degeneracy is.2747

It is a very important concept.2753

Degeneracy emerges when the system becomes symmetric.2759

Remove the symmetry and you actually end up removing the degeneracy.2762

Once again we have some rectangular box A, B, C are different.2767

You are going to get different values.2772

With them, you actually end up making the size of the box equal, you actually introduced degeneracy into the system.2776

You actually solve by introducing symmetry into the system, the cube is perfectly symmetric.2783

The square is perfectly symmetric.2788

You actually introduced degeneracy.2790

That was actually a fundamental principle of quantum mechanics.2792

It says that degeneracy is up here, they emerge, they show up when you introduce symmetry into the system.2795

If you remove the symmetry, you end up removing the degeneracy.2803

Degeneracy is not necessarily like an open door and a closed door.2807

It is not like yes it symmetrical, not it is not symmetrical.2811

There are actually degrees of symmetry and we are going to see some of that when we actually did example problems.2814

But again, degeneracy emerges as a result of introducing symmetry.2820

Take away the symmetry, you reduce the degeneracy.2826

You take away the degeneracy.2829

This is actually a very profound quantum mechanical principle.2830

Not just quantum mechanical.2833

The idea of symmetry is what permeates nature, permeates all of mathematics and physics.2835

I’m going to go ahead off with what I said earlier, when I said I would discuss more formally the idea of the particle in a 3 dimensional box2842

being a product of the functions of the 1 dimensional box and the energies being the sum.2850

Let me go back to black here.2857

We said that the ψ of N sub X N sub Y N sub Z = ψ of N sub X × ψ of N sub Y × ψ of N sub Z,2860

and that the energy of N sub X N sub Y N sub Z state is just the energy of the N sub X state +2880

the energy of N sub Y state + the energy of the N sub Z state.2888

I think I’m going to leave this for another time, I apologize.2906

I think I can go ahead and close out this lesson right here and I will discuss this formality a little bit later.2910

Perhaps, I will actually do some problems.2915

That should not be a problem at all.2917

Thank you so much for joining us here at www.educator.com.2919

We will see you next time, bye.2921

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