  Raffi Hovasapian

Probability & Statistics

Slide Duration:

Section 1: Classical Thermodynamics Preliminaries
The Ideal Gas Law

46m 5s

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

46m 2s

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

1h 6m 45s

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

1h 6m 33s

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

1h 2m 17s

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

1h 4m 39s

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

16m 50s

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

43m 40s

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

32m 23s

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

39m 15s

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

35m 52s

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

42m 40s

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

1h 23s

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

44m 34s

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

59m 7s

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

49m 16s

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

33m 59s

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

54m 37s

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

31m 18s

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

23m 6s

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

25m 42s

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

43m 39s

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

56m 44s

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

57m 6s

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

54m 35s

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

35m 5s

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

28m 42s

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

34m 38s

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

30m 50s

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

34m 6s

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

39m 18s

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

19m 40s

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

54m 16s

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

31m 17s

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

45m

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

58m 5s

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

25m 20s

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

34m 25s

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

59m 57s

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

42m 5s

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

30m 26s

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

21m 34s

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

56m 22s

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

45m 24s

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

48m 43s

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

46m 18s

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

39m 28s

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

35m 32s

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

29m 55s

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

54m 25s

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

46m 58s

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

44m 11s

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

35m 33s

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

43m 4s

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

26m 30s

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

41m 10s

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

30m 32s

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

35m 19s

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

33m 48s

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

46m 43s

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

40m

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

35m 58s

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

36m 18s

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

33m 55s

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

51m 53s

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

51m 53s

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

34m 29s

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

43m 49s

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

1h 1m 57s

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

48m 33s

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

48m 33s

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

48m 33s

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

59m 18s

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

34m 54s

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

38m 3s

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

57m 49s

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

1h 1m 20s

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

56m 34s

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

18m 24s

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

50m 2s

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

59m 47s

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

46m 22s

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

29m 24s

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

30m 53s

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

1h 1m 33s

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

33m 47s

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

1h 1m 5s

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

33m 31s

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

1h 1m 15s

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

47m 23s

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

54m 9s

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

48m 32s

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

57m 30s

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

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
• Probability & Statistics 1:51
• Normalization Condition
• Define the Mean or Average of x
• 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
• 1st Term
• 2nd Term
• 3rd Term
• Continuous Distributions 35:47
• Continuous Distributions
• Probability Density 39:30
• Probability Density
• Normalization Condition
• Example III 50:13
• Part A - Show that P(x) is Normalized
• Part B - Calculate the Average Position of the Particle Along the Interval
• Important Things to Remember 58:24

### Transcription: Probability & Statistics

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

Today, I thought we would do a little discussion on probability and statistics.0004

Probability and statistics, probability plays a huge role in quantum mechanics because quantum mechanical systems are probabilistic.0011

They are not deterministic like classical mechanics, like the physics that you learn in freshman and sophomore year.0021

Deterministic meaning we can come up with an equation that is going to tell us exactly where something is going to be.0027

How fast it is moving, what its momentum is, what its angular momentum is, things like that.0034

We know what it is going to be, it is determine, it is predetermined.0040

Probabilistic quantum mechanics, elementary particles, high speeds,0044

we can say with certainty that something is here or there or it is moving this faster, that fast.0049

We speak probabilistically there is a 60% chance that it is here.0058

There is a 70% chance that it is moving this fast in this direction.0064

A lot of what we are going to be presenting here, I would not worry too much about it.0073

If you cannot actually wrap your mind around some of this material, a lot of the probability and0079

statistical aspects of quantum mechanics, the understanding will emerge over time.0085

We will do a lot of computational problems.0090

We will just get a feel for it.0093

Again, quantum mechanics is very unusual.0095

It takes some time to get accustomed to it.0099

Do not feel bad if some of the stuff does not quite fit well with you.0102

But I do want to introduce it and see what we can do.0106

If we were to conduct an experiment that has impossible outcomes like flipping a coin, rolling the dice, rolling a pair of dice, whatever it is.0114

