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Raffi Hovasapian

Raffi Hovasapian

Entropy & Probability I

Slide Duration:

Table of Contents

I. Classical Thermodynamics Preliminaries
The Ideal Gas Law

46m 5s

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

46m 2s

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

1h 6m 45s

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

1h 6m 33s

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

1h 2m 17s

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

1h 4m 39s

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

16m 50s

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

43m 40s

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

32m 23s

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

39m 15s

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

35m 52s

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

42m 40s

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

1h 23s

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

44m 34s

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

59m 7s

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

49m 16s

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

33m 59s

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

54m 37s

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

31m 18s

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

23m 6s

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

25m 42s

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

43m 39s

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

56m 44s

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

57m 6s

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

54m 35s

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

35m 5s

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

28m 42s

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

34m 38s

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

30m 50s

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

34m 6s

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

39m 18s

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

19m 40s

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

54m 16s

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

31m 17s

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

45m

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

58m 5s

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

25m 20s

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

34m 25s

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

59m 57s

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

42m 5s

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

30m 26s

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

21m 34s

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

56m 22s

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

45m 24s

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

48m 43s

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

46m 18s

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

39m 28s

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

35m 32s

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

29m 55s

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

54m 25s

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

46m 58s

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

44m 11s

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

35m 33s

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

43m 4s

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

26m 30s

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

41m 10s

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

30m 32s

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

35m 19s

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

33m 48s

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

46m 43s

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

40m

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

35m 58s

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

36m 18s

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

33m 55s

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

51m 53s

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

51m 53s

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

34m 29s

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

43m 49s

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

1h 1m 57s

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

48m 33s

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

48m 33s

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

48m 33s

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

59m 18s

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

34m 54s

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

38m 3s

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

57m 49s

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

1h 1m 20s

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

56m 34s

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

18m 24s

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

50m 2s

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

59m 47s

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

46m 22s

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

29m 24s

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

30m 53s

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

1h 1m 33s

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

33m 47s

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

1h 1m 5s

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

33m 31s

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

1h 1m 15s

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

47m 23s

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

54m 9s

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

48m 32s

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

57m 30s

Intro
0:00
Example I: Make a Plot of the Fraction of CO Molecules in Various Rotational Levels
0:10
Example II: Calculate the Ratio of the Translational Partition Function for Cl₂ and Br₂ at Equal Volume & Temperature
8:05
Example III: Vibrational Degree of Freedom & Vibrational Molar Heat Capacity
11:59
Example IV: Calculate the Characteristic Vibrational & Rotational temperatures for Each DOF
45:03
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Lecture Comments (4)

3 answers

Last reply by: Professor Hovasapian
Thu Nov 15, 2018 7:23 AM

Post by Curtis Marriott on November 12 at 11:31:24 AM

Hello Prof. I am grateful for your ecture series. You are very clear and thorough and you for sure an amazing teacher(communicator). After listening too this particular lecture, i am  more appreciative of thermodynamics.
The question below was already done, but i may be incorrect. Could you comment on it, if it is not correct, i would like to know why? Knowing why is super important for me.
If a molecule has 5 possible vibrational states, how many microstates do10 of these molecules have?

Using S=kbln? where kb = Boltzmann constant, ln= log, and ? = energy and volume distribution.
? = Microstates.
? = 10! Divided by 3! 3! 2! 1! 1!   = 50400
The 50400 are the possible microstates.

Entropy & Probability I

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

  • Intro 0:00
  • Entropy & Probability 0:11
    • Structural Model
    • Recall the Fundamental Equation of Thermodynamics
    • Two Independent Ways of Affecting the Entropy of a System
    • Boltzmann Definition
  • Omega 16:24
    • Definition of Omega
  • Energy Distribution 19:43
    • The Energy Distribution
    • In How Many Ways can N Particles be Distributed According to the Energy Distribution
  • 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
    • Example I Summary
    • Example II Summary
    • Distribution that Maximizes Omega
    • If Omega is Large, then S is Large
    • Two Constraints for a System to Achieve the Highest Entropy Possible
    • What Happened When the Energy of a System is Increased?

