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Raffi Hovasapian

Raffi Hovasapian

Energy & the First Law III

Slide Duration:

Table of Contents

I. Classical Thermodynamics Preliminaries
The Ideal Gas Law

46m 5s

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

46m 2s

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

1h 6m 45s

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

2 answers

Last reply by: Joyce Ferreira
Wed Oct 25, 2017 12:42 PM

Post by Joyce Ferreira on October 22, 2017

Dear prof. Hovasapian,

I was asked to contrast a reversible and irreversible process in both physical and thermodynamics terms. I did not not how to explain it in physical terms. What would be the difference, please?

Thank You In Advance.

Best Regards.

1 answer

Last reply by: Professor Hovasapian
Mon Sep 7, 2015 6:52 AM

Post by Shukree AbdulRashed on September 6, 2015

Just to be clear, with all of these graphs, we're evaluating the work done from the perspective of the system (the gas under the piston)? Thank you.

1 answer

Last reply by: Professor Hovasapian
Tue Jun 23, 2015 1:39 AM

Post by Jinhai Zhang on June 22, 2015

Hello, Professor! In the text you referenced, it has the example of van der waals derivation for reversible isothermal work, and the book it use a - sign in front of the integral, and the answer it derived was -nRTIn(...)...
why it has an negative sign in front, and our discussion has no negative sign in front. is that the textbook you suggested made a mistake?

Energy & the First Law III

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

  • Intro 0:00
  • Compression 0:20
    • Compression Overview
    • Single-stage compression vs. 2-stage Compression
    • Multi-stage Compression
  • Example I: Compression 14:47
    • Example 1: Single-stage Compression
    • Example 1: 2-stage Compression
    • Example 1: Absolute Minimum
  • More on Compression 32:55
    • Isothermal Expansion & Compression
    • External & Internal Pressure of the System
  • Reversible & Irreversible Processes 37:32
    • Process 1: Overview
    • Process 2: Overview
    • Process 1: Analysis
    • Process 2: Analysis
    • Reversible Process
    • Isothermal Expansion and Compression
  • Example II: Reversible Isothermal Compression of a Van der Waals Gas 58:10
    • Example 2: Reversible Isothermal Compression of a Van der Waals Gas

Transcription: Energy & the First Law III

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

Today, we are going to continue our discussion of energy and the first law.0005

In the lesson, we have talked about the expansion of the gas.0008

Today, we are going to talk about the contraction of a gas + some other elements of work and things like that.0011

Let us go ahead and jump right on in.0018

Compression, as it turns out is going to be exactly the same as expansion in terms of the mathematics, the same equations apply.0022

Let us go ahead and recall what these equations are.0028

Let me go to blue and see here.0031

Compression, the same equations apply.0035

If we are expanding a gas under a constant external pressure, the amount of work or the numerical value is the external pressure × the change in volume.0042

If we have lots of differential elements, if in fact the pressure is not exactly constant throughout the whole stage,0055

if the pressure changes that is constant over a certain differential elements, we have this one which is the general expression v1 to v2, P external dv.0066

When we replace, when we take the differential limit as we go to more stages, we get to a point0080

where we are actually doing the expansion isothermally right along the isotherm.0087

We are not just doing it in 1 stage, 2 stages, or 50 stages.0093

We are actually following the isotherm when we do the expansion.0097

In that case, the work is going to be v1 to v2, the pressure dv, where we have actually replaced the external pressure0101

with the pressure of the system, the internal pressure of the system.0110

At that point, we are following the isotherm, the external pressure and the pressure of the system differ only by infinitesimal amount.0114

They are essentially equilibrium which is why we justified replacing this with this.0121

This is not the same and this is different.0126

This is external pressure, this pressure is the internal pressure of the system.0128

Let us go ahead and take a look at what a single stage compression looks like, the single stage compression.0136

We are taking a gas at a certain volume and we are squeezing it and making the volume smaller.0149

The final volume is going to be less than the initial volume.0155

A single stage compression and here I’m going to go ahead and draw the Pv diagram for a 2 stage compression so we can actually compare them.0159

We already have the background of expansion, the compression is exactly the same.0171

It is just reversed.0175

The paths says, underneath the isotherm that we are going to be above the isotherm.0177

It is going to look something like this.0182

We have an isotherm, in this particular case, because we are doing the compression this is our initial state down here.0186

