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Raffi Hovasapian

Raffi Hovasapian

Energy & the First Law II

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 (6)

1 answer

Last reply by: Professor Hovasapian
Thu Sep 14, 2017 4:14 AM

Post by peter alabi on September 12, 2017

Hi professor,
suppose instead of integration as follow [p(v)-Pext] like we did in this lecture, we integrate as follow {Pext-P(v)}? is the area between this function still work? and if so, what does that value means?
Thank for the great lecture, you've helped me passed my general chemistry, biochemistry, and AB calculus, now you've started saving my grade in physical chemistry. thanks a lot!

1 answer

Last reply by: Professor Hovasapian
Mon Feb 8, 2016 1:35 AM

Post by Van Anh Do on January 22, 2016

Hello. I have a question about what you said around 34:30. Why isn't the path for the gas expansion be along the isotherm? I thought as the gas expands in volume, the pressure and volume would follow that curve? Thank you.

1 answer

Last reply by: Professor Hovasapian
Tue Jul 28, 2015 11:24 PM

Post by Ayanna Hogan on July 24, 2015

What would be the are under the entire cure of the P-V diagram ?

Energy & the First Law II

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
  • The First Law of Thermodynamics 0:53
    • The First Law of Thermodynamics
  • 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
    • Systems Possess Energy, Not Heat or Work
    • Scenario 1
    • Scenario 2
    • State Property, Path Properties, and Path Functions
  • Pressure-Volume Work 22:36
    • When a System Changes
    • Gas Expands
    • Gas is Compressed
    • Pressure Volume Diagram: Analyzing Expansion
    • What if We do the Same Expansion in Two Stages?
    • Multistage Expansion
    • General Expression for the Pressure-Volume Work
    • Upper Limit of Isothermal Expansion
    • Expression for the Work Done in an Isothermal Expansion
  • 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

Transcription: Energy & the First Law II

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

We are going to continue our discussion of energy and the first law of gravity.0004

We are going to be spending several lessons on this topic, profoundly important because we are at the beginning of thermodynamics.0007

We definitely want to spend as much time as possible and make sure we understand what is happening at the basic level.0016

Later on, we will tend to move a little bit more quickly but for this topic it is important that you understand everything.0022

we are going to do a ton of example problems.0029

Not necessarily in the individual lessons as I have said before, due to the nature of the material we have to go through a fair amount of theory.0032

There are example problems in these lessons but I do not worry, we are going to do a ton of example problems in blocks,0040

in separate lessons themselves at the end of this particular unit.0048

Let us get started.0051

We are going to start with the first law of thermodynamics.0056

I’m going to state it, the first law says,0060

I think I’m going to go to my favorite color which is blue.0070

There are many statement of the first law, I’m going to give another one later.0085

I’m going to give this one here, it is probably the one that you are familiar with most.0089

It says that the energy of the universe is constant.0100

Basically, you know that energy cannot be created or destroyed, which means that the energy of the universe is constant, it does not change.0115

It moves around from system to surroundings, from surroundings to system, but it states as a whole, energy does not change.0124

We can express it like this, the energy of the universe is equal to the energy of the system + the energy of the surroundings.0133

The energy the system possesses, the energy of surroundings possesses, that is the total energy that stays a constant.0144

It moves but does not change, the total amount.0150

Let us go ahead and see what we can do here.0154

Let us say that energy 1 of the system + energy 1 of the surroundings, or initial or final I’m going to be using initial and final, 1 and 2 interchangeably.0157

The energy of the universe is a constant.0174

Some of the energies is some constant, let us call it C.0176

Let us undergo a change of state.0181

Upon a change of state, now the system has a new energy, the surroundings have a new energy.0186

Energy 2 of the system + energy 2 of the surroundings, it said it is constant so that is also equal to a constant.0194

I’m going to go ahead and subtract the first equation from the second equation and get something like this.0204

E2 of the system – e1 of the system + e2 of the surroundings – e1 of the surroundings is equal to C –C which is equal to 0.0212

Basically, it says e2 – e1 of the system that is a change in energy of the system.0244

E2 –e1 of the surroundings that is a change in energy of the surroundings that is equal to 0.0252

This is a statement of the first law of thermodynamics.0259

The change in energy that the system undergoes is equal to negative the change in energy that the surroundings undergo.0265

It is very simple.0271

If the surroundings loses 30 units of energy, the system has gained 30 units of energy.0272

