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Raffi Hovasapian

Raffi Hovasapian

Statistical Thermodynamics: The Big Picture

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)

1 answer

Last reply by: Professor Hovasapian
Mon Mar 26, 2018 7:54 AM

Post by Richard Lee on March 25 at 07:58:12 PM

Professor,

a single system can have a number of particles that occupy different quantum states - this gives rise to different total energy of the system?

1 answer

Last reply by: Professor Hovasapian
Tue Feb 27, 2018 4:28 AM

Post by Richard Lee on February 19 at 10:38:18 PM

@ 25:00 Did you mean "if you have 100 states that are available then you have 100 terms in that sum."  Instead of accessible.

Statistical Thermodynamics: The Big Picture

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
  • Statistical Thermodynamics: The Big Picture 0:10
    • Our Big Picture Goal
    • Partition Function (Q)
    • The Molecular Partition Function (q)
    • Consider a System of N Particles
    • Ensemble
    • Energy Distribution Table
    • Probability of Finding a System with Energy
    • The Partition Function
    • Microstate
    • Entropy of the Ensemble
    • Entropy of the System
  • Expressing the Thermodynamic Functions in Terms of The Partition Function 39:21
    • The Partition Function
    • Pi & U
    • Entropy of the System
    • Helmholtz Energy
    • Pressure of the System
    • Enthalpy of the System
    • Gibbs Free Energy
    • Heat Capacity
  • Expressing Q in Terms of the Molecular Partition Function (q) 59:31
    • Indistinguishable Particles
    • N is the Number of Particles in the System
    • The Molecular Partition Function
    • Quantum States & Degeneracy
    • Thermo Property in Terms of ln Q
    • Example: Thermo Property in Terms of ln Q

Transcription: Statistical Thermodynamics: The Big Picture

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

Today, we are going to start our discussion of statistical thermodynamics.0005

Let us jump right on in.0009

I want to go through this big picture of statistical thermodynamics is that we know why we are doing this.0013

What are all for our goal?0022

What is it that we are trying to achieve?0023

Our big picture goal is going to be the following.0025

Let me actually work in blue today.0028

Our big picture goal it is to find a way to express the thermodynamic properties of a bulk system and0036

by bulk we mean just a bunch of particles, like a block of wood as opposed to the individual molecules that make up that wood.0071

Of a bulk system, in other words the energy, the entropy, the enthalpy, the Helmholtz energy,0077

the Gibbs free energy, all of these things.0089

The constant of volume heat capacity and the pressure.0092

The basic thermodynamic functions of bulk system.0096

Our goal is to express these properties in terms of the properties and of the particles that make up the system.0099

That is what we are doing with statistical thermodynamics.0124

We start off the course with classical thermodynamics.0133

We moved on to quantum mechanics.0136

In quantum mechanics we are dealing with the individual energies and properties.0138

The individual particles out of a molecule, whatever it is.0141

Now that we have quantum mechanics, we want to go back and we want to explain0146

what we learned in classical thermodynamics via the individual particles.0151

That is it, we are just closing the circle like talked about in the overview of the course.0155

I will go back to black here, sorry about that.0161

Our primary tool in this investigation is going to be something called the partition function.0166

Our primary tool will be something called the partition function.0174

Let me actually come over here, called the partition function.0191

The symbol for the partition function is going to be a capital Q.0201

It is going to be the partition function of the system.0205

What I’m going to do is we are going to express the thermodynamic properties in terms of Q.0210

We will express the thermodynamic properties, the ones I have listed above.0218

We will be listing those and I would be expressing those properties in terms of Q.0228

I’m having a little difficulty talking today, sorry about that.0234

In terms of this Q, the partition function.0237

We then introduce q.0241

We then introduce q, this is called a molecular partition function.0248

What that means it is or should say that is the partition function for each particle in the system.0271

We can actually do that, we can write this thing called a partition function0280

for each individual particle of whatever system we happen to be dealing with.0286

It is pretty extraordinary, that it is very extraordinary.0289

For each particle of the system.0293

We will express Q, the partition function of the system in terms of q.0300

We will then have exactly what we wanted.0325

We will then have, finally, our direct relationship between the thermodynamic properties of the system0328

and the particles that make up the system.0365

That is our big picture goal, that is what we want to do.0367

We want to come up with this thing called a partition function.0376

And we want to come up with this thing called, we want to find the various partition functions0381

for whatever quantum mechanical system we would happen to be dealing with.0385

And again, we already talked about the particle in the box.0389

We talked about the rigid rotator, the harmonic oscillator, these are the partition functions that we are going to look at.0392