Transcription: Entropy & Probability I

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

Today, we are going to talk about the entropy and probability.0004

We are going to define what entropy is finally.0008

Let us jump right on in.0011

When we introduced the definition of entropy, when we introduced that the DS =DQ reversible/ T which was our definition of the differential element.0014

That is the definition of entropy, we did not get the structural model.0032

We did not require a structural model for the system in order to work with the entropy,0044

in order to work with this state function entropy or describe its behavior .0072

In fact, we did not even need to know what entropy was.0086

We had these mathematical descriptions and we had these constraints of temperature, pressure, volume and we saw how entropy behaves.0090

We are able to derive and calculate numerical values for it.0096

The only thing that we really did was casually refer to it as a measure of the disorder or randomness of the system.0100

I still think, in my personal opinion that disorder and randomness is actually a great way of thinking about entropy.0108

We are going to do was to define what we mean when we say disorder and randomness.0115

We are going to quantify, we are going to come up with some numerical way of explaining what is this disorder or randomness.0120

When I use the words disorder and randomness, what I’m talking about is something called the distribution.0131

When we talk about disorder or randomness, we are talking about a distribution.0145

In this case, it is going to be the distribution of particles.0148

We did not require a structural model, we did not care what a particular system was made up off.0152

Whether it is particles, chairs, it could be made absolutely anything.0156

There was this behavior that it represented, this is our empirical observation.0162

We are able to use mathematics to derive other to describe different ways of how this thing behaves, we are going to give it a structural model.0167

Here is a structural model.0176

What I'm going to do, I would go ahead and do this in blue, I think.0180

Our structural model is exactly what you think it is.0189

A system is composed of a very large number of particles.0196

Those particles could be molecules, they could be atoms, they could be ions, whatever that you have to be discussing in that particular problem.0220

Molecules, atoms, and ions.0228

Let us say some things about these particles.0234

These particles have various energies.0237

The best way to think about this is to think about a collection of gas like the kinetic theory.0243

Basically, it is just a bunch of particles in a thick space that are bouncing into each other and bouncing off the walls, that is a system.0249

These particles have various energies.0259

In other words, not each particle is flying around at the same speed.0265

There are some going faster, some that are going slower.0268

There is a large number of them that have the same energy.0272

There are others that also have the same energy so there might be 10,000,000 of them that are traveling at 500 kph.0275

There might be 20,000,000 of them that are traveling at 550 kph.0281

They are distributed if all the energy is distributed among the different numbers of particles.0285

These particles have various energies and there is a distribution of the total energy of the system DU0292

of the total energy of the system which is U first law / the various particles.0310

The total energy of the system is made up of the sum of all the individual energies of the particles.0334

If the system has 100 J of energy, those 100 J are going to be distributed among the different particles and different ways.0342

Maybe two parts might have 1 energy, two particles might have another energy, 15 particles might have another energy.0350

All the different energies, that is the distribution.0356

In other words, n sub 1 particles have energy E sub 1.0360

N sub 2 particles have energy E sub 2, and so on.0381

The number of particles, the total number of particles so n sub 1 + n sub 2 + n sub 3 and so on,0394

have to equal n which is the total number of particles in the system.0404

If I have 100 particles in the system n sub 1 might be 10, n sub 2 might be 20, n sub 3 might be 70.0409

70 + 20 + 10 =100 the total number of particles.0416

That is one of our constraints.0419

The number of particles with energy 1 + the number of particles + energy 2 + the number of particles with energy 3 and so on,0425

onto the number of particles n sub i with energy sub I, that has to equal the total energy of the system.0436

When I had all these energies, the maximum energy that I can have is U, the total energy of the system, that is the second constraint.0443

Nothing strange happening here, the system has a certain energy, our system is made up of a bunch of particles,0450

we are distributing this energy over a bunch of particles.0456

The next part is the structural model is these particles for the first one of these particles have energy.0463