If the system has lost 50 units of energy, the surroundings have gained 15.0277

Energy transfers at the total amount of energy that is contained is a constant value.0282

This is a state of the first law of thermodynamics.0290

We generally like the law like this, this is the one that you know of from general chemistry.0295

Generally, write the first law from one point of view.0305

Generally, writing from the systems point of view, in chemistry this is how you are accustomed to seeing it.0319

You are accustomed to seeing it like this, the change in energy of the system is equal to q – w.0323

You have probably seen as, the change in energy of the system is equal to q + w.0331

It has to do with conventions.0335

Again, the convention that we are taking is that when the system gains heat, heat is positive.0337

When the surroundings gain work, work is positive.0344

Under those conventions, the statement is actually written like this.0349

We are not going to be using E and we are going to be using e for the most part, we would it be using u for energy.0353

Δ u = q – w, these two ways, this is how you see that you are familiar with the first law of thermodynamics from your general chemistry course.0359

This is what this basically says that, the change in energy that the system undergoes or the surroundings0373

either one is equal to the heat that it gains or loses and the work that it gains or loses.0380

That is all this is saying.0386

Of course, we will have more to say about this in subsequent lessons.0389

This says, when the system undergoes a change of state, the initial energy of the system now there is some final energy of the system,0396

the net change is equal to the energy lost or gained as heat + the energy lost or gained as work.0427

Energy transfers in two ways.0461

It transfers as heat and it transfers as work.0463

When a particular system loses energy, it can lose it either as heat, gain or lose as heat, gain or lose as work.0466

Some of those are going to be the total change in energy of the system.0474

The change of energy of the system is equal to negative change in energy of the surrounding.0478

It is all about perspective and point of view.0483

In general chemistry, the point was not exactly hammered.0486

Basically, presented with this equation and we use to a bunch of problems.0492

A system gains this much energy as heat, loses this much energy as work.0498

What is the change in energy of a system?0502

We are going to do a problem like that in just a minute.0503

It is very important to take a step back, we want to think about things thermodynamically not just chemically.0505

There is a system, there is a surrounding, there is a very deep relationship between the two.0512

You have to make sure you do not forget about the surroundings or you do not forget that there is a system.0515

With that, let us go ahead and do a nice simple example problem just to get a sense of how this is dealt with numerically.0525

Our first example is going to be, a system does 100 J of work on its surroundings.0534

In the process, the system gains 125 J of heat, what is the change in energy of the system?0541

What is the change in energy of the surroundings?0548

Let us go ahead and take a look at what this looks like a diagrammatically.0553

Thermodynamics is like any other science, you want to draw as many pictures as possible, if you can draw a picture.0557

If you have schematics and pictures to fall back on, it will certainly help.0563

Let us see what we have got, I will go ahead and draw it here.0568

Let us go ahead and draw that as our system.0572

I will just go ahead and make a bigger box around it that is going to be our surroundings.0576

This is what the system does 100 J of work on its surroundings.0580

Work is actually leaving the system into the surroundings.0585

It is going to be positive because again that is our convention.0590

Once the work is done on the surroundings, work is positive.0593

Now in the process the system gains 125 J of heat.0600

Heat, the q, that is going to be positive because that is from the systems point of view.0605

When the system gains heat, the heat is positive.0611

Therefore, our q is equal to +125 J and our work is equal to +100 J.0615

We just go ahead and fall back on our equation.0630

The change in energy of the system is equal to q – w.0634

A change in energy of the system is 125 J - 100 J.0639

The change in energy of the system is equal to +25 J.0647

Why are we using this anyway?0656

You just have to look at, you do not have to do this equation.0658

If it gains 125 and it loses 100, it gains 125 is heat and lose 100 of work, there is 25 J left in the system.0661

That is the energy of the system.0671

We know that the δ u of the system is equal to –δ u of the surroundings.0674

Therefore, δ u of the surroundings is equal to -25 J.0679

The system has gained 25 J of energy net, the surroundings has lost 25 J of energy net.0687

When we say that the system has gain 125 J of heat, it has gained it from somewhere.0694

It has gained from the surroundings, that is was going on.0698

That is the transfer of energy.0701

Let us go ahead and then talk a little bit more about heat, work, and energy.0708

We have δ u - q – w.0717

The energy of a system changes in two ways, as a transfer of heat and or a transfer of work.0731