It is a higher volume v2 v1.0194

We are going to be going from this initial state I’m going to call s1 and s2.0198

This is going to be P1 this is going to be P2.0211

We start at a certain volume and pressure.0218

The external pressure has to be at least as large as the final pressure in order for the compression to take place.0224

We know the external pressure has to be bigger than the pressure of the system.0232

Otherwise, we would not be able to compress it.0234

The external pressure is bigger than the internal pressure, we have to compress it.0237

It has to be at least as big as this.0241

The external pressure that way, the smallest value of external pressure is going to be that.0243

What happens is this, we compress the gas, the volume, and the volume gets smaller.0249

As it gets smaller, the pressure rises and rises until it gets to a point where the pressure exactly matches the external pressure.0259

The pressure inside the system matches the external pressure.0270

The total work done is that right there, work =the external pressure × the change in volume or0275

the external pressure is here with this value, that is the height.0284

The change in volume is this value, so the work done during this compression, the amount of work that I have to do to compress this gas isothermally0291

under constant pressure is the area underneath that graph.0300

The same compression but let us do it in 2 stages.0306

This was our first path single state compression.0315

We have the isotherm, we have state 1, we have state 2, this volume 1, this is volume 2, this is pressure 1, this is pressure 2.0321

This time I'm going to compress it, I’m going to use a pressure that is going to be bigger than P1 but less than P2.0332

My first external pressure is going to be somewhere around there.0341

I'm going to go along this path and I'm going to compress the gas until I hit the volume decreases.0346

As the volume decreases, we know the pressure of the system is going to rise until the pressure hits that external value of pressure that is going to stop there.0357

I’m going to put more pressure and going to raise the external pressure to P2.0365

I’m going to squeeze it some more until the external pressure now matches or the internal pressure matches the external pressure.0372

The work that I have done is going to equal this first stage.0385

In this 2 stage compression, the amount of work that I have done is the area underneath here, the shaded area.0408

Clearly, the work done here is more than the work done here by that amount.0416

That is the difference.0427

The total work here is the work done in stage 1 + work done in stage 2.0431

This is the work in stage 1, this is a work in stage 2.0438

These are compressions under constant external pressures.0443

In this case, it was 1 external pressure.0461

In this case, it was 2 external pressures.0462

First here, and then here.0466

Clearly, these 2 stage compression requires less work.0471

We do not need the numbers, we can see it just from the area.0489

This has a lower area than this so it requires less work if I do this compression in 2 stages I do not have to work as hard to actually compress the gas.0492

Less work than 1 stage.0506

You can see where this is going, a multistage compression 50, 100, 150.0521

If I keep going it is going to require less and less and less work.0528

A multistage compression would require yet less work.0540

A multistage just turning into a calculus course.0566

A multistage compression from a higher volume to a lower volume.0575

This is volume, this is pressure, it would require less work that we do to compress the gas.0580

As we pass to the differential limit, smaller and smaller changes in volume, we find there is a minimum amount of work that0596

the surroundings must do to compress the system from an initial state to a final state.0632

This is the initial state, this is the final state, there is a minimum amount of work.0653

It is work, less work, the more stages I do the compression in, the less work I have to do.0658

At some point, the work that I do is the area underneath the graph.0664

Underneath all of these, it pass to the differential limit that work is going to achieve a minimum value.0670

It is going to be some value below which I cannot go.0678

This minimum work is achieved if we compress the gas isothermally.0683

If we actually go along the isotherm, we actually go this way.0715

Again, if you keep taking, you know this already from calculus along the isotherm.0721

We are not going to take that path or that path or this path.0732

We are going to take straight, we are going to pass right along the isotherm.0739

In this particular case, what you get is work.0745

Volume 1 volume 2, Pdv, Ihave replaced external pressure with the internal pressure of the system because0751

as we pass along the isotherm, the difference between the external pressure and the internal pressure of the system is infinitesimal.0760

It is so small that it really does not matter.0769

Therefore, I can replace the external pressure with the internal pressure.0775

They are essentially in equilibrium at all times along this path where we have replaced external pressure with the pressure of the system.0778

The pressure, internal pressure of the system itself, we do not put the subscript here but this is essentially where it is,0800