Let me work in blue here.0127

If we conduct an experiment with impossible outcomes, in the case of flipping a coin you have two outcomes, either heads or tails.0131

We actually repeat the experiment over and over and over again.0156

Just keep flipping, keep flipping and keeps flipping.0159

Repeat the experiment over and over then the probability of each outcome is the probability of event I,0167

the limit as N goes to infinity of N sub I/ N.0206

Where I = 1, 2, 3 all the way to N.0216

In the case of flipping a coin, you have got N is equal to 2.0225

You have 2 possible outcomes.0229

In the case of rolling a dice, you have 6 possible outcome so N =6.0231

Over N sub I is the number of times that outcome I occurs.0238

In other words, if you are to do 10 flips of a coin and 6 × it comes up heads and 4 × it comes up tails,0260

the N sub I for heads is going to be 6 and the N sub I for tails is going to be 4.0266

N is the total tries, you flip it 10 ×.0273

In this case, N is equal to 10 for the 10 flips.0277

N is the total tries.0282

In flipping a coin, you know the probability of getting a head is going to be 50% or 0.5.0289

Getting a tails is going to be 0.5.0296

If you are to roll a dice, you have 6 numbers, 6 faces on the cubic dice.0298

You have 6 numbers 123456.0303

The probability of getting a 1 is going to 1/6.0305

The probability of getting a 2 is going to be 1/ 6.0308

All this says that if I keep flipping it over and over again, in other words if I do the 10 and I get 6 × heads, 4 × tails.0317

That is going to be 6/10 and 4/ 10.0327

Will that is not quite 0.5, it is 0.6 and 0.4.0329

If I do it 20 ×, if I do it 30 ×, if I do it 50 ×, if I do a 100 ×, a 1000 ×, as N goes to infinity it is going to end up pretty evenly.0336

Half of the time it is going to be heads, half of the time it is going to be tails.0346

That is why this limit is there.0349

It is not that important.0354

N is the total number of tries.0356

This is just for our formality so that we actually see it but that is all it says.0359

It is that the probability of an event is the number of times that a particular outcome occurs ÷0364

the total number of times and then you just keep trying.0371

You keep trying and keep trying, eventually you are going to get the probability of the event.0374

Clearly, N is the number of times that a certain outcome occurs and N is the total number of times.0386

We know that this N sub I is definitely less than or equal to N and is greater than or equal to 0.0391

In the case of flipping a coin, 6 × heads, for this particular outcome 6 is going to be less than0400

or equal to 10 and it is greater than or equal to 0 because you are going to have that outcome.0406

If we divide by N we end up with the following.0411

0 less than or equal to the probability N sub I/ N is the probability, less than or equal to 1.0420

The probability is always expressed in terms of a percentage or a fraction.0427

I think it is best to express it as a fraction.0432

Of course, all those fractions for all of the different possibilities they add up to 1.0435

That is actually the real important thing here.0440

And we are going to formalize this in a minute.0442

If the probability, if P sub I = 1 that means that outcome is 100% certain.0449

That means it happens all the time.0464

There is no uncertainty there.0468

If the probability is 0, this means the outcome is impossible.0472

It is never going to happen.0476

Of course, most probability are going to be somewhere between 0 and 1.0482

We also know that the sum of the N sub I is actually equal to N.0488

In other words, if I have 2 outcomes and the 10 × that we actually flip a coin, N is the total number of times.0497

The N sub 1 which is heads, that is a fix.0508

The N sub 2 is going to be the tails so 6 and 4 they add up to 10.0511

That is all this is saying.0515

Again, we are just formalizing everything.0516

I = 1 to N = N.0519

The number of times, if you add up all the different outcomes, how many times each one happens,0522

you are going to get the total number of tries which is the experiment.0527

Once again, let us go ahead and divide by N.0530

If I divide by N, I get 1/ N × the sum as I goes from 1 to N of N sub I = 1.0533

I’m going to pull this back inside so I get the sum N sub I/ N, goes from 1 to N = 1.0551