These are going to be the partition functions of the molecules.0397

We are going to express the thermodynamic properties that we learned about back when we started the course.0400

Let us see what we have got.0411

Let us start up here.0412

Consider a system of N particles and N is usually just going to be Avogadro’s number.0415

Consider a system of N articles.0421

The energies of the particles are discreet, we know that already.0431

That is what quantum was all about.0435

The energies of the particles are discreet and they are distributed over various quantum states.0437

For example, if you had some rotating molecule.0465

You know whatever it might be in the J = 1 couple of 1,000,000 of them might be in J = 2.0468

A couple of 1,000,000,000 of them might be in J = 3.0473

Different energies, the particles are the same.0476

They are distributed over the various quantum states, that is all we are saying here.0480

In any given moment, if we have add up the energies, we get the energy of the system.0483

We will add up the energies of the individual particles, you get the energy of the system.0511

We will call that E sub I.0523

Also discreet because of the individual energies are discreet, the E sub I is going to be discreet.0526

If we come back to any moment of the same system, let us say 30 seconds later, whatever,0537

1 second later, does not really matter.0548

The particles are going to be in this differ distribution of quantum states.0552

Therefore, the energy of the system is going to be different.0558

In another moment, the particles being in other quantum states, this gives rise to another,0561

I will call this not E sub I, E sub 1.0593

I take my first measurement and I get E sub 1.0595

It gives rise to another energy of the system and I will call it E sub 2.0600

What we call the thermodynamic energy of the system is an average of all of these E1, E2, E3, E4, E5.0614

If I take 100 measurements, a 1000 measurements, 500 measurements of the system at different times,0622

all the particles are going to be in different quantum states.0627

I’m going to get different energy of the system.0629

I take an average of that, that is what I call the thermodynamic energy.0632

That was what we call U.0635

What we call the thermodynamic energy of the system.0639

In other words, U is the average of many observations.0659

Make sense, I think it is particularly strange here.0670

Let us take a bunch of observations, take the average and call that number the energy of the system.0673

Instead of making a 100 or 1000, or 500 observations on the same system.0680

Instead of making a multiple, let us say multiple instead of choosing a number, multiple observations.0689

Instead of making multiple observations, let us actually choose a number.0705

I’m going to choose one of the number, it can be any number but I'm just going to choose one, a hundred.0713

Let us say we take 100 observations.0717

We average that out a 100 observations of the same system, we get the energy of the system.0719

Instead of making a 100 observations on the same system, we can also just create 100 identical systems.0724

That is it, same circumstances, same surroundings, same particles, same temperature,0743

same pressure, create 100 identical systems.0748

Each system will be in a particular quantum state.0755

It might have all hundred that are in different quantum states.0774

You might have 20 of them that are in one, 10 of them in another, 2 of them that are in another.0776

Again, this could be in various quantum states.0781

A system will be in various quantum states with the energy E sub I.0788

Let us increase that number.0803

Now, instead of 100 or 1000 or 2000 or 5000, let us create a large number of identical systems0805

and we will call that large number of identical systems N.0817

Now, let us create, in other words let us just make this 100 a really big number.0821

Because we know that the bigger number we have, the better our average.0829

Let us create a very large number of identical systems.0834

Let us call that number N.0848

We call this collection of identical systems, this large number of identical systems, we call that the ensemble.0852

That is what the ensemble means.0858

Let me go to red.0866

We call this collection an ensemble.0870

It seems to always be a bit of confusion about what ensemble is and that is it.0884

We are just taking system, we are duplicating it, a 6.02 × 10²³ × and we are calling that ensemble.0888

It is the identical system, just copied.0897

The collection is an ensemble.0899

Each system will have a particular energy.0902

Each system in the ensemble will have a particular energy E sub I.0905

The energy distribution looks like this.0937

The energy distribution, we have the number of systems in the ensemble and0944

we have a particular energy of the ensemble.0965

If there are N sub one systems that have energy 1, we might have N sub 2 of the systems, we might have energy 2.0968