With the pressure of the system at any given moment along the compression.0806

This is the exact same thing as along the expansion.0818

When we do the isothermal expansion actually along the isotherm and that is the path we take, we get the maximum amount of work that a gas can do as it expands.0822

If we do an isothermal compression along the actual isotherm itself, we get the minimum of amount of work0834

that we have to do that the system of the surroundings have to do on the system in order to compress that gas.0844

You can do a lot of work to compress a gas or you can do the minimum lot of work to compress a gas.0853

You can do a certain amount of work.0859

When the gas expands, it can do a certain amount of work but the maximum amount of work that I can do is done when it expands along the isotherm itself.0862

That is what is going on here.0873

Let us see here, let us go ahead and take a look at an example and see if we can make sense of some of this numerically.0881

A little long but it is very important, a lot of things are going to be dealt with here.0892

An ideal gas occupies 4 L at 2 atm and 25° C.0897

It is compressed to a volume of 1 L, calculate the minimum work needed to compress the gas isothermally by a single stage compression at constant pressure.0906

Here is the crucial, then calculate a minimum work required to compress it isothermally in a two stage compression.0918

First to a volume of 2 L, then from 2 L to 1 L, the initial status 4 L, the final state is 1 L.0927

It is a compression.0936

Single stage we are just going from 4 to 1 directly under constant pressure.0937

In a 2 stage, we are going to go from 4 to 2 and 2 to 1, under 2 different constant pressures.0941

Finally, calculate the absolute minimum.0948

The absolute minimum work required to affect the same change of state.0952

Let us go ahead and draw the Pv diagram to help us out here.0959

We are going to go like this, we have an initial state, we have a final state.0967

The initial volume is going to be 4 L and the final volume is going to be 1 L.0974

We said that the initial pressure is 2.0 atm and this is going to be our final pressure which we do not know yet.0982

Notice that they did not give it to us here.0993

I would have to look calculate that, that is not a problem, p1 and v1, p2 and v2.0995

We already know how to do that.0999

Let us see what we can do, a single stage compression.1004

Isothermally just means that the temperature stays the same, the minimum amount of work needed to compress the gas isothermally by a single stage compression.1010

In order to compress this gas from this state to this state, from 4 L to 1 L.1021

The external pressure has to be bigger than 2 to be able to stop the compression.1028

In order to reach the final pressure that was suppose to reach, a volume of 1 L whatever that pressure happens to be, it has to be at least as big as the pressure.1035

A single stage, the external pressure has to be at least as big as a final pressure.1046

For a single stage, we have the following.1062

The minimum work needed means that the external pressure has to be at least as large as the final pressure.1071

Let us go ahead and find out what the final pressure has to be.1104

P1v1 =P2v2, we are looking for P2.1109

Therefore, P2 = P1v1 / v2.1114

The initial pressure is 2 atm, the initial volume is 4 L, the final volume is 1 L.1122

Therefore, our final pressure has to be 8 atm.1137

That means the external pressure has to be at least 8 atm in order to affect that single stage compression.1142

The external pressure has to be there in order to do this compression.1157

Now that we know what the external pressure is 8 atm, let us go ahead and calculate the work for that.1162

Therefore, our work is equal to our external pressure × change in volume.1170

Our external pressure is 8 atm and our change in volume final – initial, 1 L -4 L=8 atm × -3 L =-24 L/ atm.1177

I’m not going to go ahead and convert it to joules, it is the numerical value that matters.1199

We have -24 L atm.1203

Let us go ahead and do this in 2 stages.1208

A 2 stage compression, for the 2 stage compression let us go ahead and draw that over here.1214

The 2 stage compression is going to look like this.1225

We have this and we said this is 4 and this is 2 atm, this is 4.0 L, this is going to be a Pf.1229

That is our final pressure, it is not a problem.1244

A 2 stage it is going to go from 4 L to 2 L and that is going to go from 2 L to 1 L.1250

Let us go ahead and put the 1 L here and let us put the 2 L here.1256

It is going to look like this.1262

That path it is going to take is going that way.1271

It is going to compress until it reaches a certain pressure inside and then we are going to have a new pressure.1273

It is going to be up here and it is going to compress some more.1281

The work done is going to be that right there.1285

The work total =work 1 + work 2.1288

The area underneath this rectangle, that is going to be work 1, the area underneath this rectangle that is going to be work 2.1298

