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Raffi Hovasapian

Raffi Hovasapian

Changes in Energy & State: Constant Volume

Slide Duration:

Table of Contents

I. Classical Thermodynamics Preliminaries
The Ideal Gas Law

46m 5s

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

46m 2s

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

1h 6m 45s

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

1h 6m 33s

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

1h 2m 17s

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

1h 4m 39s

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

16m 50s

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

43m 40s

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

32m 23s

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

39m 15s

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

35m 52s

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

42m 40s

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

1h 23s

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

44m 34s

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

59m 7s

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

49m 16s

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

33m 59s

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

54m 37s

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

31m 18s

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

23m 6s

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

25m 42s

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

43m 39s

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

56m 44s

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

57m 6s

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

54m 35s

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

35m 5s

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

28m 42s

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

34m 38s

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

30m 50s

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

34m 6s

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

39m 18s

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

19m 40s

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

54m 16s

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

31m 17s

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

45m

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

58m 5s

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

25m 20s

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

34m 25s

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

59m 57s

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

42m 5s

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

30m 26s

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

21m 34s

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

56m 22s

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

45m 24s

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

48m 43s

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

46m 18s

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

39m 28s

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

35m 32s

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

29m 55s

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

54m 25s

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

46m 58s

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

44m 11s

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

35m 33s

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

43m 4s

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

26m 30s

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

41m 10s

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

30m 32s

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

35m 19s

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

33m 48s

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

46m 43s

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

40m

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

35m 58s

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

36m 18s

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

33m 55s

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

51m 53s

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

51m 53s

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

34m 29s

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

43m 49s

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

1h 1m 57s

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

48m 33s

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

48m 33s

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

48m 33s

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

59m 18s

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

34m 54s

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

38m 3s

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

57m 49s

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

1h 1m 20s

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

56m 34s

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

18m 24s

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

50m 2s

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

59m 47s

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

46m 22s

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

29m 24s

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

30m 53s

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

1h 1m 33s

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

33m 47s

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

1h 1m 5s

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

33m 31s

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

1h 1m 15s

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

47m 23s

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

54m 9s

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

48m 32s

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

57m 30s

Intro
0:00
Example I: Make a Plot of the Fraction of CO Molecules in Various Rotational Levels
0:10
Example II: Calculate the Ratio of the Translational Partition Function for Cl₂ and Br₂ at Equal Volume & Temperature
8:05
Example III: Vibrational Degree of Freedom & Vibrational Molar Heat Capacity
11:59
Example IV: Calculate the Characteristic Vibrational & Rotational temperatures for Each DOF
45:03
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Lecture Comments (7)

0 answers

Post by Joyce Ferreira on October 22, 2017

I am really enjoying your lessons. Your teaching skills are extraordinary. Thank you very much and God bless you!

1 answer

Last reply by: Professor Hovasapian
Fri Feb 26, 2016 1:23 AM

Post by Van Anh Do on February 14, 2016

For the example, I thought Cv is the heat capacity at constant volume and since the problem doesn't say that the system has a constant volume the whole time, how do we know to plug 3/2R in for Cv? Thank you.

1 answer

Last reply by: Professor Hovasapian
Fri Feb 26, 2016 1:14 AM

Post by Van Anh Do on February 14, 2016

For this lecture, are we assuming that pressure is also constant? I'm not sure why we're able to leave out pressure. Thank you.

1 answer

Last reply by: Professor Hovasapian
Wed Jun 3, 2015 7:41 PM

Post by Joseph Carroll on June 3, 2015

Exemplary teaching methods Professor Raffi! Now, I can honestly profess that all the calculus courses I took are being utilized in a quite a profoundly and complementary way in relation to the physical world. Thanks for the excellent break down of the total differential of U(T,V) :-). I am using the Atkins 10th edition of physical chemistry, and although I read the section 2D before your lesson and understood the physical relation behind the mathematics roughly, you made the ideas come together effortlessly for me.

Changes in Energy & State: Constant Volume

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
  • Recall 0:37
    • State Function & Path Function
  • First Law 2:11
    • Exact & Inexact Differential
  • Where Does (∆U = Q - W) or dU = dQ - dU Come from? 8:54
    • Cyclic Integrals of Path and State Functions
    • Our Empirical Experience of the First Law
    • ∆U = Q - W
  • Relations between Changes in Properties and Energy 22:24
    • Relations between Changes in Properties and Energy
    • Rate of Change of Energy per Unit Change in Temperature
    • Rate of Change of Energy per Unit Change in Volume at Constant Temperature
    • Total Differential Equation
  • Constant Volume 41:08
    • If Volume Remains Constant, then dV = 0
    • Constant Volume Heat Capacity
    • Constant Volume Integrated
    • Increase & Decrease in Energy of the System
  • Example 1: ∆U and Qv 57:43
  • Important Equations 1:02:06