We have N sub 3 of the systems are in energy 3 and so on.0977

N = the sum of all of these N sub I.0984

If I add all of the N, I'm going to get the total number of systems in the ensemble.0991

Let us say I have 6.02 × 10²³ ensemble, it does not really matter.0995

It is a really large number 500,000, whatever.1000

If 10,000 of them are energy 1, if 50,000 of them of the systems are in energy 2, that is all it says.1003

There is a distribution of energies.1009

Now, the probability.1012

The probability of finding a system in a system with the energy E sub I, the basic probability,1020

you take a number of times of something can happen over the total number.1038

What is the probability of rolling a 5 when you roll a single dice?1045

There is only one way to get a 5, you roll a 5.1049

How many different possibilities are there when you can roll 1 through 6?1052

Yes, you have 6 possibilities that you can roll but only one way to roll a 5.1057

The probability of rolling a 5 is 1/ 6.1061

You remember this from algebra class.1064

The probability of finding a system with energy E sub I is symbolized by P sub I,1067

it = the number of states in that energy with that energy divided by the total number of systems.1076

That is it, very simple, very basic.1084

It just = the number of systems having E sub I energy divided by the total number systems.1087

It is the basic definition of probability.1101

Total number of systems.1103

We are going to define Q, the partition function.1108

Q is equal to the sum E to the - β E sub I, where β = 1 / K × T.1118

I’m going to go ahead and put this 1 / TT into the β.1141

Q is actually equal to the sum / I of E to the - E sub I divided by KT.1145

This division is up in the exponent.1157

This whole thing is up in the exponent.1159

In this particular case, T is the temperature in Kelvin, the absolute temperature.1162

K is something called the Boltzmann constant.1167

K is equal to 1.381 × 10⁻²³ and the unit is J/ K.1172

This is the partition function.1188

I will tell you what it is in just a minute.1190

My best advice is just deal with the mathematics.1197

This looks complicated because of the summation symbol, it is not.1200

All you are doing is adding a bunch of terms together.1202

You taking the first energy, the second energy, the third energy, you are dividing it by KT.1205

You are exponentiating it and that is just one term of the sum.1210

If we said out of the first 5 terms for the first 5 energy states, you have 5 terms of the some.1213

That is all it is and I will tell you what the partition function is in just a minute.1218

Once again, we said P sub I was this thing.1223

With respect to the partition function, P sub I is this.1230

It is equal to E ⁻E sub I / KT/ Q.1236

The probability of finding a system in a given energy state is equal to E raised to the energy state1246

divided by KT divided by the sum of all the possible energy states.1257

The part / the whole, the probability.1263

It is a fraction, that is all this is.1267

Let us go ahead and tell you what the partition function actually is.1270

Q, the partition function is a measure.1275

It is a numerical measure of the number of energy states that are accessible to a system or by a system1297

depending on which partition you want to use at a given temperature, at a given T.1327

Let us talk about this.1335

There is always this sense of what is a partition function?1337

I’m still not sure what it really means.1343

This is what a partition function is.1345

Let us also talk about a system with a given set of energies.1347

At a given temperature, let us say there are 100 available energies for a given system.1355

There are 100 energies that could have at a given temperature, let us say only 5 of that energy levels are actually accessible.1366

The partition function is 5.1377

It is very important that we differentiate between accessible and available.1379

You might have, like for an example the rotational states of the diatomic molecule.1384

There is an infinite number of rotational states in a diatomic molecule.1387

Not infinite but a really large number if the molecule flies apart.1390

It will spin faster and faster and faster and faster into different quantum states.1394

J could be 50, 60, 70, 100, 200, but not all of those are accessible.1398

At a given temperature, let us say maybe only 30 of those rotational states are accessible.1405

That is what a partition function tells us.1411

A partition function is going to give you some number.1414

That number gives you roughly the number of states that are accessible to a system at that temperature.1416

As I raise the temperature more of the states become accessible, that is what is happening, that is all that is happened.1423

It is all a partition function is.1430

Again, we have P sub I is equal to E ⁻E sub I / KT all / Q which is equal to E ⁻E sub I/ KT.1433

I think all these exponentials and summations, fractions on top of fractions, it tends to look really intimidating.1455