This is going to equal the external pressure × the change in volume for the first stage and the external pressure and the change in volume for the second stage.1306

Let us go ahead and find out what these are, put it in, and see if we can come up with some numerical value for that.1319

For stage 1, it looks like this.1326

Stage 1, I need to find a certain pressure that is going to be bigger than 2 atm but is going to be less than the actual final pressure that we want,1337

The pressure that accompanies this 1 L.1345

It has to be somewhere in between here, in order to bring it here.1348

It has to be that value, the value of the pressure for the first stage has to match the pressure along the isotherm that takes us to 2 L.1351

Stage 1, we have P1v1 =P2v2, therefore, P2 =P1v1/ v2.1363

The initial is 2.0 atm, I’m going to go ahead and use my units here.1385

The initial pressure is 2 atm, the initial volume is 4 L, the final volume in the first stage is going to be 2 L.1393

I'm left with 4.0 atm, this is going to be my external pressure for stage 1.1407

The change in volume = 2 L -4 L =- 2 L.1419

I have the first stage, I have an external of 4 atm, I have a change in volume of 2 L.1444

I will do the actual multiplication of the n.1453

Let us do stage 2, for stage 2 my external pressure that is equal the external pressure from the first part, the single stage expansion.1455

We are still going from an initial state to a final state.1472

The final state is 1 L, that 1 L that is just the final state.1475

We already calculated what the pressure is going to be, that is going to be 8 atm.1479

Because now we need the pressure to be up here, in order to affect this compression, to go from 2 L to 1 L.1491

As we go from 2 L to 1 L, the pressure is going to rise until the pressure reaches whatever pressure this is for 1 L, which happens to be 8 atm.1501

The δ v of the second stage is equal to 1.0 -2.0 =s-1.0 L so we have that and we have that.1513

We can go ahead and calculate our work.1529

so the total work for this 2 stage is going to be work 1 + work 2.1533

Work 1 = the external pressure for stage 1 is 4 atm and the change in volume is -2.0 L.1544

This is going to be 8.0 atm × -1.0 L =-8 + -8 =-16.0 L atm.1556

Notice this is less than the single stage.1575

The single stage is -24 L atm.1579

In other words, the surroundings did 24 L atm of work on the system.1584

If we do it in 2 stages, the surroundings does 16 L atm.1590

It is dropping, it is confirming everything.1595

Let us go ahead and calculate the absolute minimum.1598

The absolute minimum is achieved, this is expansion along the isotherm itself, not this way or this way but along the isotherm itself.1603

This is expansion along the isotherm and that work is equal to the integral from the initial volume of Pdv.1623

This is an ideal gas.1644

We will put that in there, work =v1v2 nRT / vdv = when I do this integration and our T is a constant, I pull it out.1657

I have to do the integration in a previous lesson or we end up with the following nRT × log of v2 / v1.1675

Let us see what we have here, how can we solve this?1694

A couple of ways that we can do this, I think I’m actually going to do both ways.1700

We know what R is, we know what T is, we need to find what n is.1712

We already know what the final volume is 1, we know what the initial volume is 4 L.1717

We need to worry about the moles.1722

Let us go ahead and find n, Pv =nRT.1725

If we rearrange this to solve for n, I get n =Pv/RT.1728

The initial pressure is 2.0 atm, the initial volume is.1754

This is fine, it is not a problem.1769

We have the work equal to this, we do the integration, this is what we want to solve.1771

We have v1, v2.1774

We have T which is 25℃, the 298 K, we have R which is the 0.08206 if we are dealing with L atm, we need to find the number of moles.1775

For Pv =nRT, n =Pv/ RT.1789

In terms of Pv, we can either use the initial pressure, the initial volume or the final pressure and the final volume.1795

It actually does not matter what it is.1802

If you happen to notice, you do not have to go through this to find what n is.1805

Let us go back to Pv =nRT, nRT = the pressure × the volume of the system.1811

The pressure × the volume P1v1 =P2v2.1823

It does not matter, it is equal to P1v1 or I can use P2v2, it does not matter.1827

This nRT, this value right here, I can just take P1 × v1 if I need to.1834

If I want to do P2v2, that is not a problem.1841

In this particular case, nRT = let us use P1v1.1846

The initial pressure was 2.0 atm and the initial volume was 4.0 L.1851

Therefore, nRT = 8 L atm.1862

The work is equal to 8 L atm × nat log.1872

The final volume was 1 L, the initial volume was 4 L, when I do this calculation I get -11.1 L atm.1888