This N sub I is just N sub I/ N, that is just the probability.0563

This just says that when I add up all of the probabilities, for all of the outcomes, I get 1.0575

This is very important, this is called the normalization condition.0584

All of the probabilities have to add up to 1.0600

In the case of a flipping a coin, 0.5 + 0.5 = 1.0602

In the case of the dice, 6 faced dice, 123456, 1/6 1/6 1/6 + 1/6 1/6 1/6 all add up to 1.0606

All the probabilities have to add up to 1.0617

Profoundly important.0619

This idea of the normalization condition is going to be very important in quantum mechanics later0621

when we introduce the wave function because we are going to normalize this wave function.0626

Because we are going to interpret the wave function as a representation of the probability that is something is here or there.0632

Or the probability of the energy being this or that, things like that.0640

Again, this is the important relation, the sum of the probabilities = 1.0652

If you do not take away anything from this lesson normalization condition.0656

Suppose that some number X is associated with probability P sub I.0667

Let us say if I had a probability of 0.2 and let us say that 0.2, there are some number 10 associated with that.0698

In other words, the number 10 is going to show up 20% of the time.0706

Let us say the number 20 is going to show up 30% of the time, that is 0.3.0710

And the number 30 is going to show up 50% of the time, that is 0.5.0717

That is what we are saying, that given a particular probability of a certain event,0722

that there is some number associated with that particular event, we are going to define the following.0726

We define the mean or average, we will use both words more often than not.0734

We will probably talk more about the average than we will the mean.0747

In quantum mechanics we can use the word average more than mean.0752

We define the mean or average of all the X is this way.0756

The average value of X is equal to the sum of the each X sub I × the probability that X is actually shows up as I goes from 1 to N.0772

What we are saying is, it will make more sense if we do an example in just a minute here.0785

X is the particular number and X sub I is a particular number and P sub I is the probability that, that number occurs.0795

In the case of rolling a dice, 1, 1/6, 2, 1/6, 3, 1/6, 4, 1/6, 5, 1/6, 6, 1/6.0806

In this particular case, all of the probabilities happen to be the same.0818

There is no guarantee that the probabilities will actually be the same.0822

You might have a number in some situations that might end up showing up 20% of the time,0825

and another number shows up 70% of the time, and a third number shows up 10% of the time.0830

The probabilities add to 1, you have 3 numbers represented.0836

The average of those numbers is not going to be right down in the middle.0839

You do not just add them and divide by 3 because one of the numbers is going to end up showing up more often,0844

the average is going to be weighted more towards that number.0851

And that is what this definition of average actually takes into account.0855

It takes into account the weight that a particular number actually occurs.0858

Again, angular brackets this is the symbol for the average quantity.0865

Later on, we are going to see something like this.0869

E is going to be the average energy of that particular particle.0872

Let us go ahead and do an example.0878

I think it will start to make sense.0879

Very important definition, this is the definition.0881

The definitions are profoundly important.0884

If you ever lose your way in something, go back to the definitions and start again.0887

That is why definitions are very important.0891

Let us see if we can do an example here.0896

Given the following discreet probability distribution, calculate the mean of X.0900

When we talk about what discreet probability distribution means.0905

A distribution is exactly what you think it is.0909

In a dartboard, if I start throwing darts at it for an hour, there is going to be distribution of holes on that dartboard.0916

That is what a distribution is.0925

It is the attempts and how they have distributed themselves over a particular interval or over an area, or over a volume.0926

Whatever it is that a particular situation calls for.0934

Discreet probability distribution, discreet means there are specific numbers 2, 5, 7, 12.0937

When we talk about a continuous distribution, we talk about the entire number line.0945

Every single number, every single fraction, every single decimal, is represented.0950

It is continuous.0954

That is what continuous means vs. discrete 0, 1, 5, 10, 15 nothing in between.0956