It is not intimidating, it is just math.1464

Over the sum of the E/ KT.1467

The partition function is just adding up all these energy values and1474

then the probability of finding it in one of those energy values as you take a part / a whole, where Q is equal to this thing.1479

Let us go ahead and write it out again.1488

Q is equal to sum I / E ⁻E sub I / KT.1491

If you have 100 energy states that are accessible, you have 100 terms in that sum.1497

That is a partition function, very important.1504

We sometimes leave θ and write Q = the sum / the index I of E ^- β E sub I.1507

Sometimes, we will just go ahead and leave the β and expressed in terms of that,1534

in order that they do not have to deal with the fraction in the exponent.1538

That is fine, you will see it both ways.1542

Again, where β = 1 / KT.1545

I'm not really going to say more about this β = 1/ KT.1552

If your teacher wants to give you reason for why that is the case, they can but I would say just take it on faith at this point.1556

U, the energy of the system, we said it is the average of the energy.1568

The average of the energy is you add up all the energies and you divide by the number of systems,1576

the energy of the ensemble.1584

The sum / I N sub I E sub I / N.1587

The number of states/ a given energy × the energy itself.1598

Add up all of those and divide by the number of systems in the ensemble.1603

That would give you an average energy.1608

I pulled the N sub I/ n out, N /N sub I.1611

N/ N sub I that is equal to P sub I.1621

N sub I/ N is equal to P sub I.1628

U, which is the average energy is equal to the probability of finding it in a given energy state × the sum of the actual thing.1634

I will go ahead and put this P sub I back in here.1647

U equal to the average energy is equal to sum of the probability of finding1650

the system of the ensemble in a given energy × that energy.1657

That is one of our basic equations.1668

We found an expression for the energy in terms of the energy, in terms of the probability.1671

The probability is a function of the partition function.1676

We have expressed energy in terms of the partition function.1679

We will get better , do not worry about that.1682

In the ensemble, the systems are distributed over the various quantum states.1691

Each specific distribution is called microstate or a complexion of the ensemble.1729

What we mean by this is the following.1763

Let us say I have I have 10 systems, let us say 3 of them are in one energy, 3 of them are in another,1765

3 of them in another, and one of them is in the fourth.1773

That is one distribution, that is one microstate.1775

Let us go to another distribution.1779

What if I have 5 in one, 5 in another, and nothing in the other 3.1782

That is another distribution, that is another microstate.1789

In other words, in microstate is if I have certain number of bins, energy baskets, our certain number of systems,1792

how can I distribute the different energies among those various systems?1806

Each different one is called a microstate.1809

The number of possible microstates is denoted as capital ω.1814

We define the entropy of the ensemble.1835

When we want the entropy of the system, we just divide the entropy by the total number of systems in that ensemble.1845

In other words N, that is it.1850

We are always talking about the ensemble.1852

Anytime we talk about a system, we just take what we have and divide by the number of particles in it,1854

the number of systems in the ensemble.1859

The entropy of the ensemble is, and you have seen this before.1861

Except now, we are talking about ensemble instead of the system.1865

S = K × the natural log rhythm of O.1868

This is the definition of the entropy of an ensemble.1874

We have seen this equation before.1877

We have seen this definition before back when we talked about classical thermodynamics.1879

We talked about entropy first empirically but then we go ahead and gave this statistical definition of entropy.1885

And we talked about what it means, we talked about this idea of complexions, and a number of possible microstates.1892

I'm not going to state too much more about it now.1898

If you want, you can go back to that particular discussion and it will talk a little bit more1900

about what these individual things mean, in any case.1905

The entropy of the system is the entropy of the ensemble divided by N, the number of systems in the ensemble.1910

Therefore, S of the system is equal to S of the ensemble divided by N.1948

S of the ensemble is K × the natlog of this thing called ω divided by N.1959

In order words, to find the entropy of the system that we are dealing with, the system that we are interested in1966

which has happen to have made billions of copies of that system to create an ensemble.1971

In other words, to find the entropy of the system, we need to find this Boltzmann constant, we know.1976

How many systems we have in an ensemble, we need to find LN of ω.1982

ω is defined as N!/ N sub 1! N sub 2! N sub 3!, and so on.1987

To find S of the system, we need to find the natlog of O.2009

That is it, we are just doing some math here, that is it, nothing too crazy.2022

The natlog of ω is the natlog of N!/ what we said, N sub 1! N sub 2!, And so on.2027