The single stage compression took 24 L atm of work.1906

A 2 stage compression took 16 L atm of work.1912

I can take more stages 3, 10, 50, 100 this number is going to keep dropping.1917

At a certain point, it is going to hit a minimum.1922

That minimum is the area underneath the isotherm from volume 1 to volume 2.1924

The minimum amount of work that I have to do in order to compress this gas is 11.1 L atm.1932

Notice, because the final volume is less than the initial volume, the sign automatically takes care of itself.1939

This -11.1, the amount of work that is done is 11.1.1947

It is negative, it is a work done by the surroundings on the system.1953

That means energy is transferring as a work from the surroundings to the system.1957

This is the minimum amount of work that I have to do with the surroundings has to do in order to compress this gas.1963

Let us see what we have got here.1973

A gas can expand or be compressed isothermally along many paths.1978

When we see that they are doing expansion or a compression isothermally, that does not mean we are doing it along the isotherm.2004

We can do it along any path.2011

Isothermal just means that we are keeping the temperature the same as it moves from one state to the next, initial to final or any intermediate stage.2013

An isothermal expansion or compression is not the same as an isothermal expansion or compression along the isotherm.2024

When the isotherm itself is the path, that represents a maximum or minimum quantity, that is the difference.2032

When we say isothermal, isothermal just means that it means they are keeping the temperature the same.2039

You can follow any path you want.2044

When we say isothermally along the isotherm, that represents the absolute minimum and absolute maximum quantities.2046

Gas can expand or compressed isothermally along many paths.2058

The path that follows the isotherm is the one that gives the maximum work done during expansion.2065

It also gives the minimum work required for compression.2105

In both cases, the numerical value of the work done is = to the integral from the initial volume to the final volume of the pressure × the differential volume element,2125

Where P is the pressure of the system at any point along the transition.2146

Along the isotherm, if the path that we actually take is along the isotherm, the external pressure and the internal pressure of the system may differ infinitesimally.2164

We are justified in replacing P external by P itself.2212

It is very important.2227

Let us go ahead and talk about reversible and irreversible processes.2233

You are going to hear the word reversible used all the time throughout thermodynamics.2238

Let us have a change of pace, let us go ahead and try red.2248

Reversible and irreversible processes.2257

We are going to subject gas and we are going to subject it to a cyclic process.2270

We are going to start at an initial state.2276

We are going to take it to a final state.2278

We are going to do something else would bring it back to the initial state.2280

We are going to do it in two different ways.2283

We are going to have some initial state s sub I, which is going to be pressure 1 volume 1.2288

It is going to be isothermal so the temperature is going to stay the same.2297

We are going to have s2 or s final and this is going to be a pressure 2 volume 2 and T.2300

What we are going to do is we are going to get 2 stages.2311

We are going to start at stage 1 and go ahead and expand that gas.2313

We are going to expand it to a final state.2319

What we are going to do is we are going to compress it.2321

Under compression, we are going to bring it back to its initial state.2326

We are going to do it in 2 different ways.2329

Process 1, we are going to do it as a single stage expansion with the external pressure = P2 and we are going to do a single stage compression.2339

We are going to bring it back with the external pressure = P1.2369

The second process, I will do the same thing.2376

I’m going to start with state 1, we are going to take it to state 2.2381

We are going to take it back to state 1 but this time the path it would take would follow the isotherm itself.2383

The process 2, this is going to be the infinite multistage expansion.2392

We know from calculus or from what we just saw, the infinite multistage expansion up and down, that differential infinitesimal amount,2406

that is the actual path along the isotherm.2413

An infinite multistage expansion and compression then compression along the isotherm.2415

Let us see what we got here.2440

Let us go ahead and look at process 1.2443

Process 1 looks as follows.2447

Let me draw it here because I want some room.2456

This is state 1, this is the initial state, and this is the final state.2467

It is P1 P2 v1 v2, this is v1 v2 P1 P2.2479

A single stage expansion going from here to here in a single stage expansion would look like this.2487