It is very important.0766

In thermodynamics, when we start to getting the problems, when we just start to speak normally about what is going on a particular process,0770

that has to be a little bit of an abusive language.0776

We speak of heat, we speak of work, almost in the same way that we speak of energy.0779

They are not.0785

A system possesses energy, it does not possess heat or work.0787

Heat or work quantities that are transferred only during a change of state.0792

A system in a given state does not have a certain amount heat.0798

It does not have a certain amount of work.0800

When it transfers its energy, either taking in some or giving out some, that transfer manifests as a heat or work.0803

It is very important to keep that in mind.0812

Systems possessed energy, energy not heat or work.0816

To label the point but it tends to fall by the wayside as we precede and this sort of abusive language has an effect on how we think about problems.0829

We want to be very clear which is why I keep we are reiterating it.0840

Work and heat are quantities that are transferred during a change of state as the only time.0849

Remember from the previous lesson, they show up only during a change of state.0871

Consider the following analogy.0884

Let us say, energy is money, energy is dollars.0887

Heat is paper, and work is coin.0904

When you go to the bank, you can deposit or withdraw money.0913

You could deposit or withdraw it as paper money or coin money, that is the thing.0915

Energy is a money, energy is the dollars, and that is what you have, that is what you possess, a certain amount.0920

The coin is like work, you could deposit money as coin, you could deposit it as paper.0930

Heat and work are the paper and the coin, respectively.0939

There are different ways of actually transferring money, of moving money.0943

They are not actually things that the bank account possesses.0948

The bank account has a certain amount of money.0952

The energy is the money, the heat is the paper, the work is the coin, that is generally how it works.0955

Let us go ahead and take a look at quick scenario here.0960

So in the first scenario, I have an initial state, I have $100 in the account.0963

Let us say I deposit $150 as paper and let us say I withdraw $50.00 as coin, then I withdraw another $25.00 as paper.0975

My final state equals $175.1000

The second scenario, my initial state, I’m going to start off with $100 in the account.1006

I’m going to deposit $1000 as coin.1016

I’m going to withdraw $900 as paper.1023

I’m going to withdraw $50, again as paper.1033

I’m going to go ahead and deposit $25 as coin.1040

My final state again $175, initial state $100, final state is $175.1049

Second scenario, initial state $100 and final state $175, this is energy.1057

Energy went from 100-175, energy went from 100 to 175.1063

It did not do it in the same way, different amounts of heat and work were transferred.1069

Clearly, the initial state and final states are the same but that paths taken to get from the initial state to final state are not the same.1076

Energy is what we call the state property.1091

Heat and work, I’m just going to go ahead and call them q and w.1101

They are called path properties.1107

Let me go over here, energy is a state property q and w are called path properties.1108

Most of the time they are also called path functions.1126

We are using the word function and properties interchangeable.1130

I, personally, do not like the use of the word function, path function or state function, you can call it state function as well.1133

The reason I do not is because we already have a specific idea what a function is.1138

Somehow when you think about a state function or path function, for me I'm looking for some kind of function.1143

When I say state property or path property, it is telling me that at some property, that is something is happening.1151

A state property is energy, it is something that has a specific numerical value, q and w.1159

Although they are not properties of the system, they are properties quantities that actually show up during a change of state.1165

They have actual numerical value, algebraic they are positive or they are negative.1172

Path function that does not really matter.1177

You are going to see them refer to as both.1179

Energy is a state property, q and w are path properties.1181

Now here is the difference, the value of state property does not depend on the path taken to get there and the path taken.1188

It depends only on the initial state and the final state.1222

The value of the path function, the value of a path property absolutely does depend on the path.1233

We will have more to say about paths in just a minute.1254

We are going to be talking about paths a lot.1256

The value of the state property, think about this example.1260

If I’m here at Los Angeles and I want to go to Denver Colorado which is roughly a mile above sea level.1263

Here is my initial state and here is my final state.1271

The only thing that matters is the fact that I’m going from here to here is the height, let us say 1 mile.1274

I have gone up above sea level 1 mile.1280

There is a thousands of paths that I can take to get there.1284

When I talk about the change in state, the change in state is 1 mile.1287

When I talk about the path that I have taken to get there, I can go from California to Miami.1292

I can go from Miami to New York.1298

I can go from New York at a space station and at the space station back down to Colorado.1300