That is all discrete probability distribution means.0962

We want to calculate the mean of X.0964

In this case, we have 2, 5, 7, and 12, we have 4 numbers.0967

Here is the X value and here is the probability of that number showing up.0976

2 is going to show up 20% of the time.0980

This is the probability distribution.0983

5 shows up 50% of the time, 7 shows up 35% of the time, and 12 shows up 30% of the time.0987

Notice that the sum of the probabilities is equal to 1.0995

We want to find the average.0999

You have learned that the average is just add 2, 5, 7, and 12, and divide by 41003

It is not going to happen here.1007

Once I talk about the relation of this definition that we gave to the definition of average that you learn ever since you are a kid in school,1008

but notice the probabilities are not the same.1016

Each number has a different weight.1019

This 7 shows up more than this 5 does.1022

The average, you should represent the fact that average is going to be a little bit closer to 7.1025

The 7 has more weight.1030

Let us just go ahead and use the definition, use the math.1032

You do not necessarily have to understand everything that is going on conceptually.1036

It is going to be a lot of the case with quantum mechanics and in other things too.1040

But if you at least trust the math and let the definition of the formulas work for you,1044

then you will at least get accustomed to just you doing it mechanically.1050

There is no problem with that.1053

There is no sin in that, in just doing things mechanically, that is how we get a sense of what is going on.1054

We have this average value definition which is the sum as I goes from 1 to N of the X × the P sub I for that particular X.1060

Well, that is going to equal I = 1 to 4 because we have 4 values of the X sub I and the P sub I.1073

Let us just go ahead and do it.1084

It is going to be the X value × this probability.1086

2 × 0.2 + 5 × 0.5 + 7 × 0.35 + 12 × 0.30.1089

I will go ahead and write it out 0.4 + 0.7 + 2.45 + 3.6 the answer ends up being 7.2.1112

There we go, that is the average given the probability that these numbers show up.1128

Notice ,7 has the highest probability followed by 12 and 2.1131

Certainly, it is going to be that is what this takes into account.1137

That is what the definition takes into account.1142

This is how you find an average.1145

If you are wondering about why this definition of average looks different than the definition that you have been using ever since you are a kid.1156

In other words, take all the numbers, add them together, and divide by the number that there are.1164

In this case, 2 + 5 + 7 + 12 ÷ 4.1169

Here is what is going on.1173

The one that you learned in school is this one.1175

You take the sum of all of the X sub I, you add them all together, and you divide by N, the number that there are.1177

This is N up here.1186

This is the one that you have learned in school.1190

This definition is actually this definition that you learned in school and is actually the same as the definition I just gave you.1192

The only difference is this definition presumes that the probability of each number showing up is equal.1200

In other words, if I have 15 numbers, the probability of each one is just 1/15.1210

If I have 36 numbers, the probability is 1/ 36 for each number.1215

And I will show you.1220

If I have the numbers 1, 3, 5, 7, 9, 11, and 13, and if I said take the average of these numbers, just take the average.1223

Notice, I have not set anything about the probability of each one of these numbers.1243

The presumption is, the natural assumption that we have to make is that the probability is going to be the same for each.1247

The average is going to be this.1254

The average is going to be based on what you have learned from school.1255

It is going to be 1 + 3 + 5 + 7 + 9 + 11 + 13 all over 7 because there are 7 numbers.1265

Let us pull out the 7 as 1/7.1280

It is going to be 1/7 × 1 + 3 + 5 + 7 + 9 + 11 + 13.1284

I will distribute so I get 1 × 1/7 + 3 × 1/7 + 5 × 1/7 + so on and so forth until + 13 × 1/7.1296

That is equal to the sum of each X sub I, the 1, 3, 5, 7, 9, 11, and 13 × the probability.1311

The probability is 1/7.1320

Notice the probability is the same.1323

That is what they do not tell you in the definition of average that they teach you in school.1326

The presumption is that the probability is the same but it turns out to be the same thing.1332

It is just that the probability is 1/7 but it is the same definition.1336

In this case, Y = 1/7.1341

It is still just a particular number × this probability added together, that is all.1344