This is going to be the work done by the expansion.2501

A single stage compression, now we go backwards.2505

Single stage compression has to follow this path.2508

The work of the compression =the work under this whole rectangle.2516

The work of the expansion is a work under this rectangle.2521

Notice that the work of the compression is a lot bigger than the work of the expansion.2526

Therefore, the total work = the work of the compression + the work of the expansion = the work of the compression is going to be P1 × v1 - v2.2537

In the compression, this is the final state and this is the initial state.2557

It is going to be v1 - v2.2562

The work of the expansion is going to be P2 × v2 - v1 v2 - v1.2566

I’m going to switch this around, v2 - v1.2575

Therefore, this is going to become -P1 × v2 - v1.2578

Whenever I switch these, I factor out a negative.2583

The factored negative is going to be put there + P2 × v2 - v1.2586

I have v2 – v1 v2 – v1.2593

This is going to = P2 - P1 × v2 - v1.2596

Pressure 2 - pressure 1, pressure 2 - pressure 1, this is negative, v2 - v1 this is positive.2604

What I end up is a negative total work done.2612

The total work done in a cyclical process I start here, I expand the gas in a single stage.2621

The work is this much.2628

I compress the gas through the path, the work is this much.2631

The total difference in work, the total work done actually ends up being negative.2637

Notice, I end up losing that much work.2641

The surroundings ends up having to do that much more work in order to compress the gas,2646

it does not recover the work that was done along the expansion.2650

The total work is negative, mean work has been destroyed in the surroundings.2657

Work is negative that means work has flowed from the surroundings to the system.2677

It is negative, its float from the surroundings to the system.2684

This cyclical process we would expect it to be the same but it is not.2688

You end up doing more work in your compression than you get back in the expansion.2692

Or you end up doing less work in the expansion but it takes more work to actually compress the gas back to where it was before, back to its initial state.2697

the total work of the process is not the same.2707

Work has left the surroundings.2711

There is a weight in the surroundings that is actually lower than it was before.2715

The surroundings has lost work net.2719

Let us go ahead and look at process 2.2727

Now process 2 looks like this.2732

Here is the isotherm, here is the initial state, here is the final state, it is going to be P1, P2, v1, v2.2741

We are going to do this along the isotherm.2755

We are going to expand along the isotherm and we are going to compress back along the isotherm.2758

The work of expansion along the isotherm = the integral from v1 to v2 of Pdv.2768

That is this way.2782

We will go back to our initial state, the work of compression that is equal to v2 v1.2784

We went from v1 to v2, we are going from v2 to v1 of Pdv.2795

It is the same, the internal pressure of the system at any point along the compression of expansion so these are the same.2802

Therefore, the work total is equal to the work of the expansion + the work of the compression is equal to the integral from v1 to v2 of Pdv + the integral from v2 to v1 of Pdv.2810

I’m going to switch this integration and turn this into a negative sign.2832

Equals the integral from v1 to v2 of Pdv - the integral v1 v2 of Pdv =0.2838

A single stage expansion followed by a single stage compression to go from state 1 to state 2, then back to state 1.2854

We ended up with a total loss of work.2861

Work is negative it is 0, the surroundings lost work.2862

That is done isothermally, we kept the temperature the same.2868

In this case, instead of following this path and this path.2873

We follow this path during the expansion and this path during the compression.2881

We see that the work is actually 0.2885

There HAs been no net loss of work by the surroundings.2889

The work is 0, this is rather extraordinary.2893

No net work has been lost.2903

In other words, the system has been restored to its original state which is no different than the first process.2910

That process the system has return to its original state.2931

In its initial state, we find s1 and s2 and back to s1.2935

There is no difference here.2939

In both cases, the system has been restored to its original state and the surroundings have been restored to their original state.2940

The surroundings have been restored to their original state.2955

The work is 0, the surroundings have been restored to their original state.2959

We call this irreversible process.2974

Any process that restores the surroundings to their original state is irreversible process.2981

Any cyclical process that ends up bringing the system back to its original state but where something is not the same in the surroundings, that is an irreversible process.2988

Let us go ahead and write this down.3001

Reversible process, when a system undergoes a change of state by a specific process that is restored to its original state by the same sequence in reverse,3003

then if the surroundings are also restored to their original state, the process is called reversible in both directions.3007

When a system undergoes a change a state by a specific process, goes from an initial state to another state and is restored back to its initial state.3117