These particles occupy space, in other words, volume.0471

I'm not saying that they themselves have volume, I'm saying that they are contained in a volume.0482

There is some fixed volume that they are in.0487

What ever that volume is, they are occupying that space.0490

There is the distribution of these particles, this is the most intuitive clear one, these particles over the volume available.0498

We have these hundred particles and we have a 1 L flask.0521

I have this hundred particles in 1 L flask, the particles are going to arrange themselves in all kinds of different ways.0525

Maybe these particles here, these particles there, a 100 all over the place.0531

They are a bunch of different ways that these particles can distribute themselves in that 1 L flask, that is the volume distribution.0536

They are going to arrange themselves in the volume up to the maximum capacity of the volume.0543

You know this intuitively.0549

Let us recall the fundamental equation of thermodynamics.0552

Our fundamental equation was the following DS = 1/ T DU + P/ T DV.0571

Notice DU DV this is energy, this is volume, there are two different independent ways of affecting the entropy of the system.0581

I can change the energy or I can change the volume or I can do both, not a problem but they are independent of each other.0596

Let us write it down.0605

There are two independent ways of affecting the entropy of the system.0608

One, I can change the energy that is U.0630

Two, I can change the volume which is V.0641

It is very important equation, the fundamental equation of thermodynamics is a relationship between all the various properties of the system,0649

the entropy, the pressure, the temperature, the volume, the energy.0656

It is very important.0660

If it is expressed in terms of entropy we would write it this way DS = something.0662

We see that there are two ways to affect the entropy, energy, and volume.0666

Since, there are two ways affecting this entropy of the system, the energy in the volume.0675

Therefore, it makes sense to define the entropy of the system which is S in terms of these two properties energy and volume.0685

In terms of these two properties U and V, both one gave the following definition.0717

Profoundly important equation, it is the fundamental equation of statistical thermodynamics.0753

so they have the following definition, he said that the entropy of the system is equal0756

to this constant which is Boltzmann constant × the natural log of something called O.0763

You can use either one you want, O happens to be the classical, that is just what we use.0771

KB is the Boltzmann constant and KB is equal to the gas constant divided by all the numbers.0777

If we take 8.314 if we do J/ °K mol and if we divide 6.02 × 10²³ particles / mol which is just / mol of the actual unit of it.0800

The mol and mol cancel and which you end up with is this value of KB = 1.381 × 10⁻²³ J/ °K.0819

Notice, it is has the same units as entropy J/ °K.0835

You can memorize 1.381 × 10⁻²³, you cannot memorize it all and just have your book in front of you and look it up or you can think of it as R÷ n.0840

This is the best way to think about it, the gas constant divided by other number.0851

We have taken care of this KB, what about this O?0857

This is the one that we are going to spend the lesson talking about.0860

This O represents the energy and the volume distribution.0863

Let me just say a couple of quick words about this and then I will actually get into what this is and get much more detail about this.0864

We said that we have these particles in the system and we are going to distribute these particles.0891

There is an energy in the system and that energy is distributed over the particles.0896

Different particles have different energies.0899

These particles also distribute themselves in this space available to them.0901

O represents the number of ways that this distribution is possible.0906

If I give you 10 particles and if I said there are 100 J of energy, distribute those 100 J among those particles and0913

then if I gave you a 1 L flask and if I said how many different ways can you take those 10 particles and0920

put them in that 1 L flask if I divided the 1 L flask into 20 volume elements, 20 spaces.0927

There is some statistical probabilistic number, some combinatory number that you can come up with.0936

The numbers actually is going to be very large, that is what O is.0944

O is a measure of how I can distribute my energy and my volume / the number of particles that I have given to me.0949

That number gets very huge, that is O.0958

When I take the log of that number and I multiply it by both constant, I'm going to get some number, that number is the entropy.0965

That number is the statistical entropy for that particular system in that state.0973