I have done a lot of work to get there and I have spent a lot of heat to get there.1305

Or I can just take a single trip straight shot from Los Angeles to Denver.1309

I have not expanded as much work.1317

I have not expanded as much heat.1318

The initial and final states are the same but the paths are very different.1321

The quantities of work and heat expanded are different.1324

This is why.1328

Height is a state property in this particular case.1329

Elevation, the difference in elevation but the paths that I take to get there, the amount of gas that I spend,1333

the amount of work that I do, the height of for which I go and come.1340

All of that is actually different so it depends on the path.1346

Mathematically, it is just going to become very significant.1349

We will have more to say about when we cross that.1353

Let us talk about work, let us talk about pressure, volume, work.1360

When a system changes volume, consider a gas in a cylinder gas and a container.1374

When the system changes volume, we better think of a balloon.1383

Either by being compressed by an external pressure or by expanding against an external pressure, work is done.1393

Work is done, in other words work is transfer.1429

Let us see here, we say that when gas expands, the volume of the system increases.1444

The gas expanding, the volume is increasing.1458

So volume of the system increases.1461

This is the system doing work on the surroundings.1478

Or work is flowing from the system to the surroundings.1490

Work is positive.1507

When gas is compressed, you know that the volume of the system decreases.1514

System this is the same as the surroundings is doing work on the system, which is the same as work flowing from the surroundings to the system.1549

In this particular case, work is negative because it is flowing from the surroundings.1581

Work or taking the point of view of the surroundings.1584

The magnitude of the work that is done in expansion or compression is equal to that external pressure × the change in volume.1590

If you want numerical values for the work that is done in gas expanding, the work that a gas does on the surroundings1605

is equal to the external pressure × the change in volume that the gas experiences.1612

If I compress the system, the work done by the surroundings on the system, the magnitude for numerical value of the amount of energy1617

that transferred is going to be that external pressure × change in volume that the system experiences.1626

Let us look at what an extension looks like on a pressure volume diagram.1638

Analyzing the pressure volume diagram, analyzing all these diagrams is going to be very important.1645

You probably did maybe a little bit of it in general chemistry, I’m guessing not altogether too much.1651

In thermodynamics it is going to be profoundly important.1655

Definitely, take your time and make sure you understand everything is happening on these diagrams.1658

These diagrams will help you solve your problems.1662

They will allow you to follow certain paths to say this is happening, this is not happening.1665

It is very important.1670

Let us see what an expansion looks like on a PV diagram.1672

Let us go ahead and do this.1678

I’m going to go ahead and draw the diagram here.1681

I will go ahead and draw this like that.1687

I will go ahead and mark my things.1691

This is going to be the initial state.1694

This is going to be some initial pressure P1.1697

Some initial volume v1 and T.1700

We are going to actually keep T constant.1703

This right here, this is an isotherm.1706

For our purposes, you want to deal with only a couple variables at a time.1712

Again, we have pressure up here and we have volume on this axis.1716

We want to be able to make things easier on us.1721

We are going to do all of our things around an isotherm.1724

It is all that it means, is that we are keeping the temperature constant when we do this expansion or this compression.1727

This is the initial state, P1 v1 T.1733

This is going to be our final state.1737

This is going to be P2 V2 and again T, temperature is constant.1739

What we are doing is following.1743

Let us say we have some cylinder with a piston, I will move this a little bit further down here.1745

We have some cylinder and we have some piston arrangement.1753

On top of that piston, we can go ahead and put a mass.1758

We have little pegs here, there is a gas in here.1761

This is a P1 v1 initial pressure, initial volume.1765

There is a certain pressure, there is a certain mass.1770

That mass accounts for the external force, external pressure.1773

What is going to happen, this is the initial state.1777

What we are going to do is remove those kegs and if the internal pressure is actually going to be bigger than the pressure outside, the gas is going to expand.1779

It is going to push the piston up.1789

That is what is going to happen.1791

It is going to go to end up looking like this.1795

It is going to end up rising until the internal pressure which is now P2, with a new volume V2, is equal to the external pressure that comes from the mass.1803

When the external and internal pressures are the same, the piston is going to stop.1812

That is what is going to happen.1817

This is what is happening physically.1818

I have a gas in the cylinder, it is under pressure, I go ahead and I release the kegs, it undergoes an expansion.1821