It goes from s1 to s2 and goes from s2 to s1 by following the same sequence of steps that it went from s1 and s2 but in reverse.3124

If the surroundings are also restored to their original state, the process is called reversible.3136

If during the change, in going from s1 and s2 then back to s1, if the surroundings are not restored to their original state, the process is irreversible.3144

Reversibility represents an idealization.3155

In this particular case, it represents the expansion along the isotherm.3159

It represents the compression along the isotherm.3164

When that is the case when we actually perform a process on expansion and a compression, when we do it isothermally along the isotherm,3168

there is no net change, there is no loss of work, that is what is important.3178

A reversible processes are important.3183

You saw that we can actually do the expansion and compression isothermally along any path.3186

There are specific paths, mainly one specific path along the isotherm both this way and that way.3192

That makes the process reversible where everything is restored exactly to where it was before.3201

It is as if nothing happened.3206

Reversible processes are important because they represent maximum and minimum quantities or values.3208

We already know before we talk about reversible processes, we talked about how if you travel along the isotherm,3238

you are going to end up getting the maximum amount of work in the expansion of the gas isothermally3245

or the minimum amount of work required for you to actually compress that gas isothermally.3251

We are giving it a name, when you are doing this isothermal expansion reversibly, in this case along the isotherm that is when you achieve your maximum and minimum values.3257

We will go back to blue, to make it look a lot better.3274

The isothermal expansion or compression of a gas along the actual isotherm is the reversible isothermal expansion or compression.3286

As opposed to just an isothermal expansion or compression which can happen along any path in a single stage or in multiple stages.3329

The maximum work obtained by an expanding gas is irreversible isothermal expansion.3367

The minimum work required to compress the gas is the reversible isothermal compression.3402

Again, when we say that something is happening isothermally, all we are saying is that the temperature is being kept constant for the process.3433

We are not saying that you are actually moving along the isotherm.3441

If we mean we want to move along the isotherm, we will say you are explicitly along the isotherm.3445

We will just use the term reversible that is what is going on here.3451

Isothermal can be any path but reversible, isothermal is one path.3455

It is along the isotherm itself.3460

It is what gives you maximum and minimum quantities of work.3462

Of course, net work again is equal to the integral from an initial volume to a final volume of Pdv.3467

Let us go ahead and finish up with an example here.3485

One more page, example number 2, the van der Waal’s equation states is a correction to the ideal gas law to account for deviation from ideal behavior in real gases.3490

The ideal gas law you know is Pv =nRT or something which is a little more instructive nRT/ v.3502

The van der Waal’s gas says that the pressure does equal to nRT =v.3514

There is a pressures that have to be make, P = nRT/ v - nb – an²/ v².3518

Or a and b are constants that have to do with a particular gas in question.3527

We want you to find an expression for the work done in the reversible isothermal compression of a van der Waal’s gas from an initial volume to a final volume.3533

Let us go ahead and see what we can do.3545

We already know the work is equal to initial v final Pdv.3547

We have an expression for P, it is right here.3563

Let us go ahead and put that in there.3566

It is equal to the integral from vi to to vf of nRT/ v - P – an²/ v² dv.3570

Let us go ahead and solve this integral now.3591

We will go ahead and separate that out.3593

The work is equal the, I’m going to integrate this first term right here and our T is a constant.3596

That is v – P, that has to stay in there.3606

It is going to = nRT × the integral of v1 to v2 of dv/ v - b – nb.3609

V – nb – n² ×the integral from v1 to v2 of dv/ v².3624

Dv/ v – nb, this is a logarithm, dv/ v² is v² is the same as v -2 dv.3637

You can do that the way you integrate any other thing, what you get is the following.3646

I will go to the next page here, work =nRT × log of v2 - nb/ v1 - nb – an² -1 / v.3652

v1 to v2 which we will end up with the final expression of nRT × log of a2 - nb/ v1 – nb.3680

It comes out here so you will get +an² 1/ v2 -1/ v1.3695

There you have that.3707

If we are dealing with van der Waal’s gas in a particular situation, we decide to use the van der Waal’s equation instead of the ideal gas equation.3708

If we want to find out what the amount of work done by the isothermal reversible expansion or compression of the van der Waal’s gas is, this is the expression for it.3716

Thank you so much for joining us here at www.educator.com.3731

We will see you next time for further discussion of energy in the first law, bye.3733

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