I hope that makes sense.1352

Let us go ahead and calculate the second moment of the data in example 1.1358

We are going to define something else called the second moment.1365

Here is our definition for that.1369

Define the second moment.1373

It is symbolized this way and that is exactly what you think it is.1381

Here, this time we are going to add from 1 to N.1385

The X sub I, the squares of the X sub I × their probabilities.1389

It is called the second moment.1393

Do not worry about what it means, just take it as it is a mathematical definition, plug the numbers in.1399

In this particular case, we get the second moment is equal to 2² × 0.2 + 5² × 0.15 + 7 squares × 0.35 + 12² × 0.30.1407

When we actually do the math here.1444

I have written 164.9.1448

We are not going to assign any meaning to it now.1454

It will be coming up later when we talk about quantum mechanics.1456

Important thing to note here.1463

Notice that the average value was 7.2 from example 1.1468

We are calculating the second moment of the data from example 1.1473

The second moment ended up being 64.9.1476

Notice, the average value², in other words 7.2² does not equal the average value of X².1488

The square of the average value of X does not equal the average value of the square.1504

The mean² is not equal the second moment.1509

Be very careful with that.1515

Be very vigilant about what the exponents on the outside or on the inside.1517

They are 2 different things.1521

Let us go ahead and define the third entity.1528

This is called the second central moment or the variance.1533

Define the second central moment or variance.1539

It is symbolized this way, as a σ².1555

The variance of a set of data.1560

The symbols right here.1582

As those moves us like that so that is equal to the sum = 1 to N of our individual X sub I - the average value of X² × the probability.1585

I find the mean of the set of values.1605

I take the difference of each individual value from that mean.1612

In other words, the difference from the distance of a particular number from the mean and I square it.1616

I multiply that number by the probability of the number and I add that up.1625

Do not worry about where these come from.1629

I know in statistics there is a real sense of some of these things just sort of dropping out of the sky.1631

To be completely honest with you, sometimes I would say that is exactly where they come from.1638

It just dropped out of the sky.1641

I wonder about some of these definitions myself.1643

Do not try to assign any meaning just now.1645

Just deal with the mathematics.1648

We also have the symbol.1658

If we take this value which is the variance and if we actually take the square root of it, we get that and this is called the standard deviation.1661

This is the one that you are actually more familiar with.1678

When we talk about standard deviation, the standard deviation of a set of data is going to be the square root of the variance.1682

This is the variance and you end up getting this number.1692

You add all these together and then you take the square root of it.1696

You do not add the square root of it.1698

You do this number then you take the square root and you get this.1701

These are just symbols, they do not have mathematical value.1704

We do the square and here without the two, simply to differentiate symbolically.1708

Show there is some relationship between them, that is all this is.1714

Now either of these variance or the standard deviation, does not matter which one you use.1718

Either of these is a numerical measure of the overall deviation of the points X sub I from the mean X.1725

It just gives you a measure of the example that we just did, we found an average value of 7.20 for the numbers 2, 5, 7, and 12 given the particular probability distribution.1762

The average was 7.2 and it gives you a measure of how far each individual point, the 2, 5, 7, and 12 are from the 7.2.1778

2 from 7.2, 5 from 7.2, 7 from 7.2, and 12 from 7.2, that is all it is.1787

It gives you some numerical measure.1794

It tells you how far apart is, how close it is to the mean.1796

We say that is a measure of the spread of the data.1800

How spread out are they or how close are they, actually to get to the mean value of all that data.1805

That is all that we are talking about here.1811

Let us go ahead and do some math with this.1815

Σ² of X is equal to the sum of the X sub I - the average value of X² × P sub I.1819

Let us go ahead and multiply this out.1834

This is just something² and we have a binomial so we can multiply this out.1836

This is going to be X sub I² -2 × X sub I × X + X² and P sub I.1840

This is a linear so I can go ahead and separate these into 3 sums.1860

This is actually equal to the sum of the X sub I² × π.1864

This π distribute over each.1872

This is going to be - 2 × the sum of X sub I × X × π + the sum of the average value of X², above the average value, × π.1877