That is equal to the natlog of N! – the sum because this is a product.2039

The natlog of N sub I!.2048

After some math, we also designated as math, which I'm not going to go through here.2050

What we get is the natlog of ω is equal to -N × the sum of the I P sub I LN P sub I.2058

S of the system is equal to K LN ω / N.2080

LNO is this thing.2088

We put this thing into there, we end up getting - K × N × the sum / I P sub I LN of P sub I O divided by N.2092

The N cancel and we get an expression for the entropy of the system.2121

The entropy of the system is equal to - K which is Boltzmann constant × the sum / I, the probability of I × the log of the probability sub I.2125

We found an expression for the energy, in terms of the probability.2143

We found an expression for the entropy, in terms of the probability.2146

The probability is expressed in terms of the partition function.2150

We are getting to where we want to go.2154

This is our second major equation.2157

Let us go ahead and rewrite what we have.2163

Our first major equation was U = which is the average energy, which is equal to the sum of the probability sub I × E sub I.2167

And our second major equation which is entropy that is equal to -K × the sum / I, the probability of I.2180

These are our two basic equations that we are going to start with and derive everything else.2190

Again, where P sub I is the probability of finding a system or ensemble.2197

Probability of finding a system in that particular energy state.2209

Or it is also a fraction of the systems in that energy state, that is the best way to think of P sub I.2214

It is a fraction of the systems in the ensemble that are in a given energy state E sub I.2222

If I have a total of 1000 systems in the ensemble and if I have 100 of those systems2230

in given energy state E sub 1, 100/ 1000 that means 10%, 0.10.2238

My P sub I is 0.10.2245

Where P sub I is the fraction of the systems in the ensemble having energy E sub I.2248

If all of these do not make sense, do not worry.2276

Really, do not worry, what matters here are the results.2279

But again, I go through this as a part of your scientific literacy.2281

If you go through this, if you see this, and you go in your book and read it, it will make your book make more sense.2286

I think it works better that way, or perhaps you read your book and you did not quite get it,2293

and now that you are seeing this lecture, it might make more sense.2296

It is just another way of looking at it.2300

We have P sub I is equal to E ⁻E sub I/ KT/ Q, that is one equation that we have.2305

We have an expression for the partition function which is the sum / I/ E ⁻E sub I/ KT,2324

very important partition function.2333

We have an expression for the energy U which is the actual average energy.2336

That is equal to the sum of the index I of the P sub I E sub I, the fraction in energy state I × the energy itself.2342

And we have expression for the entropy.2353

Entropy = - K × the sum/ I P sub I LN P sub I.2354

These equations, if these 4 equations all of the thermodynamic properties, all of the thermodynamic quantities,2362

all the thermodynamic functions can be expressed in terms of Q.2387

In terms of Q, all we need is this, this, the energy and the entropy and2398

we can express all the other thermodynamic functions in terms of this thing we call the partition function.2403

Partition function, very important.2409

Let us start first of all with Q, let us start with that equation.2415

Q = the sum of E ⁻E sub I/ KT.2419

We differentiate with respect to, I will go ahead and differentiate with respect to T.2428

Therefore, DQ DT and we will hold volume constant.2442

When you take the derivative of this, you get 1 / KT² × the sum I E sub I E ⁻E sub I/ KT.2453

We took the derivative of Q with respect to T.2476

Let us go ahead and go over here.2479

P sub I is equal to E ⁻E sub I/ KT/ Q which means that if I multiply Q which means that π × Q is equal to E ⁻E sub I/ KT.2483

It is just mathematical manipulation.2502

If I put this back into the other equation, if I do KT² × DQ DT under constant V,2504

that is going to equal this sum E sub I P sub I × Q = Q × the sum of the E sub I P sub I.2517

This is U and U = that, the sum/ I of P sub I E sub I.2547

KT² DQ DT is equal to Q × U.2571

I solve for U.2582

U is equal to KT² / Q DQ DT V, which is the same as if I take, instead of taking the derivative of Q2585

with respect to T, if I take the derivative LN Q.2602

The derivative of LN Q is 1 / Q DQ DT, I get the following.2606

I get KT² D LN Q DT constant V.2611

I have an expression for U directly in terms of Q or LN Q.2624

In this last part because D DT or LN Q = 1 / Q DQ DT, this is energy in terms of the partition function.2628