Let us say some more about it.0981

I’m going to say a few more words and I’m going to start quantifying this.0986

A given system is in a given state, that is in microscopic state.0991

This is the state that we experience, see, and measure.0996

In other words, the temperature, the pressure, and the volume, that is the microscopic state of the system.0999

O is the number of individual ways that the particles making up the system can distribute themselves over the given volume and1008

over the given total energy in order to achieve that particular state.1017

The temperature, pressure, volume etc, whatever it is that I happen to be measuring.1022

These individual ways of the distributions are called microstates.1026

When I come across a system that has a certain temperature, pressure, and volume, the particles,1032

the energy of the system and the volume of the system are distributed among those particles.1038

The particles spread themselves out of the volume in different ways and there is an energy distribution.1043

There are different particles of different energies.1050

However, that is not fixed, the particles are indistinguishable.1051

If I have particle A here and particle B here, if I switch them and put particle B here and particle A there1056

because they are indistinguishable they represent the same thing.1061

Because they represent the same thing, because the particles are indistinguishable, there are millions and billions and trillions1065

of ways of achieving the same state, the same temperature, pressure, and volume, the same microscopic state.1072

We have a bunch of microstates, a bunch of different ways of distributing it to achieve the same state, that is what we are saying.1080

O is the number of microstates.1089

It is the different individual arrangements giving rise to a single microscopic state.1092

Let me say that again, it is the different individual arrangements giving rise to a single microscopic state.1098

The more individual ways, they are achieving a given state, the greater the probability of finding the system in that state.1116

If I have 15 different ways of achieving a certain temperature, pressure, and volume, another distribution gives me 500 ways1126

of achieving that same temperature, pressure, and volume, or if I come up on that state chances are the probability says that the 500 ways,1135

I’m probably did run across 1 of those 500 ways more than I’m going to run across 1 of those 15 ways.1143

You know this intuitively, it is that simple.1149

This is why you never see a gas collected in one corner of the room, instead it spreads out and occupies as much of the room as possible1153

because there are more individual ways to fill up a large space than there are filling up a small space.1161

That you know this intuitively, let us quantify this.1168

Let us put some numbers to it.1173

The first thing I want to talk about is the energy distribution and then in the next lesson I’m going to talk about the volume distribution and1177

I’m going to put them together.1183

The energy distribution is first.1185

We have energy distribution.1189

We had n particles now what we are going to do is we are going to divide U the total energy into compartments of various energies.1197

I got energy 1 J, energy 2 make it 2 J, energy 3 that is 3 J, and so on.1223

Just different compartments with different energies and I will just put E sub I right there.1235

When I add up all the energies they have to equal U.1246

Divide U compartment of various energies.1257

It is just a bunch of different energies.1260

Specify how many particles have which particular energy?1268

Specify how many particles n sub i have energy e sub I.1274

How many particles n sub 2 have energy e sub 2, and so on.1303

5 particles have energy 1, 30 particles have energy 2, 30 particles have energy 3, 1 particle has energy 4, that is the energy distribution.1317

n sub 1, n sub 2, n sub 3, + so on + n sub i is equal to n the total number of particles.1330

This is the energy distribution when you specify the n's.1344

This is the energy distribution.1362

It is the actual specifying of how many particles n sub i have energy e sub i.1367

The question is in how many ways can n particles be distributed according to the energy distribution n sub 1 n sub 2 n sub 3 and so on.1390

I have some energy distribution n sub 1 n sub 2 n sub 3, these are the particles that have a particular energy e sub 1 e sub 2 e sub 3.1433

If I have a total of n particles is there a way for me to count how many different ways1442

I can actually distribute the energy over this many compartments? how can I do that?1447

Let us go ahead and do it.1454

Let us do this with some small number examples first.1460

Let us suppose that the n sub 1=3.1464

We have 3 particles that have an energy 1 so n sub 1 is 3, 3 particles energy e sub 1.1472

The n particles, the question is all those n particles how many different ways can I actually put 3 of those particles into the first energy level?1481