Transcription: Changes in Energy & State: Constant Volume

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

Today, we are going to start talking about, or to continue our discussion of the first law and energy and work and heat and things like that.0004

We are going to start talking about changes in energy and state.0014

In this particular lecture, we are going to be talking about constant volume processes.0018

We are going to start putting some constraints either to constant volume, constant pressure, constant temperature, things like that,0022

to see if we can learn some things about what is going on with the first law and with the energy.0029

Let us jump right on in.0035

Let us recall, this will go with black today.0038

U energy is a state function or a state property.0045

It does not depend on the path that you take.0055

It just depends on your beginning state and your ending state.0058

If I start at 0 feet sea level and ago up to 500 feet above sea level, the difference is 500 feet.0063

It does not matter whether I drop down to negative 300 and go up to 800, then come back to 500.0072

All that matters is the beginning and ending state.0078

Work and heat are not state functions, they are path functions.0082

W is a path function or path property.0087

Heat is also a path function.0107

A path function, its value, the value of work or heat depends on the path that we take.0111

I’m going from here to here, this symbols less work than this or this.0117

The amount of heat from here to here is going to be less than the amount of heat like that.0124

It depends on the particular path that we take.0129

The first law is this, it says that the change in energy, the total change in energy of the system is equal to0132

the heat gained by the system - the work loss by the system.0145

That is all that is.0151

We will often use this form.0154

This is the fully integrated form, we will often use the form, the differential form because0156

we will be leading much more on the mathematics and more sophisticated mathematics.0161

So du= dq – dw, the differential change in energy = the differential change in heat -the differential change in work.0167

A quick word about this, U is a state function.0179

W and Q are path functions.0186

The symbolism that I have used is a DDD and tends to imply that they are the same.0189

They are not the same.0193

Many books will differentiate notationally a state function and the path function.0195

They used this differential notation to standard, when you are used to the d for a state function.0200

They will use some variation of that for a path function.0206

You often see something like dw, that is Greek δ and dq or you will see d with a line through it.0209

For dw and dq, this let us know that these are path functions.0219

Most specifically, for mathematical point of view they are an exact differentials.0224

A state function is an exact differential and a path function is an inexact differential.0228

And we will be talking more about what we mean by exact and inexact and the properties of exact and inexact differentials.0234

I, myself, I do not like to differentiate because I believe, I mean physical chemistry and0240

Thermodynamics, there is already just an abundance of symbolism as it is.0245

To introduce new symbolism, it just confuses me.0250

To me, personally, it is just easier to remember that work and heat are path functions and energy is a state function.0253

Other than that, it should not cause you any problem.0262

I’m going to go and use the same, but they are not the same.0265

This is an exact differential, these are inexact differentials.0267

Let us go ahead and talk about integral from an initial state to a final state of this equals U2 – U1.0272

This is the fundamental property of a state function but it integrates the way you are used to according to the fundamental theory of calculus.0285

It is equal to δ U, this is an exact differential.0293

Exact differentials integrate like this.0297

You take the final state - the initial state of the integral of the particular function.0299

Exact differential, this is a state function.0305

When we integrated heat, dq, we do not get Q2 - Q1, we just get whatever the value of Q is.0313

This is not δ Q, this is an inexact differential.0325

Inexact differentials cannot integrate the same way but exact differential do.0335

You do not follow this final – initial.0339

It depends on the path so you just get a quantity.0345

dQ and dQ, it does not make any sense because a system does not possess heat.0361

Heat is something that shows up during a change of state.0368

Heat is actually a process, when there is a temperature differential of two of the system in the surroundings,0371

or two systems whenever it is, heat is the transfer.0377

Heat happens at the boundary.0382

That is what is going on, there is a system does not possess heat.0384

The system has a certain temperature.0387

Temperature and heat are not the same thing.0389

Heat is what happens during the change of state.0391

It makes no sense to say that there is a certain amount of heat at the beginning, a certain amount of heat in the end,0394

and change in heat is the difference between the two.0400

That does not make sense the system does not possess heat.0404

Granted, we tend to be little loose with our language usage when we talk about heat as if the system does possess it.0407

That is just part and parcel of the historical discussion of thermodynamics.0414

That sort of looseness in language that existed, signs in general, it is not a problem as long as you remember these things.0417

δQ makes no sense because a system does not possess heat.0424

And the same thing with work, when we integrate the differential of the work from one state to another,0444

all we are adding up all the work done along the particular path, we get the final work its equal to W.0450