We have our first part.2654

Now, the entropy of the system = -K × the sum of P sub I LN P sub I.2655

We said that P sub I is equal to the E ⁻E sub I/ KT/ Q.2671

LN of P sub I = -E sub I/ KT – LN Q.2681

LN P sub I, if I take this thing and put it into here.2702

Therefore, S is equal to -K × the sum P sub I - E sub I/ KT – LN Q.2715

I get S = -K × -1 / KT, the sum P sub I E sub I – LN Q × the sum of the P sub I.2735

Therefore, S is equal to 1 / T × the sum / I of the P sub I E sub I + K × LN Q.2758

And this is because this thing is actually equal to 1.2779

The sum of the probabilities, the sum of all of fractions always = 1.2783

S = this is U.2788

U / T + K LN Q.2793

We already found U, U equal to KT² D LN Q DT constant V.2804

Let me go ahead and put this in for that and when we do, we end up with S2819

is equal to KT D LN Q DT under constant volume + K LN Q.2825

We found an expression for entropy directly in terms of the partition function.2839

Very nice.2847

Let us see, with energy, entropy, temperature, and volume, all of the other thermodynamic properties can be derived.2851

All of the other thermal properties can be derived.2880

Let us begin with, let us go back to black.2895

Let us begin with Helmholtz A = U - TS.2902

This is the definition of the Helmholtz energy.2907

We have an expression for U and we have an expression for S.2910

This is equal to KT² D LN Q DT under constant V - T LN Q - KT² D LN Q DT constant V.2913

The Helmholtz energy is equal to -KT LN Q.2947

There you go, that is an expression for the Helmholtz energy.2960

That one was reasonably straightforward.2966

I just put the value of U and S in here and solve, and I end up with this.2968

One of the fundamental questions of thermodynamics,2973

if you remember from towards the end of the classical thermodynamics portion of the course.2979

One of the fundamental equations of thermodynamics says DA is equal to - S DT - P DV.2985

That means P, let me go to red, P is equal to - DA DV under constant temperature.3010

That is what this says, this is the total differential equation.3029

This P is just the partial derivative of this with respect to this variable.3034

That is it, because the DA DV × the DV.3040

The DV DV cancel, you are left with the A.3043

That is what this means.3048

S would be partial of A of DA DT, holding V constant.3048

Therefore, P is equal to - DDV constant T of A.3060

A was this, - KT LN Q.3075

Therefore, the pressure of the system is equal to KT D LN Q DV holding temperature constant.3082

That is quite extraordinary.3095

You are just knocking out all these thermodynamic expressions in terms of partition function.3099

Let us go ahead and do the enthalpy of the system.3106

The enthalpy of the system is defined as the energy + the pressure × the volume.3109

We are using these or sometimes look exactly alike.3119

I will just put them in, we have expressions for these.3122

We have KT² DLN Q DT constant V + the P which we said was KT D LN Q DV constant T × V.3124

Therefore, this is equal to KT × T D LN Q DT under constant V + V × D LN Q DT constant T.3146

Very beautiful, absolutely stunningly beautiful.3167

That is enthalpy.3172

Let us go ahead and go to Gibb’s free energy which is the most important for chemists.3176

K = U + PV - TS.3184

If we put all of these all in, I have it here, I will just write it all out.3192

U was KT² D LN Q DT under constant volume + KT × D LN Q DV at constant temperature3198

× V - T × K LN Q + KT × D LN Q DT under constant V.3214

This is equal to, when I multiply this, when I multiply that and add some terms,3235

I end up with G = KT D LN Q DV under constant temperature × V - LN Q.3240

This gives me an expression for the Gibb’s free energy.3263

Heat capacity is very important.3270

The heat capacity is the partial derivative of the energy with respect to temperature under constant volume.3275

We have an expression for the energy of the system.3284

We have got DDT constant volume of this expression which is KT² D LN Q DT constant volume.3288

This is going to end up equaling K × T² D² LN Q DT² under constant volume + D LN Q DT under constant volume × QT.3306