When it undergoes an expansion, we are going to look at what this looks like on the pressure volume diagram.1828

Here I’m going to mark P1 and here I’m going to mark P2.1833

Here I’m going to mark volume 1, volume 2.1838

We are starting up here.1842

Here is what happens, in order for there to be an expansion, the pressure on the inside has to be bigger than the external pressure.1844

Here is what happens, if this mass is a single mass, the external pressure does not change.1852

The external pressure is constant.1859

This external pressure here is what happens.1863

We started at volume 1, I pull up the pegs.1866

Now the internal pressure is going to cause the thing to expand so therefore, the volume is going to change.1871

The volume is going to increase.1881

The volume is going to change, the volume is going to change, it is going to keep increasing as the volume changes.1882

You know that the pressure on the volume have an inverse relationship.1889

As the volume of the system increases, the pressure inside the system decreases.1893

It decreases along this line here, the isotherm.1897

We are keeping the temperature constant.1902

As the volume increases, the pressure is going to decrease.1905

It is going to go down.1910

It is going to come to a point when the volume has reached such a point, that the external pressure is now the same as,1914

This external pressure is a constant pressure that comes from the mass.1924

When it actually reaches the value of P2, P2 the piston is going to stop.1928

That is what happens.1936

The pressure inside is bigger than external pressure.1938

Well it is going to expand.1941

As it expands, the pressure starts to drop when the pressure on the inside P2 reaches the external pressure which comes from the mass is going to stop.1944

The pressure and this pressure is the same so we get this.1954

I’m going to go ahead and shade this in.1963

We said that the work done during expansion, the numerical value is equal to the external pressure × that change in volume,1966

The external pressure happens to be this.1973

It happens to be the same as P2 because that is when the piston will actually come to a stop.1975

That is this value right here, external pressure.1984

The change in volume is v2 – v1, that is this length right here.1988

External pressure × change in volume, the area underneath that rectangle is the work done during this expansion against the constant pressure.1992

That is what is happening here.2003

The path taken in this process is this one, we start with a given pressure.2006

We release depends it starts to expand until it reaches this point.2015

We went from an initial state to a final state via this path right here, that is the path that we took.2022

The work done during this expansion is the area underneath that rectangle.2028

Let us go ahead and see.2035

Work is equal to P external × change in volume which is P external × v2 – v1.2040

Again, this is just the area underneath that.2053

This is a single stage or one state expansion.2059

All that means is that you have a single constant external pressure in a single volume expansion.2073

We get our expansion in one stage.2093

We went from initial state to final state, initial state to final state in a single state which we allowed to go this way.2095

That is the path that we took.2105

When we talk about path, this is actually where the word path comes from.2108

It comes from these pressure volume diagrams and the particular path that the system is following in order to get from an initial state to a final state.2111

What if we do the same expansion, in other words, going from P1 v1 to P2 v2.2129

Same initial state, the same final state, this time what if we do it in two states?2139

What if we do the same expansion in two stages?2144

Here is what it looks like physically.2151

We have a cylinder, that is there, this one is a little higher.2154

This one is going to be up here.2170

Basically, what we do in this case is we put a certain mass on there.2171

When we put a total mass on there, this is going to be P1 and this is v1.2175

This is going to be P prime, this is going to be v prime, and this is going to be P2, and v2.2179

We put a certain mass there.2186

The pressure in here is bigger than the external pressure, it is going to expand a little.2188

It is going to expand a little until it reaches a certain height and it is going to stop.2192

What we do is we take this mass off and we replace it with another mass.2197

The mass that is a little bit lighter.2206

It is going to expand some more until it reaches another height.2208

Let us see what this looks like in the pressure volume diagram.2214

These PV diagrams there are very useful and very important.2219

Again, we have the same initial state and the same final state.2224

This is the initial state which is P1 v1 and this is the final state which is P2 v2.2228

Our initial pressure, our final pressure, our initial volume, and our final volume.2238

Here is what the expansion looks like, this two stage of expansion.2246

Here is what happens, I'm going to put a mass on there such that the external pressure is actually less than P1 but is bigger than P2.2250

I’m going to go ahead and put it right there.2260

I’m going to call it P external prime, here is what happens.2263

The minute I pull out these pegs, the expansion is going to start.2272

I pull out the pegs, it is going to start at volume 1.2277

What is going to happen is it is going to expand until it reaches that point.2281