It is not equal to dW because the system does not possess work.0456

It does not make any sense.0462

Work inexact path function.0465

The dW makes no sense for the same reason, a system does not possess work.0471

Here is what is interesting, the Q and W are path functions their difference is a state function.0479

That is extraordinary, that is truly amazing.0488

Q and W, they are path functions but their difference or sum in a different convention, the chemists convention usually write Q + W.0492

Our convention is Q – W but their difference, I will put sum here, is a state function, that is amazing.0511

State function, energy.0525

Where does this come from?0534

The question where does δ U = Q –W come from?0536

Or more often we use this with a differential = dq - du come from.0545

Let us talk about a cyclic process.0557

A cyclic processes is where you start with some initial state, you have a change of state,0559

you go to state 2 and then from state 2, you come back to state 1.0563

Exactly, it sounds like it is a cyclic process.0567

You go to some final state, you come back, and end up where you started.0569

The work of the cyclic process is equal to the integral of all the work done along the particular path that you take.0575

The integral dW, we will often put a circle around it to let us know that it is a cyclic process.0586

This is sort of an older notation, you probably do not see all that much anymore in modern books.0590

Cyclic just means cyclic, that is all it is, nothing strange about it.0597

In the Q, the heat lost or gained during a cyclic process that is equal to all the integrals of all of the dQ.0602

All of the differential heat elements along that.0615

Now these are not usually 0, in other words the cyclic work for a processes is not usually 0.0618

The cyclic heat is not usually 0.0626

These integrals are usually not equal to 0.0633

If they are, it is strictly a coincidence that is it.0644

In general, the cyclic integrals of path functions, the cyclic integrals of inexact differentials are not equal to 0.0651

In general, the cyclic integral of path functions / inexact differentials does not equal 0.0667

Now the cyclic integral of a state function does equal 0.0686

The cyclic integral of any state function does equal 0, as it must.0696

A state function depends on its initial and final state.0713

In a cyclic process, the final state and initial state are the same.0716

State - state you get 0.0724

In a cyclic involve any state function and the exact differential is equal to 0, as it must.0726

Initial, final, you go from initial to final and then you come back.0735

The cyclic integral of some dz = 0, where z is a state function.0743

Our empirical experience of the first law is the following.0753

This is what this says, this is the science, this is thermodynamics, it is all empirical.0761

Our empirical experience of the first law is this, if the system is subjected to a cyclic process or cyclic transformation,0767

either one, the work transferred to the surroundings, in other words,0802

the work gained by the surroundings to the surroundings = the heat transferred from the surroundings.0817

Now mathematically, this says this.0846

Our convention is we said that heat and work, these are effects that manifested in the surroundings.0850

We never measured something directly in the system.0857

We know what is happening in the system based on what we observed in the surroundings.0859

And we can take the systems point of view but what we are directly measuring is, is what is happening in the surroundings.0863

It is not a big deal as long as you think of the surroundings.0870

And if you want to talk about the system, you just switch your point of view.0873

When we talk about heat gain by work gained by the surroundings, we are talking about work done by the system.0876

Work that is lost by the system.0882

You just have to switch around.0884

Mathematically, it says this, a cyclic process dQ = dW.0887

This is our empirical experience of the first law.0898

Now let us see what happens here.0902

This is our mathematical or empirical experience of the first law.0905

What is going to happen is the following.0908

This is the same as, let me just move one of the integrals over to the other side which you end up with is this.0911

The cyclic integral of dQ - the cyclic interval of dW is equal to 0.0919

This is just the same as cyclic integral of dQ -dW = 0 because integration is a linear operator.0929

It is linear so I can just pull out the linear and the sort distributes.0941

We will talk about that little bit in calculus.0946

Now, I have a cyclic integral of something right here, this in the grand underneath the integral sign, the cyclic integral of it is equal to 0.0949

I know that the cyclic integral of any state function equal 0.0961

The cyclic integral of any exact differential = 0.0965

Therefore, dQ - dW must be some state function.0969

We call the state function energy, that is where this comes from.0973

We drop the definition on you but this is where it comes from becomes it comes from the empirical experience.0978

Empirically, this is what happens.0982

We want to give this a name we call it the energy of the system.0984

Any cyclic integral that = 0 is some state function.0991

We need to get the state function a name.1006

We call this state function which is dQ - dW or Q – W, we call this state function energy, the energy of the system.1014

This is where it comes from.1031

dU= dQ – dW, the differential change in energy of the system comes from1036

the differential change in heat - the differential change in work or the integrated form δ U= Q – W.1045