How many different choices do I have for my first particle n?1498

There are n ways to choose particle 1.1502

If I have 10 particles I can choose any of those 10 as my first choice, to put in to be number 1.1506

I have n -1 ways to choose particle 2.1514

I have n -2 ways to choose particle 3.1523

If I choose particle 1 that is going to be at 10 ways to choose that and I have I chosen out and 9 particles was left.1529

I have 9 ways of choosing from the other particles, I have 8.1535

I have the following so 10 × n -1 × n -2 ways of choosing those particles.1538

The particles are indistinguishable so it does not matter whether I choose particle 1 first or 2 first, or 3 first, they are indistinguishable.1552

This number of ways of choosing is actually going to have redundancies.1559

Therefore, I have to divide because it does not matter whether I choose 1, 2.1564

Again, I'm choosing 1, 2, 3 to put them into bin number 1.1568

I can choose 1, 2, 3 or I can choose 2, 1, 3 or 3, 2, 1.1572

I can choose 3, 1, 2 or 3, 2, 1 as it turns out there are three factorial ways of arranging three particles that are indistinguishable.1578

The particles are indistinguishable so the order of choosing is unimportant.1605

This n × n -1 × n -2 has redundancies and otherwise, if I choose 1 and 2 and 3 and they end up in this bin.1637

It is going to be the same as if I end up choosing 2, 1, and 3, it is the same particles ending up in the same bin.1654

I have repeated myself that is what we mean by the redundancies.1660

If I choose 3, 2, 1 it is still the same particles 3, 2, 1 in that bin.1663

It does not matter, the order, if it would all end up in that bin I cannot just keep counting those ways.1668

There is only one way of getting that.1672

For 3 particles, there are 3 factorial permutations.1677

Therefore, we take this n -1 × n -2 and we divide by 3!.1695

This gives us the number of ways of taking n particles and choosing 3 of them to actually go into bin number 1.1704

There are these many ways of doing it, whatever n happens to be.1714

If n is 10 it would be 10 × 9 × 8 ÷ 3!, whatever that number is.1718

We will see some examples in just a minute.1724

That is the first level, that is the first part, let us deal with n sub 2.1726

n sub 2, that is 2 that means there are 2 particles in the second energy level.1731

How many different ways now that I have chosen my 3 particles from my n I have 10 × n -1 × n -2.1742

I have n -3 particles leftover.1750

All those n -3 particles I'm going to pick two of them to put into the second energy level.1752

How many different ways can I do that?1758

If I have n -3 particles well I choose one of the particles that leaves me with n -4 for the second particle.1765

I have n -3 particles to choose from, I choose one that leaves me with n -4 particles but there are redundancies because I can choose 1 and 2 or 2 and 1.1777

I divide by 2! that takes care of the second bin.1786

The total so far which is for 3 n energy compartment 1 and 2 in energy compartment 2, we multiply those two numbers.1795

We have n × n -1 × n -2/ 3! × n -3 × n -4 / 2!.1818

What if we continue?1842

If we continue with n3, n4, n5, and so on, we get the following.1848

We get the O is equal to n! divided by n sub 1!, n sub 2!, n sub 3!, and so on.1863

This is the general expression for the number of ways, the number of individual arrangements,1880

the number of microstates that allow n sub 1 particles in e1, n sub 2 particles in e2, etc.1906

Given a particular distribution n sub 1 n sub 2 n sub 3 n sub 4, the total number of ways of distributing the energy of those particles1936

over the number of particles of a number of energy compartments is this expression right here.1946

The total number of particles factorial divided by the number of particles in each bin, each factorial and then divided.1951

Let us do some examples and there we go.1967

Let us do some examples, given 10 particles and four energy states e1 e2 e3 e4, in how many ways can the following distribution be achieved?1974

This is n1, this is n2, this is n3, this is n4.1984

In this case, we have four energy compartments.1991

We are saying we have 10 particles total so n= 10.1994

We are saying that n1= 10.2000

In other words, we are taking all those 10 particles and we are putting all of them into bin number 1.2003