This point is S prime, this is P prime, V prime.2288

As the volume expands, the pressure inside the system decreases because as volume expenands, pressure decreases, that is the relationship.2292

Doing this along the isotherm and so the pressure goes down.2301

Again, the pressure goes down until the pressure inside the system is the same as the external pressure that comes from the mass and then stops.2306

Now what I do, is I'm going to go and say this is our V prime.2318

I would take that mass off and replace that with a lighter mass.2325

If I go with a lighter mass, now the external pressure is less than the P2.2329

It is going to expand some more.2335

It is going to expand some more.2338

It is going to keep expanding until now the external the pressure inside the system matches the external pressure at which point it will stop.2340

Again, we have gone from initial system to final system as P1 v1.2350

The P2 v2, the same weight but now we have taken a different path.2356

We have done it as a two stage expansion.2362

Well the first expansion, the amount of work done,2367

I’m going to have these lined up properly, sorry about.2374

Let us go ahead and change this form.2377

This is v1, the work done during the first stage, this expansion that is that area.2384

The work done during the second expansion, that is that area.2395

This is the isotherm, I think I have everything here.2403

I will say this is work 1, first stage, and this is work 2 that is the second stage.2410

The total work done in this two stage expansion is equal to work done during the first stage + the work done during the second stage.2418

Nothing new there, let us compare the two diagrams.2426

In the single stage, this was the work that is done.2444

This is the one stage expansion, this is the two stage expansion.2470

This area right here, that is the same as this area right here.2477

The two stage expansion did more work.2482

It did more work by this amount right here.2485

This is what is important.2490

We see that the two stage expansion has done more work.2492

The area is larger, the initial state and the final state are exactly the same but the paths taken are different.2512

Because the paths taken are different, the work is going to be different.2542

Now this path vs. this path.2548

They are very different paths.2557

This is why we call work a path function, this right here.2562

It is based on the Pv diagram.2575

Schematically, it is following a different path to get from an initial state to a final state, an initial state to a final state.2578

It can take any path it wants but the fact that a different path gives me a different value of the amount of work that it did.2585

This is why we call that a U path function.2591

Its value depends on the path we take.2599

As a reminder, heat is also a path function.2611

Q was also a path function.2617

Not just work but heat and work are path functions.2619

Let us take a look at a multistage expansion.2631

A multistage expansion, which I will go ahead and draw it here, I’m not going to do much analysis on it.2635

It is going to look exactly like what you think it looks.2647

A multistage expansion is going to be, if this is our initial state and this is our final state, one stage, two stage, multistage looks like this.2650

That is the total work done is going to be the area underneath everything.2663

That is what a multistage expansion looks like.2674

Exactly what you think.2678

You can see where this is going.2681

This looks a lot like a calculus course, one more stages and one more stage, tinier rectangles this way.2683

More and more area.2689

You can see where this is going.2695

If we keep increasing, the number of stages, we get this.2707

We can keep increasing the number of stages until the steps that we take, the paths that we take, in other words,2729

the change in volume until the changes in volume now become a differential length.2741

You know where this is going.2752

Instead of writing work equals P external, change in volume which is true.2754

This is valid and this is actually valid equation for constant pressure process, this is how we get the work.2761

If we do in multiple stages, it is working differentials instead of in large quantities like this P external × the differential volume element.2768

This little increments differential volume increases.2783

The differential work that is done in going from let us say this point to this point,2788

is going to equal the external pressure from here to here × the differential volume element.2794

We are just breaking this up into a bunch of little rectangles and adding up all of that.2800

Now so this is going to be a very important relation.2809

We are often to be working with differentials as opposed to large scale stuff like that.2814

If we assume and it is a pretty fair assumption to make that our external pressure remains constant over the differential volume change.2820

Since, DW is the differential amount of work in going from here to here, if I want the total work done in going from here to here.2852

I integrate all of those, I add them up, I add up all the differential work elements.2860

Our total work is equal to the integral from state 1 to state 2 of the differential work element.2871

The differential work element is equal to this, v1 v2, P external dv.2882

That is it, we just gone from δ v to dv.2889

We are just working smaller and smaller, this is it.2893

This is the most general expression for the pressure volume work done by a gas as it expands isothermally.2906

This is an isothermal expansion, we are keeping the temperature the same.2938

If we do not keep the temperature the same, the amount of work that is actually different.2943