Note also, this σ², the variance is a sum of positive terms.1904

These are positive, the probability is positive.1916

What we want you to notice is that the variance is actually greater than or equal to 0.1920

It is going to be very important.1925

Now we have first term, second term, third term.1928

First term, the sum of the X sub I P sub I² that is just equal to, by definition, the second moment.1940

That is the first term.1957

Our second term is -2 × the sum of the X sub I × the average value of X.1960

This average value of X is just a number. Because it is a number, I can pull it out of the summation symbol.1976

This sum of the X sub I π, this is just the definition.2000

It = -2, that is just the definition of average value so it becomes -2.2004

That is that one.2017

Let us do the third term.2023

I was able to go from here to here because this is just the number.2026

It is an average, that is a number, it has nothing.2034

I’m not adding so this stays the same.2035

It is these that change, this is the one that is indexed.2038

This I here, I goes from 1 to N.2041

This is the number that changes so this just pulls out of the constant.2045

The third term, we have the sum of the X² P sub I.2048

Once again, I can pull this out, this is just a number.2063

This is the average value of X² × the sum of the P sub I.2066

Normalization condition, the sum of the probabilities is always equal to 1.2073

Therefore, this is just equal to that².2078

That is our third term.2084

Our σ², our variance is equal to the second moment -2 × the square of the average value + the square of the average value.2089

Therefore, our variance is equal to our second moment - the square of our average value.2107

Now this is of course, σ² is greater than or equal to 0,2120

Which means that the second moment is going to be greater than or equal to the square of the average value.2125

You do not particularly have to know this derivation.2136

We are just throwing some things out there so that you see them and start to become comfortable with them, familiar with them.2138

Let us talk about some continuous distributions.2148

I will go back to blue.2151

Now, we discuss continuous distributions.2156

We said discreet earlier.2170

A discreet, we have a number line, this is 0, and number here, as few or as many as you like.2173

Only specific values are possible, that is what discreet means.2185

Only specific values are possible.2190

Now the difference of continuous distribution, not all values on a real number line.2199

All values along the real line.2212

They all have a shot, they all have a chance.2220

That is the only difference.2227

Recall from calculus, if you do no recall, this is what you actually learned when doing the integration.2232

When we go from discreet to continuous, we go from the summation symbol to the integral symbol.2254

That is what the integral symbol is.2267

It is a limit of discreet sum.2270

That is what we are doing, we are taking the limit.2272

We are just taking the integration smaller, smaller.2274

We are taking the limit of an actual summation which is a discreet.2278

We are adding a finite number of areas.2281

They are discreet numbers that we are adding.2285

When we passed the limit, we all of a sudden get this new object.2287

We get the integral of a function.2291

When we go from discreet to continuous, we go from summation to integration.2293

Let me see, do I just introduce them or do I want to actually talk about probability density.2301

I will talk probability density.2315

All of the definitions that we just gave, the average value, the second moment, and variance, notice we gave them in terms of sums.2318

Those were for discreet distributions of specific number of numbers.2325

10 numbers, 20 numbers, 50 numbers, 1000 numbers.2330

If now all the numbers on a real line are possible and infinite in any direction,2333

we are going to use the same formulas but now we are going to define them with integrals instead of sums.2338

Other than that, they actually stay the same.2343

We will get to those in just a moment.2349

But before I do, I want to talk about something called probability density2350

which is going to play a huge role in quantum mechanics and the wave function.2354

As it is going to turn out, we interpret the square of the wave function as a probability density.2361

And let us talk about what that means.2368

I want to introduce it here.2373

Probability density, first and physical analogy.2376

You already know what density is.2387

I’m going to use the physical analogy of a mass density.2391

Now, I'm going to use this equation right here DM = D of X × DX.2395

What this says is that a differential linear amount of mass, let us just call it an amount of mass K is equal to the linear mass density.2408

In other words, the mass of something per the unit length × the differential length element.2437