Remember, δ Q and δ W they make no sense.1055

Differential form and integrated form, we will be called the finite form.1062

Only δ U is defined.1073

This is the differential form, when we integrate this, when we integrate both sides, we end up getting is δ U = Q- W.1079

We do not get U, we get Q and W because they are path functions but du is a state function, we get the δ.1087

Only the δ U is defined, we do not measure energies of the system.1096

We measure changes in energy of the system, that is what we do.1101

Science best measures changes in things.1104

Only dU is defined, not U.1107

Let us see here, we have that δ U = Q – W.1117

If we measure the heat loss by the surroundings then subtract from it the work gained by the surroundings, both of which are easily measurable.1135

We get δ U, the energy change of the system.1186

If we measure the heat lost by the surroundings, the heat of the surroundings loses.1202

Once it loses something that goes into the system.1209

Then subtract from it the work gained by the surroundings, we get the change in energy of the system.1212

Basically, it always says that if you have a system and if you have the surroundings, if heat goes into the system,1218

if energy goes into the system as heat and a certain amount of energy to leave the system as work,1225

what you get is just the net change in energy.1231

It is just simple arithmetic is what it is.1234

I give you $100, take back $25, you are left with $75.1237

That is all this is, this is a simple accounting of energy in terms of work and heat, in terms of paper or coin money.1242

The change in state means changes in properties of the state of the system such as temperature, pressure, and volume.1254

The things that are easily measurable and they are properties of the state.1286

Also, temperature, pressure, and volume, these things are state functions also just like energy.1290

They do not depend on the particular path that you take to get there.1297

If you start with a system at 2 atm and then you increase it to 10 atm, and I take it down to 0.2 atm,1300

and take it back to 5 atm, the difference is from 2 to 5.1308

The change in pressure is 3 atm.1311

It does not matter how you got there, same with volume and the same as temperature.1314

These are state functions.1317

What we want to do, we are going to find relations between the changes in state,1320

the changes in these properties, and the change in energy.1325

The first law was work and heat, now we want to get a little bit more direct.1328

We want to express it in terms of things that the properties of the system itself, the temperature, the pressured, the volume.1333

What the changes in does, the quantities and say about the change in energy.1339

Let us find relations between changes in these properties and changes in energy.1353

Again, that temperature no b a P, temperature, pressure, and volume are state functions.1382

They are exact differentials, dT dP dV.1398

These are exact differentials.1401

We can start by assuming that energy is a function of temperature and pressure, temperature and volume, pressure and volume.1406

We have to start anywhere we like and this is certain how it appears.1415

What you are seeing us do here on a page, here is where mathematics starts.1418

You are seeing the final result of this.1426

You are not seeing the process the sort of let up to it.1428

When a scientist sits down on a theoretical, he starts playing with mathematics.1431

He does not necessarily know where he is going.1437

He just starts playing with derivatives.1440

He starts playing with equations and by starting to take derivatives and taking other derivatives, integrating and moving this here.1441

You will see something that looks like it is important.1451

Using the final result of that, you are not seeing the process.1456

If it seems like we are pulling things out of the air, when we are doing these mathematics, we are not pulling things out of the air.1459

What you are seeing is the final result of all the work that is been done.1467

The truth is when somebody has done it for the first time, it is all over the place.1471

You go this way, and this way.1475

That is the real nature of science.1477

You do not see that in your experience and from your books.1479

It just seems like one day you are a scientist, woke up, and goes like that.1482

It did not happen like that.1485

Do not worry about it, it does not entirely make sense to you all at once.1489

Let us start with this.1494

We are going to let energy equal, we are going to take two variables, temperature and volume.1497

We are going to say that energy is a function of temperature and volume.1504

In other words, if I change the temperature, if I change the volume, or if I do both, the energy of the system changes.1507

If I can express it this way, you get the following.1514

I can express the differential change in energy this way, du = du dt vdt.1518

Do not worry, I would explain everything here.1530

DU dv, these are partial derivatives under constant V.1534

This is constant temperature DV.1540

This is called a total differential of this variable.1547

You have a function which is a function of two variables.1552

It is differential can be written like this, the total change in energy is equal, du = partial du dt × dt + du dv under constant temperature × dv.1555

Let us talk about what all this actually means.1569

You might want to go ahead and go back to the second lesson of the series, one that discusses partial differentiation1572

and actually introduces this thing called the total differential.1579

This is called the total differential for du.1582

It is based on the presumption that is a function of two variables.1588

First of all, we want you to notice that the differential of any state property which is an exact differential.1595