This is nice, this is an integral.2949

If we happen to know how the pressure changes with volume, we can go ahead and solve the integral in order to get the work done along that particular path.2952

If we happen to know how the external pressure changes as a function of volume then we just evaluate the integral.2965

Let us go ahead and redraw.3003

As we one more stages, one more stages, we get higher and higher work.3009

There is only so many stages you can go.3015

There is a maximum amount of stages that you can have.3019

Basically, when you take the differential and when you take the differential volume element to 0, what you can end up getting is the integral under this.3022

Clearly, there is a maximum, there is an upper limit on the amount of work that a gas can do upon isothermal expansion.3031

There is an upper limit because there is an upper limit on the area.3052

Basically, that is it.3059

You are going to get more and more until you basically have the area underneath the curve.3062

There is an upper limit on the area under the isotherm.3069

There is a maximum amount of work a gas can do upon isothermal expansion.3085

This max is achieved when the path we take is directly along the isotherm.3114

Instead of going to little differential elements when we take the limit of the differential, now what we do is we just move right along that isotherm.3138

That is our path, not this, but right along the isotherm.3151

Straight down this way and straight up that way when we do our compression.3159

Let us see what we got here, in order for a gas to expand, the pressure inside the system at any given moment has to be bigger than the external pressure.3167

That is how expansion takes place.3183

If a pressure the same, there is no expansion.3185

It has to be greater that.3187

For this differential changes in volume, for the small changes in volume, the pressure inside has to be slightly bigger than the external pressure3189

or the external pressure is has to equal the internal pressure - from differential amount in pressure.3202

This says that the external pressure is going to be just slightly less, slightly by differential amount, an infinitesimal amount less than the pressure inside.3213

If this is the case, the work equals change in volume 1 to volume 2 of the external pressure × dv, that is equal to the external, that is equal to the pressure of the system.3223

The internal pressure - dp dv that equals the interval for v1 v2 of Pdv - the interval from v1 to v2 of dp dv.3240

This is just mass and will actually goes to 0, this is the second order differential.3260

A differential × a differential and it goes to 0.3265

Do not worry, what you are left is with is this.3268

The work is actually equal to the integral from v1 to v2, the pressure dv.3276

The pressure here, this is the pressure of the system.3284

Before, under constant pressure conditions, the initial equation we said is that it is the external pressure that defines the work that is done.3287

In this case, if we are traveling along the isotherm, with only really infinitesimal amounts we no longer have to use the external pressure.3296

We can use the internal pressure, the pressure of the system.3305

Because the pressure of the system and the external pressure, in order for there to be small infinitesimal changes, they are almost the same.3308

When we are moving along this isotherm, the pressure outside and the pressure inside are essentially equilibrium, that is what is going on here.3316

We have managed to eliminate the external pressure term and deal only with the pressure of the system, the pressure inside.3323

P external has been replaced by P which is a pressure of the system at any given moment along the isothermal expansion.3333

Let us go ahead and do an example here.3373

Example 2, we want to find an expression for the maximum work done by an ideal gas upon isothermal expansion from an initial volume to a final volume.3380

Ideal gas, we already know what an ideal gas is.3394

We know the equation state for the ideal gas, it is Pv=n RT.3397

We also know this, we also know that the work is equal to the integral from v initial to v final of Pdv.3405

We need to find an expression for P as a function of v.3415

We have an ideal gas right here, let us just rearranged it.3419

The pressure of an ideal gas is equal to nrt/v.3422

I’m just going to go ahead and put that in there and what I get is the following.3426

Work equals the integral from the v1 to v2.3430

I will just use v1 and v2.3434

Dp dv which is equal to now nrt/ vdv.3437

Nrt is a constant so I’m going to pull that out.3445

Nrt v1 to v2 of dv/ v, 1/ vdv.3448

I know what the integral of dv or v is, it is the natural logarithm of v.3457

What we get is work is equal to nrt × log of v2 – log v1.3463

I will go ahead and rearrange that using the properties of logarithms.3477

The log of v2/ v1, there you go.3481

For an ideal gas expanding isothermally, this is the expression for the work done by that gas on its surroundings.3488

This is the maximum amount of work in an ideal gas can do as it expands, n × R× T × log of the final volume divided by the initial volume.3496

This is a very important relation to understand.3510

This is the isothermal expansion, the maximum amount of work that an ideal gas can do upon isothermal expansion.3513