All those particles have an energy whatever energy e sub 1 happens to be, there is nothing in bin sub 2, nothing in bin sub 3, nothing in bin sub 4.2008

How many different ways is it possible if I let 10 particles to arrange themselves according to this distribution?2018

We use our equation, we have O= n !/n1! n2! n3! and n4!.2028

N is 10 and so this is going to be 10!, n1=10, this is 10!.2041

N2, n3, n4 are 0 so this is 0!.2049

0! by definition is 1.2056

Therefore, what you have is 10!/ 10! you get 1.2062

If I have 10 particles and I take 4 and 4 energy states available, there is only one way that I can put those 10 particles into 1 energy state.2068

There is only one way, all the 10 have to go into that one spot.2077

You know this already.2082

Let us change the distribution.2086

Same thing, we are given 10 particles and we are given 4 energy states, it is the same old basic situation.2088

I still have 4 energy states available and I have 10 particles.2094

In how many ways can the following distribution be achieved.2097

I want 9 particles in bin 1, 0 in bin 2, 1 particle in bin 3 and 0 in bin 4.2102

How can I choose in how many different ways can I achieve this one distribution?2112

That is the question.2117

In how many different ways can I achieve this one distribution, this is the microscopic state.2118

The number of individual ways of achieving this are the different ways, they are the microstates.2126

This is the one, the ways of achieving that there is several ways of achieving one state.2134

There are several microstates, there are ways of achieving the on microstate.2140

Let us do it.2146

We have this is n1, this is n2, this is n3, and this is n4.2148

n does not change that is equal to 10.2153

This is what changed the distribution.2157

O=10!/ 9!, 0!, 1!, 0!.2160

What we end up with is 10!/ 9! which is equal to 10 × 9!/ 9! 10.2172

This distribution I still have 10 particles, I still have 4 energy states but now the distribution is different.2186

How they different ways can I do this?2192

10 different ways because there are 10 different ways of achieving this one distribution, if I come across this system 10 particles 4 energy levels,2193

chances are I’m going to find in this state for this distribution instead of all 10 and packed into energy level 1.2204

There is only one way of achieving that but there is 10 ways of achieving this.2213

The system is going to shift and achieve all those 10 states more often.2216

There is 10 different ways so chances are the probability is that I'm going to run across 1 of these 10 instead of that 1.2223

That is what we are saying.2230

Change the distribution again, given 10 particles and 4 energy states and how many ways can the following distribution be achieved?2236

With 6220 so n=6, n2=2, n3=2, and n4= 0.2242

O = 10!/ 6! 2! 0!.2255

When you do this under calculator you end up with 1260 ways.2264

Clearly this is jumping, we went from 1 to 10 to 1260 just by broadening the distribution.2272

Taking it all from 1 energy level and just letting a few more drift off into some of the other energy levels.2280

If I come across 10 particles in 4 energy levels of the 3 distributions which 1 more likely to come across?2285

The only one way to do, the on with 10 ways to do it, or the one with 1260 ways of achieving that distribution.2294

If I take a snapshot of the system in any given moment , these 10 particles with this 4 energy levels2301

chances are that I’m going to run across one of these 1260 ways.2307

Chances are that when I look at that, I'm going to have 6, 2, 2, 0 that this is the distribution I’m going to see.2311

This is going to be my microscopic state.2318

The particles are going to be in that distribution because there are so many ways.2322

Example 4, given 10 particles and 4 energy states, in how many ways can the following distribution be achieved 3, 3, 2, 2?2330

Let us see what this one gives us.2340

We have 10!/ 3! 2! 2!.2343

We have 25,200 ways, from 1 to 10 to 1200 to 25,000.2352

If I have 10 particles and 4 energy states in how many ways if, I take a snapshot, if I just sort of come across the system2364

that has 10 particles and 4 energy states available, which distribution I’m most likely going to see when I take a picture of it?2372

I’m going to see this one because there are 25,200 ways of achieving that distribution.2381