Therefore, the direct expression for the constant volume heat capacity is KT × T D² LN Q DT² under constant volume + 2 × D LN Q DT constant volume.3328

This is a direct expression for the constant volume heat capacity.3352

Normally, what we would be doing is we are going to be finding an expression for the energy.3357

I'm just taking the partial derivative of that with respect to temperature directly.3360

We are not going to be using this expression.3364

And again, the almost important of the thermodynamic functions,3367

we are mostly to be concerned with the energy and the constant volume heat capacity.3370

Occasionally, we will deal with pressure.3374

If for any reason you need to go to the other thermodynamic functions, that is fine.3376

But again, this is an overview.3380

I wanted to show you what our big picture goal is.3381

When we get the expression for the energy in the individual cases, we will just differentiate with respect to T.3384

We will write that down.3390

In general, we will be concerned with U and CV.3393

We will normally find an expression for U then differentiate directly, and differentiate with respect to T directly.3419

We have actually done it, we have done what we are set out to do.3446

Let me go ahead and go to blue here.3451

We have done it, thermodynamic properties expressed in terms of Q or LN Q.3454

Thermodynamic properties expressed in terms of Q.3471

Q, by definition is related to the energy states of the system E sub I.3488

These are the E sub I.3517

The E sub I are related to the energies of the individual particles making up the system, the small E sub I.3521

E sub I are related to the energies of the particles making up the system.3533

Let us say that, of the particles making up the system, the small E sub I.3550

We expressed these thermodynamic functions in terms of the partition function of the system.3564

Now, we are going to express it in terms of the partition function of the individual particles, the molecular partition function.3571

We will now express Q in terms of Q, in terms of small q, the molecular partition function.3583

Because we have expressions in terms of Q, if we have an expression for Q in terms of small q,3595

we put that in wherever we see a Q and we have an expression for thermodynamic properties in terms of the small q.3600

The molecular partition function.3608

The E sub I that we talked about above that is made up of the energies of the individual particles,3623

E sub 1 + E sub 2 + E sub 3, and so on.3631

The energy of the system is the sum of the energies of the individual particles.3640

The partition function of the system is therefore going to be a product of the partition functions of the individual particles.3648

That is how this works, some product.3658

That is the whole idea behind the log, the exponential.3661

That is why the log shows up in these problems.3665

Let us say this again.3671

The partition function of the system Q can be written as the product of the partition functions for each particle in the system.3675

The molecular partition function is Q.3719

For indistinguishable particles which is going to be pretty much all that we talk about when we have a liter of nitrogen gas.3724

You can tell one molecule of nitrogen gas from another molecule of nitrogen gas.3732

For indistinguishable particles, Q is equal to Q ⁺N!, this is the expression.3736

If I have N particles, 6.82 × 10²³, I find a partition function for each particle.3762

I raise that partition function to the nth power, I divide by N!, that would be give me the partition function of the system.3769

That is what this is, very very important equation right here, for indistinguishable particles.3778

Is there another expression for the distinguishable particles?3784

Yes, it is just that without the denominator.3787

And if your teacher feels like discussing that, if it comes in a problem, we will deal with it then, not a problem.3789

But for the most part, it is going to be indistinguishable particles.3795

N is the number of particles in the system.3801

Let us see what we have got here.3805

N is the number of particles in the system.3808

Q, the small partition function, it is the same definition as Q except now we use the individual energies not of the system.3823

The individual energies of the particles, the atoms, and molecules.3832

It is going to be the sum / the index I of E ⁻I E sub I/ KT.3837

We are taking the sum / quantum states.3848

You will see a minute in the quantum levels.3855

If I have a diatomic nitrogen molecule and I want to find the partition function of its vibrational partition function.3857

The vibrational partition function as you know, first quantum state R = 0, R =1, R =2, R =3, those are the different quantum states.3865

Each one has an energy, I put those in here for the E sub I and add it all up.3877

That is how get my vibrational partition function for that molecule, for vibration.3883

If I want the partition function for rotation, there is a difference of energies.3890

If I want one for translation, it is a different set of energies.3894

We will get to that in subsequent lessons.3897

The molecular partition function is exactly the same as what is qualitatively is this.3900

The molecular partition function is a measure, it is a numerical measure.3909

It actually gives you the number of states that are accessible.3922

A partition function is a measure of the number of quantum states,3926

the number of energy states that are accessible to the particle at a given temperature.3940