I will just say × a length, × a differential length.2449

Basically, what is happening here is this.2456

We are saying gram is equal to g/ cm.2458

Let us take cm as our unit of length × some cm.2466

A density is just that.2473

A density is just the amount of something per unit something else.2474

It does not matter what that is.2480

In chemistry, we normally do g/ cc or g/ ml.2482

We can do g/ L, we could do mcg/ ml.2488

It does not matter, it is the amount of something per something else.2495

When we multiply that by the another unit of the denominator, we actually get what we want.2501

It is just a question of units.2509

In this particular case, the grams of something = the grams per length,2511

whatever the density happens to be in that particular case × the particular length.2518

Let us write this out in words.2525

In general, this is very important.2531

A quantity of something of a given unit, does not matter what the unit is,2544

of a given unit equals the density of that unit per another unit × this other unit.2549

And by unit, I'm talking about grams, centimeters, liters, things like that.2585

The gram of something is equal to the density in g/ cc × a cc.2592

Another example, the number of particles of something, in other words, a quantity of something of a given unit.2601

In this case, the number of particles is equal to a number of particles per in square × that particular unit in square.2609

You know this already, this is just units of cancellation.2623

2 mi/ hr × 3 hr is 6 mi.2626

That is all that is going on here.2631

In this case, mi/ hr you might call it a length density per time.2633

That is all we are doing. That is all that is happening here.2639

Let us use this analogy to talk about something called a probability density.2644

The probability that some particle is between X and X + δ X is in some interval.2653

Δ X is going to equal the probability as a function of X × the differential X element.2665

This is the same exact thing that we did before.2678

The analogy is DM = D of X × DX.2680

Except now, we are talking about the probability density.2686

The probability at this particular point in space.2690

The probability is now a function of X at this point in space.2695

This point along the real number line, the probability of me finding something is this.2700

There is some probability per some length element.2704

This is a probability density, that is what is going on here.2709

The probability that some particle is between A and B, I integrate this thing.2718

In other words, if I were to integrate this, I would end up with my total mass.2740

It is just the integral of the density.2744

I’m just adding up all of the mass elements.2749

Here, if I want the probability in some small length, it is this thing.2753

It is the probability density × the particular length element.2760

This is what is important right here.2764

This is the probability density.2766

If I multiply by some length element, this whole thing is actually the probability.2770

The whole thing is probability, this particular thing is probability density.2780

If I want the total mass, I just integrate it.2786

If I want the total probability that something is between this point and this point, I just integrate this thing.2789

In other words, I add up all of the probabilities.2795

Let us go ahead and do our integral definitions.2799

Again, do not worry if this does not really make sense, if you cannot wrap your mind around it.2802

We will be discussing it more and more especially when we talk about quantum mechanics and the wave function.2805

But I do want to introduce it to you.2809

Let me actually stay with red.2812

The normalization condition we said that the sum of the probabilities has to equal 0.2818

Will now that we are talking about continuous distributions, the sum becomes an integral.2830

The integral for - infinity to infinity of this PX DX which is the probability has to = 1.2836

Profoundly important equation.2845

This is the normalization condition.2847

The integral over the particular region in space that we happen to be dealing with,2850

it is the most general form, from -infinity to infinity of the probability density × a length element = 1.2853

This is the probability.2861

Let us give our definitions.2865

Let me write that again.2872

Normalization condition, we have the integral from -infinity to infinity of the probability is equal to 0.2874

This is just the continuous version of the sum of the probabilities = 1 not 0.2896

The average value of a continuous distribution = the integral from -infinity to infinity of X ×2902

the sum we have X × the probability of X.2914

Here the probability of X is this, the second moment = the integral from -infinity to infinity of X² × the probability which is PX DX.2917

We have a final one which is the σ² of X which we call the second central moment or the variance.2938

It is the integral from - infinity to infinity of X sub I.2947

That is fine, I will just do this X – I cannot do X sub I here, this is continuous distribution.2959