In fact, given the constraints of 10 particles and the constraint of 4 energy levels, this number right here the 3 3 2 2 represents2388

the maximum number of ways that I can actually achieve the maximum number of ways that I have in achieving a given distribution.2399

With these constraints, this achieves the distribution that I will most likely see.2410

The probability is that because there are 25,200 ways of achieving this one distribution, that is the one that I'm going to see.2416

If I take a snapshot, I’m going to find 3 in bin 1, 3 in bin 2, 2 in bin 3, 2 in bin 4.2423

Or I might find 3 3 2 2 but it is always going to be 3 and 1, 3 another 2, to another.2431

That is what we are talking about here.2439

I hope that makes sense, these are the microstates, this is O, this is the macrostate given the constraints of distribution,2442

the number of particles and the total energy.2451

Let us see what we got.2458

Clearly, as the distribution broadens, the number of ways to achieve that distribution increases massively 1 to 10 to 1200 to 25,000.2461

With just a small change in the distribution the system will appear in the state distribution that offers the greatest number of ways of achieving that state.2472

This is very important.2483

The system will appear in the state that offers the greatest number of ways of achieving that state because2487

the probability of finding a system in a given state depends directly on the total number of ways that that state is achievable.2494

In example 1, there is only one way of achieving the particular distribution.2504

The chance of finding a given system of that arrangement are 10 particles with 4 energy levels available is very slim.2508

In any given moment you probably not going to run across that distribution.2515

In other words, these 10 particles will not crowded to one compartment and stay that way.2519

If they have other compartments available to them as far as energy is concern, within the constraints of total energy and total number of particles.2524

Example 4, offers a distribution as a huge number of ways of being obtained.2533

Therefore, chances are very high that if we come up on the system it will be in a state that is consistent with this distribution.2538

Within the constraints of the sum of the total, the sum of the individual n sub i equals the total number of particles n.2547

The sum of the number of particles in a given energy level × the energy of that level equals U the total energy.2556

There exists a distribution, there is always one distribution that completely maximizes this O.2562

The sheer number of microstates for this distribution is so huge that it completely dominates the landscape of probabilities.2570

You will certainly find the system with this distribution in the state.2578

In our example we had 1, we had 10, we had 1260, I think and we had 25,200.2581

Within the constraints of the particular problem, the 10 and 4, the distribution is 25,200 is so massive, it is so much bigger than this and this.2590

If we ever come across the system with 10 particles and 4 energy levels, we are going to find the distribution 3 3 2 2 or 3 2 3 2 or 2 2 3 3.2601

We are going to find 3 in 1 bin, 3 in another, 2 in another, and 2 in another.2611

That is the distribution the particles will arrange themselves in the way that offers the greatest number of ways of achieving that distribution, that microscopic state.2614

Sorry if I keep repeating myself, but this is profoundly important.2627

Let us look again at this O.2633

O we set this as n!/ n sub 1! n sub 2! n sub 3!, and so on.2650

We set that S, let us go back to entropy because we are talking about entropy here.2652

S= Boltzmann constant × nat log of this thing called O.2657

If O is large, the log of O is large, the entropy is large.2663

If O is large then S is large.2670

The smaller the n sub I, these individual n sub I, n sub I, the smaller these numbers are the larger O becomes because this is the ratio for the numerator/ denominator.2686

If I have a certain number of particles that is fixed, the smaller the denominator is the bigger my ratio is, the bigger O is going to be.2709

The smaller the n sub I, the larger O becomes which means the larger the entropy.2718

Small n sub i means distributing as many particles in as many compartments as possible to lower that number.2731

Instead of 10 particles in 1 bin, it is a lot better to have a fewer number in this the 3 3 2 2 for the particular arrangement.2743

That maximizes it, that lower the n sub 1 n sub 2 n sub 3 n sub 4, as low as they will go.2753

As low as they will go that raises the O to as high as it will go which in that case was 25,200.2764