Let us say there are 250 vibrational states available for carbon monoxide that are available.3962

At a given temperature, let us say 300 K, or I just say 298 K, room temperature.3970

Let us say that only 5 of those states are accessible.3977

In other words, the molecule does not have enough energy to get to the 50th state of the 49th state, or the 10th state.3981

It only has enough energy to occupy state 1, 2, 3, 4, 5, two different degrees.3989

Most the molecules might be in state 1 and 2, and maybe a couple in 3, 4, 5.3995

But at a given temperature, it just cannot vibrate anymore than that.3999

They are the states that are available at a given temperature, this is what is accessible.4003

That is what the molecular partition function does when you calculate this, when you actually get a number like 3.4.4008

That is telling you that at that temperature, there is really mostly about 3.4 states that are accessible.4015

If in a 3.4 that means that the 4th state is, there is a couple of particles in that state and maybe even in the 5th.4024

But in general, it is going to be the 1st, 2nd, 3rd.4032

That said, that is all the partition function is.4035

It is a numerical measure of the number of quantum states that are accessible to a particle at a given temperature.4037

Accessible not available.4043

Quantum states can be degenerate, as we now.4047

For example, the rotational degeneracy is 2J +1.4050

The degeneracy is the number of quantum states that have that particular energy.4056

The degeneracy of 5 for a given level means that 5 different quantum states have that same energy.4062

Quantum states can be degenerate.4069

In other words, have the same energy E sub I.4083

If we include degeneracy in our definition of the molecular partition function, we get the following.4097

This is the one that we are going to be using.4102

Including degeneracy, our molecular partition functions as follows.4105

Q is equal to the sum, the index I G sub I, the degeneracy × E ⁻E sub I/ KT.4115

Our sum is over the energy levels.4128

Our sum is over energy levels not states.4136

This degeneracy takes care of all the states.4150

Q, we said is equal to Q ⁺nth/ N!.4157

In the expressions for the thermodynamic functions, we used LN Q not Q.4169

Therefore, let us take the log of this and see what we get.4198

LN Q is equal to LN of Q ⁺N/ N!.4213

That is equal to N LN Q - LN N!.4222

By sterling's formula, we have LN of N! is actually equal to N LN N – N.4231

We have an expression for this so we can substitute back into that.4250

LN³ is equal to N LN³ - this thing N LN of N – N.4256

Therefore, we have LN Q = N LN – N LN + N.4269

LN Q, we can express Q in terms of q.4281

Using this expression for LN Q, we put it back into the expressions for the thermodynamic functions4287

and we have our thermodynamic functions now in terms of q.4295

Using this expression for LN Q, we can now express all of the thermodynamic functions in terms of LN q.4301

Our connection is complete.4338

We had the thermodynamic properties, a classical thermodynamic properties.4350

The bulk properties of a system that we developed empirically back in the 19th century.4358

We related that to the partition function of the system Q and related that to the partition function, the particles.4366

The circle is closed.4377

We began with classical thermodynamics, we went on to quantum mechanics.4378

Quantum mechanics deals with particles.4383

We have this thing called partition function.4385

We can use properties of the particles to express the thermodynamic properties of the bulk system.4388

The circle for physical chemistry is closed.4395

Let us go ahead and do an example here so that we see.4399

An example of a thermodynamic property in terms of q, in terms of LN q.4404

Let us go ahead and talk about energy.4423

Energy is equal to KT² D LN Q DT V.4425

We said that LN Q is equal to N LN q - N LN N + N.4434

We put this expression into here.4447

We get U is equal to K × T² × D DT under constant V of N LN Q – N LN N.4450

Everything is basically in dropout, when you take the derivative, these are constants.4469

We take the derivative of them with respect to temperature, they just going to go to 0.4472

What you end up with here is N KT² D DT of LN Q.4476

We will just leave it as D LN Q DT constant V + 0 + 0.4484

Therefore, the energy of the system is equal to the number of particles in the system × K × T² ×4494

the temperature derivative of the natlog of the molecular partition function, holding volume constant.4505

There you go, that is it.4512

Thank you so much for joining us here at www.educator.com.4516

We will see you next time for a continuation of statistical thermodynamics.4518

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