Sorry if I keep repeating myself, it is important though.1618

The differential can be written like this, it can be written in this form.1623

Any state property that you deal with, if you know that state property is a function of two variables,1632

you can write that state property the differential like this, always.1637

This is a proofing, any state property, any exact differential can be written like this on the right hand side of the equality sign.1642

The question is what does this mean?1651

Let us examine what this means.1653

What does this mean?1658

What if you are faced with something like this, what if you are faced with some mathematical equation1663

and you definitely want to give it a habit of stopping and examining what every single term means1668

and what every single term says and you want to get physical meaning.1673

In the case of du dt, this is the rate of change of energy with respect to temperature.1678

In other words, as the temperature changes by 1 unit, how much does the energy change?1684

This is a rate of change.1689

This is a derivative, is it like a derivative like any other derivative.1690

This happens to be partial derivative because we are dealing with a function of two variables instead of 1.1693

This really is just the same as du dt.1698

When you take that rate and you multiply it by the change in temperature, you get the total change in energy for the increment.1700

The same thing here, this is the rate of change of energy per unit change of volume.1707

Time and change in volume of the system experiences using a total change in energy.1713

You knew you want to get in the habit of doing this, a lot of work that we do specially in thermodynamics.1718

It is going to be strictly mathematical.1724

Actually, throughout all of physical chemistry, quantum mechanics, spectroscopy, you need to be able to assign physical meaning.1727

After you have done this a couple of times, pretty soon you are not intimidated by the mathematics anymore.1732

It makes sense what is going on.1738

There is nothing here that is counterintuitive.1741

It seems that way simply because you just unaccustomed of the mathematics.1745

You can get used to it like anything else.1749

This is something new that we are going to be using with some rather sophisticated mathematics.1751

But it is nothing that you do not understand, that you have not really been exposed to.1755

What you have exposed to, we will go over but do not just pull away from it, assign physical meaning to it.1759

Eventually, you will get them very quickly of knowing exactly what is going on and be able to relate this math1766

to relate what is happening physically in a system, which way temperature is moving, which way a system as the surrounding.1773

It says this way and that way, you will be able the see what is happening physically.1780

When you see what is happening physically, that is when you understand what is happening mathematically.1783

Let us see what this means.1792

The first term du dt, du dt with this little subscript V on it, this is the rate of change of energy per unit change in temperature.1796

That is it, it is just a rate, it is just a derivative per unit change in temperature.1818

Look at the units, that is where I start.1829

I look at the units to help it make physical sense for me when I see a derivative like this.1833

We are talking about physical things, this is not just theoretical.1842

Well energy is in joules, temperature is in Kelvin, and this is J/K.1846

The V subscript that tells us that something is happening at a constant volume.1854

In thermodynamics, any subscript let us know that it is happening as we keep that variable fixed.1859

The rate of change of energy per unit change in temperature, just look at the units.1867

Therefore, if T changes by an amount, if the temperature changes by some differential amount dt,1873

and this du dt sub V × dt, it gives the energy change for the particular incremental change in temperature.1889

It gives the energy change, holding V constant.1904

Again just look at the units, du dt that is J/K.1917

If you multiply that by a change in temperature, a temperature is in Kelvin, we are left with joules.1924

You already notice this is heat capacity and you will see in a minute that it is.1930

We are just interpreting what this is right now.1937

We are just getting a sense of the mathematics.1940

Attaching meaning to the partial derivatives, what they mean, we do not just want them to be scrolls and scribbles on a paper, that mean nothing to us.1944

We know how to manipulate this so we have done calculus.1952

We are very good at calculus.1955

This is a simple calculus.1956

For the next one, du dv sub T well this is exactly what you think it is.1962

It is the rate of change of energy per unit change in volume at a constant temperature.1972

Therefore, if the volume of the system changes by some differential amount,2006

dv changes by an amount dv then this du/ dv sub T × dv.2017

It gives the total energy change for the differential energy change, du.2042

It gives the incremental energy change.2047

Again look at the units, energy is in joules, volume always take deci³, dv deci³.2057

Volume cancels volume, leaving you just energy, that is what is happening.2068

Therefore, this expression du=du dt sub V dt + du dv sub T dv says the following.2076

It is absolutely imperative that you do this, that you assign physical meaning, otherwise this stuff is going to get away from you very quickly.2099

Because from this point on, it is going to be essentially mathematical.2108

It is physical, we assign physical meaning to it but we are expressing these physical changes mathematically with partial derivatives.2112

This expression says the following.2120

If T changes by an amount dt and volume changes by an amount dv simultaneously, the amount dv2127

then the energy change which is du as the sum of the two.2161

This is totally intuitive.2170

It is just as sum of the two, if I change the temperature, this amount, and the rate at2172

which the energy changes per unit change in temperature, that is going to give me the total energy change for temperature movement.2183