That is an increase in entropy.2771

This is what we mean by this order or randomness having 25,200 ways of achieving given distribution2774

is kind of chaotic vs. only one way of achieving a distribution.2782

It is a highly ordered system having 25,200 ways of doing the same thing that is random, that is disordered, that is chaotic, that is what we mean.2788

It represents the distribution, these are the numbers.2798

Back to this, the larger the n sub I, the larger these numbers, the smaller O is.2803

The smaller O becomes which implies that the smaller entropy.2814

The system will try to achieve the highest entropy possible thus the broadest distribution subject to the two constraints.2831

Broad means we want to spread out the particles in as many bins as possible to lower the number of particles in each bin.2881

The lower these numbers, the higher O, the higher the entropy.2888

A system is going to try to achieve the highest entropy possible thus the broadest distribution subject to the two constraints.2891

And I will go ahead and put those constraints again.2900

The sum of the individual n sub i =n, this is the sigma notation and the sum of n sub i × the energy of that particular bin is equal to the total energy.2902

Within these two constraints, the system will try to achieve the broadest distribution possible that it can.2918

In the case of the 10 particles and 4 bins it was 3 3 2 2.2925

That was its broadest distribution possible.2930

When the energy of a system is increased, the energy distribution broadens and the particles occupied this broader distribution.2945

In other words, O rises which means the entropy arises.3000

If I have a certain energy of the system and all of a sudden I pump some more energy in the system, if I increase the energy of the system,3008

by increasing the energy of the system now I have allowed more energy, a lot more bins.3013

Therefore, if I have more bins, the particles that are in these other bins are going to filter off and move into those other bins.3026

They are going to be fewer particles in each individual bin.3034

Again, if there are fewer particles in each individual bin that means the number of ways that3038

the denominator of the O is to get smaller which means O goes up.3043

Numerically, if O goes up the entropy goes up.3048

If I have the 123 and 4, all of a sudden if I pump some more energy into that, now all of a sudden I introduced.3053

We are going back to the example where I have 10 particles and I have the 4 bins.3063

We said we had 3 3 2 2 under the constraints of 10 and 4, this was the broadest distribution I could have which is going to be 25,200 microstates.3068

Let us say if I increase the energy of the system now introduce let us say 3 more energy levels in here, these particles can now move to those.3079

It is going to move as much as possible.3088

Maybe one of these particles end up going over here so that goes to 1.3091

Maybe one of these particles that goes here and maybe about one of these particles ends up coming over here because I have added energy.3095

I have broaden the energy distribution .3104

I have allowed more bin, now they are going to distribute themselves in such a way that its broader.3106

It is going to achieve a broader distribution.3110

If I do 10! / 3! 2! 1 1, now it is going to be a lot higher than 25,200.3113

The entropy is going to go up but you know this already.3125

Here is how you know this.3127

You know this already from the work that we did in the previous lessons.3132

You know the DS =1/ T DU + P/ T DV.3139

This 1/ T is positive, it is always positive because the Kelvin temperature is always going to be about 0.3149

1/ T is always positive which means that if you increase the energy of the system, you increase the entropy.3153

This is a simple math.3160

You know this already from your experience, you have dealt with this.3163

What we have given is a statistical, we have given a probabilistic explanation for these increase in entropy.3167

We know why when we increase the energy we are increasing the energy distribution because we increase the energy distribution O goes up.3175

When O goes up, the entropy goes up.3185

That is what is happening here.3188

What we have done here, we just given.3190

I have to slow down.3199

The statistical and I will go ahead and say probabilistic.3202

In other words, the microscopic reason for our classical observation which is our macroscopic observation.3215

I hope this has made sense, increasing the energy of the system broadens the energy distribution.3237

As you broaden the energy distribution, the number of particles that can achieve the distribution become the spread out, they themselves broaden out,3243

when that happens the numerator of our O ends up getting smaller so O gets higher.3252

When O gets higher, because S is equal to KB LN O S gets higher.3259

I hope that makes sense.3268

Thank you so much for joining us here at www.educator.com.3270

We will see you next time for a continuation of this discussion, bye.3272

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