× PX DX.2974

The PX of X DX is the probability.2982

P of X alone is the probability density.2993

Let us do a final example.3014

Let us go back to blue here.3017

If a particle is constrained to move in 1 dimension only, the interval from 0 to A.3020

0 to A in a real line, here is 0 and here is A.3031

It is constrained here, back and forth.3037

It turns out that the probability that the particle will be found between X and DX is given by this function right here.3040

We will see this again, do not worry about that.3049

Later, when we talk about this thing called a particle in a box, we will see this again.3051

That turns out to be very basic and elementary.3055

The probability that the particle will be found between X and DX is given by this right here.3062

Notice PX, that is the probability density.3070

This whole thing is the probability.3073

This is the probability density and it is going to be equal to 1, 2, 3, and so on.3075

We want you to show that PX is actually normalized.3079

We want to calculate the average position of the particle along the interval.3084

They want us to show that PX is actually normalized.3096

The normalization condition is from -infinity to infinity of this PX DX = 1.3100

We need to show the PX is normalized.3118

We need to show the following.3123

We need to demonstrate that this integral, in this particular case are space that we are dealing with.3125

It is a -infinity to infinity, it is just to 0 to A of this function 2/ A × the sin² of N π X/ A DX.3132

We need to actually show that it is equal to 1.3149

Let us go ahead and do this integral and see if it is actually equal to 1.3153

This integral, it is going to equal, I’m going to pull this 2/ A out.3159

0 to A of sin² × N π X/ A DX.3169

I can use math software or I can go ahead and look this up in a table of integrals.3183

I get the following.3188

I get 2/ A × this integral is X/ 2 - the sin of 2 N π X/ A ÷ 4 N π/ A.3189

We are taking this from 0 to A.3211

When I do this, it is going to be 2/ A.3216

I will put in A, put 0 in here, I'm going to end up getting A/ 2 -0.3222

When I put A in here and here, I get A/ 0 -, now I put 0 in here.3235

0 -0, so I end up with 2/ A × A/ 2 which = 1.3240

Calculate the average position of the particle along the interval.3258

Let us just use our definition of average.3264

Nice and straightforward.3267

Part B, the average position of a particle or the average of whatever is going to be equal to 0 to A of X × the P of X DX.3271

It is going to equal the integral from 0 to A of X × 2/ A × sin² of N π X/ A DX.3289

Again, I can use math software or I can look this up in a table of integrals.3308

I get the following 2/ A ×, this is going to be X² / 4 - X × the sin of 2 N π X/ A / 4 N π/ A -,3313

this is going to be the cos of 2 N π X/ A ÷ 8 × N² π²/ A².3334

I'm going to do this from 0 to A.3352

When I put this in, I get the following.3355

I get A²/ 4 -0 -1/ 8 N² π²/ A² -0 -0 -1/8 N² π² / A².3359

- this, - and - that becomes +, these cancels.3390

And I'm left with 2/ A × A²/ 4 which is equal to A/ 2.3396

And the average position.3411

It says is the following.3412

This says that the particles spends half of its time to the left of A/ 2 and half of its time, the other half of its time to the right.3415

The average position between 0 and A is A/ 2.3453

That is what this is saying.3460

On average, some× it is going to be here, sometime just going to be there.3462

Overall, the average is going to be right down the middle because it is going to spend3467

an equal amount of time to the left and to the right.3470

From your perspective, to the right.3474

It says that the particle spent half of its time to the left of A/ 2 and half of its time to the right of A/ 2.3480

This average is 2A/ 2.3488

This confirmed this averages to A/ 2.3492

We will definitely see this again when we talk about the particle in a box.3499

The things that I would like you to actually take away from this lesson.3505

The important things to remember.3509

The P of X is a probability density.3526

P of X DX is the probability of finding something between the X and DX.3540

Of course, the normalization condition -infinity to infinity of this probability = 1,3569

which is completely analogous to this one for the discrete probability.3582

The sum of the probabilities = 1, this is the continuous version of it.3588

Thank you so much for joining us here at www.educator.com.3594

We will see you next time, bye.3597

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