And if the volume changes also, that change in volume × the rate of change of the energy of the volume gives the energy change for the volume.2190

If I add those two together because the total energy change, that is it.2198

That is all this is saying, you know this already.2201

When you know this intuitively, you know this since you were a kid.2204

There is nothing here that is strange.2206

It is just the symbolism that can look a little intimidating.2210

Let us see here, if we happen to know what these partial derivatives are, in other words2216

if we happen to know what du dt and du dv sub T are, we just integrate the expression.2231

We just put into the differential because we just integrate the expression.2245

We can just integrate this expression to get the total energy change.2251

We have the du, we have the expression on the right.2278

If we know what these partial derivatives are, we just put them in and integrate that function that gives us the total energy change.2280

Nice and simple.2287

Let us see if we can express these partial derivatives in terms of things that we know, in terms of things that we can measure.2292

We know what they mean.2312

Let us see if we can actually measure them somehow.2313

Let us see if we can express these partial derivatives, this and this, in terms of things we can measure.2317

Let us see, we have the du is equal to du dt sub v dt + du dv sub T dv.2342

We also have the first law which says du=dq – dw which is equal to dq and dw is equal2362

to the external pressure × the differential change in volume.2374

Right, that is the definition of work, pressure × volume.2379

I’m going to go ahead and put this expression over here on the left side so we get dq - P external dv2382

is equal to the right side du dt sub v dt + du dv sub T dv.2394

I'm going to ask that you actually confirm this.2407

There are a lot of symbols that I’m writing on the page and all of the subscripts and letters so it does get to be a little confusing.2409

Please make sure I’m actually writing this correctly.2417

In any case, this is where we start.2423

Let us see what we can do with this.2425

We have the first law, we have the expression of the change in energy in terms of the properties temperature, volume,2427

we sent them equal to each other and we are left with this.2435

Let us see if we can make sense of this.2437

This is the equation we start.2440

I apply this equation to our changes of state.2442

We apply this equation to various changes of state.2447

Here is where it begins, various changes of state.2461

The first change in state we are going to do is we are going to change of state under conditions of constant volume.2469

A system goes from state 1 to state 2 but in that process, the volume stays constant.2484

We have dealt with the volume change.2488

If V remains constant and then this dv equal 0, V final V initial, the volume stays the same.2498

There is no differential change in volume so dv = 0.2513

Let me write this expression again, we have dq - P external dv = du dt sub V × dt + du dv sub T × dv.2517

If dv = 0 then this goes to 0, that goes to 0, what we are left with is the following.2540

Dq, we are going to go ahead and put that V there because we are under constant volume process, = du dt sub V dt.2548

Let us see, it gets to U = dqv and we write the other equation also, du = dq - the external dv.2569

This goes to 0 so we are also left with du = dqv.2589

What we have here is the following, under constant volume, the change in energy of the system = the change in heat.2599

It is equal the heat that is a lost by the surroundings or that heat gained by the system, if you want to take the systems point of view.2607

Again, we are taking the surroundings point of view for the most part.2615

For a constant volume processes, the heat that the surroundings loses happens to equal the change in energy.2619

If I want to know what the change in energy is, I just have to measure how much heat the surroundings loses.2627

Over here, change in heat, the amount of heat that the surroundings loses which is equal to the change in energy2632

is equal to the rate of change of the energy of the temperature × the temperature increment.2641

This equation which relates dqv which is the heat withdrawn from the surroundings, the heat lost by the surroundings.2651

The heat withdrawn from the surroundings to an increase in temperature of the system which is the dt.2669

Well, both of these are easily measurable, the change in temperature and the change in heat.2697

We can measure the change in heat lost by this, we can measure the heat lost by the surroundings.2711

We can measure the change in temperature of the system.2717

The ratio which is dqv/ dt it is called heat capacity.2722

The change and heat over a change in temperature, that is the definition of heat capacity.2733

In this particular case, because it is a constant volume process, this ratio of the heat of the surroundings loses2738

divided by the temperature by which the system, the temperature increase of the system,2745

this is defined as the constant volume heat capacity.2754

Constant volume heat capacity is defined as heat capacity in this particular case because2760

the process is happening under constant volume it is called the constant volume incapacity.2766

Dq sub V dt which is defined as a constant volume heat capacity C sub V.2776

That happens to be associated, identified, with this partial derivative.2784

A partial derivative which is the change in energy or change in temperature at constant volume.2792