Let us go ahead and do another example.3526

Example 3, an ideal gas occupies a volume of 1.5 deci³ at 2.0 atm.3532

If this gas is to expand isothermally to a new volume of 3.5 deci³ under a constant external pressure,3546

what is the largest possible value that the external pressure can have and how much work is done by the gas in this expansion under this value of external pressure?3555

We have an ideal gas, we know it occupies a volume of 1.5 in deci³ at 2 atm.3569

If this gas is to expand isothermally to a new volume under a constant external pressure, the expansion is constant external pressure here.3578

What is the largest possible value of the external pressure can have if it is going to actually affect this change?3589

Let us go ahead and see what this looks like.3596

Amount of volume of 1.5 deci³ at 2.0 atm.3612

I need to go to a volume of 3.5 deci³ and I’m going to do it isothermally under a constant external pressure.3622

Constant external pressure this means a single stage expansion, there is going to be some value.3633

Basically, it is like this, I know that this is some final pressure here, I need to get from this state to state.3638

I need to get from this pressure to whenever this pressure is.3646

I need to go from 1.5 to 3.5 this is done isothermally, we are going to move along here but this is done under constant pressure.3649

Because it is constant pressure, it is going to have to happen as a single stage expansion.3659

In order for the gas to expand, the external pressure has to be less than the initial pressure.3667

In order to expand, and now in order to reach this state, I could have any external pressure I want to up to the final pressure of the system.3677

In other words, I can have a external pressure here, external pressure here.3693

Any of those values will be just fine because the expansion will take place.3700

Remember, it will go this way, this is the amount of work done.3704

This way, this is the amount of work done.3708

Whatever the value is, they want to know the largest possible value that the external can have.3709

What the largest possible value that the external pressure can have is actually going to be P2.3721

In order to reach that state, I have to reach an internal pressure of P2.3733

In order to reach an internal pressure of P2, I have to be able to get to P2, that is the largest value.3736

So the external pressure it can be bigger than 0 but it has to be less than or equal to the final pressure.3740

Therefore, the largest value that the external pressure can have is the P2.3748

Let us go ahead and calculate that.3753

That is very easy.3755

P1 v1 =P2 v2, I have P1 v1 and I have v2, it was very easy for me to calculate.3759

My final pressure P2 is going to equal P1 v1/ v2.3769

It is going to equal, the initial pressure is 2 atm, the initial volume is 1.5 deci³, my final volume is going to be 3.5 deci³.3778

I do not have to make any conversions here because units are going to go ahead and cancel.3794

My P2, my final pressure is going to end up being 0.857 atm.3800

This is the largest value that the external pressure can have in order for this to happen.3808

It can be anything less than that.3813

If it is less than that, less work is going to be done.3815

If it is a little more than that, more work is going to be done.3817

At a certain point, when the external pressure is the same as the final pressure of the system that I want,3820

that is going to be the largest value that I can have in order to actually reach P2.3828

My external pressure is equal to P2 is equal to 0.857 atm.3837

In order to calculate the actual work done, the work is equal to the external pressure × the change in volume.3846

Here is where we have to make some conversions.3855

Let us just go ahead and do this.3860

We can go ahead and do it this way.3865

The external pressure we said is going to be 0.857 atm, the change in volume is going to be from 1.5 to 3.5.3868

It is going to be 3.5 - 1.5 which is going to be 2.0 L.3877

We get 1.714 L/ atm that is a perfectly valid number.3885

Let us go ahead and convert it to J just for the heck of it.3891

1.74 L/ atm × 8.314 J / 0.08206 L/ atm.3895

This is the relationship between a L/ atm and a joule.3912

When I do this calculation, I end up with 173.7 J.3915

If you are wondering where this relationship came from between L/ atm and joule, these are the two values of R.3922

R is the same.3930

I have 8.314 J/ mol K.3932

I have mol K 0.08206 L/ atm.3941

Mol K cancels mol K, I have myself a conversion factor, J L/ atm.3953

L/ atm is unit of energy.3959

Here is the conversion factor, 101.3 is what this number is but I like to write it that way.3961

There you go, that is the maximum amount of, this is the work done by the maximum value that the P external can have3967

in order to undergo this expansion under constant external pressure.3981

Thank you for joining us here at www.educator.com.3987

We will see you next time for our continuation of energy in the first law, bye.3989

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