We were able to associate some partial derivative with something that we can easily measure,2799

this P capacity which happens to be under constant volume.2807

The heat capacity just happens to be the heat withdrawn from the surroundings divided by the change in temperature of the system.2811

You are in chemistry, you are accustomed to thinking about it as a heat gain by the system divided by the temperature increase of the system.2822

That heat change of the system divided by the temperature change of the system, that is fine, it is the same thing.2831

Again, we are taking the surroundings point of view.2836

This is the heat lost by the surroundings.2838

It happens to be the heat gained by the system.2841

It is just a question of point of view, as long as you know what is happening.2844

That is the very definition of heat capacity, that is how we define it.2848

Let us just go ahead and do what we do with differentials.2853

We can just move this over here so dq sub V = cvdt.2856

We just move that there.2869

This is the differential form and we know this already.2872

The change in heat is equal to the heat capacity × the change in temperature.2879

We know this from general chemistry, constant volume.2883

Let us go ahead and write the integrated form here.2888

Our integrated form is, let me integrate that function, we get the energy = the integral from temperature 1, temperature 2, and CVDT.2895

We said that dqv=cvdt, dqv is equal to du.2923

That is what we get as the first law tells us when there is no change in volume.2932

We just put this over here, we will get this thing and we integrate it.2936

In other words, du is equal to cvdt and we integrate both sides.2941

The integral of du is δ U, the integral is that integral from temperature 1 to temperature 2.2947

From du= dqv, we also get this integrated version which is δ U= QV.2958

Q is a path function, so there is no δ Q, the change in energy of the system happens to equal at a constant volume processes,2974

the change in energy of the system happens to equal the heat withdrawn from the surroundings.2982

If we are measuring temperature, the change in energy of the system happens to equal the integral of the change in temperature,2989

the integral of the heat capacity of the system × the change in temperature from one temperature to the other.2997

These are the equations that are important.3003

If this is heat capacity happens to be, if T is constant over the range of temperature increase then we have δ U = CV δ T.3010

In other words, over a certain temperature increase, let us say 25 to 50, if the heat capacity does not change.3047

We do not need to integrate, we can just take the change in temperature.3052

This is how we seen it before in general chemistry.3055

Our assumption was that heat capacity is constant over a range of temperatures.3057

As it turns out, heat capacities temperature dependent.3063

The hotter something gets, the heat capacity changes.3066

This is the wheel equation, that right there.3070

These two equations, in other words, δ U= the integral from temperature 1 to temperature 2 of the constant volume heat capacity × that.3078

The change in energy = the heat withdrawn from the surroundings, these two equations,3094

they express the energy change of the system, in terms of measurable quantities.3104

That is what we wanted.3122

In terms of measurable quantities, that is good, that is what we want.3125

In science, we measure things.3135

We need to be able to measure things.3137

The theory that we got from manipulating to get a partial derivatives, that tells us one thing3140

but we need to associate these partial derivatives as something we can measure.3146

We have something that dudt V this, we can associate it with the heat capacity.3150

Heat capacity is easily measurable.3160

We measure the heat lost by the surroundings or the heat gained by the system and we divide by the temperature change in the system.3162

We have something mathematical that is associated with something physical, something we could measure.3168

This is beautiful, it is what we want.3174

Mathematically, δ U = QV.3179

Δu QV they have the same sign.3198

They have the same sign.3206

Now given our particular convention, a positive value of heat means that heat is flowing from the surroundings to the system.3209

Heat flows from the surroundings, in other words through the system.3231

If heat is positive and change in energy is positive.3255

A positive QV implies that δ U of the system, this is what all these means.3263

I’m just explicitly writing at this time in general.3272

It means that δ U is positive.3278

That means the system has increase in the energy of the system.3284

A negative QV means that heat is flowing to the surroundings.3295

In other words, the heat is flowing from the system.3302

The system is losing heat, therefore, the energy of the system decreases.3305

Δ U is negative.3311

If δ U is Q, Q is positive W is positive.3314

If Q is negative δ U is negative.3317

This is a decrease in energy.3321

The constant of volume heat capacity is always greater than 0.3330

A positive temperature change implies that δ U is positive.3339

It is an increase in energy, a drop in temperature implies that the δ U is negative.3348

If the temperature of the system increases, the energy of the system increases.3358

If the temperature of the system decreases, the energy of the system decreases.3361

That is what is going on here.3365

This is actually really important.3367

Therefore, at a constant volume, at constant V the temperature is a direct representation of the energy of the system.3369

Let us see what temperature actually is.3398

Temperature is a measure of the energy, the average energy of the system.3400

OK let us go ahead and do an example.3407

It is a very simple example.3410

A lot of these lectures I’m only going to have a lot of these preliminary lectures.3412

I already have 1 or 2 examples, do not worry about that.3417

In subsequent lessons and for several lessons, after we get these preliminary theoretical discussions out of the way,3423

that is where we are going to do the bulk of our examples.3429

It is not going to be a lesson where we have a little bit of discussion and a handful of examples.3432

I’m going to set aside complete lessons, several of them at the end of this unit, to do all of the example problems that we need.3437

Do not worry there is only 1 or 2 showing up in these particular lectures.3446

We are going to do a lot and when I say a lot I mean a lot.3449

We definitely need to get familiar with this material.3454

Handling the first law, we need to set a good foundation.3456

We are going to do a lot of problems, I promise you.3459

Let us take a look.3464

The first example we have, 2 mol helium gas, they are taken from a temperature of 25° C to a temperature of 55° C.3466

The molar heat capacity happens to be 3 ½ R, notice molar heat capacity.3477

This is the amount of heat J/ K/ mol.3485

Molar heat capacity, if I just said heat capacity it would be J/ K/ C.3491

If I say specific heat capacity, we are accustomed to in general chemistry it would be J/ K/g or J/℃/ g,3497

because specific heat does talk about mass, lower heat capacity, J/ mol/ K/ mol.3506

We want to find the change in energy of the system and we want to find the heat that is lost by the surroundings for the transformation.3512

We know that δ U is equal to the integral from temperature 1 to temperature 2 of CVDT, we know that already.3524

We also happen to know that δ U is equal to QV.3537

We find this and we find that.3541

Let us go ahead and work this one out, it is really simple.3544

Δ U is equal to the integral of T1 to T2, the CV or constant volume heat capacity is 3 ½ rdt.3548

The 3 ½ R is not a function of T, it is a constant so we can pull it out, = 3 ½ R × the integral of T1T2 dt.3561

That = 3 ½ R × δ T, notice I have not put the values in.3579

I may have to change this to 298 and change the 55 to whatever it is, 55 goes to 73.3584

In this particular case, because this is constant I do not to have to evaluate the integral.3593

This right here becomes R δ T, temperature is a state function.3598

The integral of the state function is δ of the T.3605

This is really simple, let us just go ahead and put the values in.3610

We get 3 ½ × R, the R value we are going to take is 8.314 J/ mol/ K.3614

The change in temperature is going to be the 55 - to 25, so 55° - 25°.3624

The δ T, in terms of Kelvin and Celsius is the same.3634

A difference of 1° C is the same as 1° K.3639

I do not actually have to convert to it to K when I’m doing δ.3642

This is going to be 30 K, K and K cancels and I end up with is, if I did my arithmetic correctly which more often that I do not, 374 J/ mol.3646

We are left with J/ mol.3663

Since we have 2 mol of helium, we multiply that and we get a total of 748 J.3667

748 J that is the change in energy.3680

So δ U= 748 J.3683

QV happens to equal that, QV = 748 J.3688

Under these circumstances of constant volume, this 2 mol of helium gas that goes for 25° to 55° C, the energy change was 748 J.3697

The energy of the system, the heat lost by the surroundings is 748 J.3707

The 748 J went from the surroundings to the system, that is what happened.3715

Let us go ahead and close this out.3726

The important equations for this particular lesson, we have the heat capacity which is defined as dq/ dt,3731

the change in heat over the change in temperature.3749

The heat lost by the surroundings divided by the temperature increase of the system or3751

the heat gain by the system divided by the temperature increase of the system.3756

Totally your choice, as long as you are consistent.3760

This happens to equal, it is associated with the derivative, this partial derivative,3763

the change in energy or the change in temperature at constant volume.3770

Du = CVDT, du = dq, δ U + integral from T1 to T2 of CVDT, this is just the differential form, this is the integrated form.3776

Δ U = Q not dq.3805

At constant volume, the heat that is lost by the surroundings = the change in energy of the system.3814

If I want to know what the change in energy of the system is, all I have to do is measure how much heat is lost by the surroundings.3827

And that is what I do, I do not measure directly into the system.3835

I measure what is happening in the surroundings which is why we keep saying surroundings.3838

I know that in chemistry we are accustomed to thinking about the system but where we can take our measurements3843

in order to find what was happening in the system is in the surroundings.3848

The change in energy of the system is equal to at under constant volume heat capacity ×3854

the differential change in temperature for the particular increment.3861

If I integrate all those increments over the temperature change, I get the change in energy of the system.3865

Thank you so much for joining us here at www.educator.com.3873

We will see you next time for a continuation of discussion, bye.3876

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