  Raffi Hovasapian

Entropy Example Problems III

Slide Duration:

Section 1: 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
Section 2: 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

35m 52s

Intro
0:00
0:10
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
18:08
18:09
21:54
22:34
Fundamental Relationship Equation for an Ideal Gas Under Adiabatic Expansion
25:00
More on the Equation
28:20
Important Equations
32:16
32:17
Reversible Adiabatic Change of State Equation
33:02
Section 3: 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
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
Section 4: 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
Section 5: 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
Section 6: 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
Section 7: 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
Section 8: 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
Section 9: 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
Section 10: Quantum Mechanics Preliminaries
Complex Numbers

34m 25s

Intro
0:00
Complex Numbers
0:11
Representing Complex Numbers in the 2-Dimmensional Plane
0:56
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
Schrӧdinger 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
Schrӧdinger 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 Schrӧdinger 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
Section 11: 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
Section 12: 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
Section 13: 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
Section 14: 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
Section 15: 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
Section 16: 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
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
Section 17: 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
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
28:02
Example: 1s Orbital
28:34
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
Section 18: 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
Section 19: 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
Section 20: 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
Section 21: 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
Section 22: 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
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
Section 23: 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
Section 24: 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
Section 25: 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
Bookmark & Share Embed

## Copy & Paste this embed code into your website’s HTML

Please ensure that your website editor is in text mode when you paste the code.
(In Wordpress, the mode button is on the top right corner.)
×
• - Allow users to view the embedded video in full-size.
Since this lesson is not free, only the preview will appear on your website.

• ## Related Books 1 answer Last reply by: Professor HovasapianMon Nov 26, 2018 5:08 AMPost by Kimberly Davis on November 25, 2018So in general, is the Standard Enthalpy of Formation temperature dependent? in example V you talked about the std Entropy of formation, the "o" sign just for 1 atm pressure with no information about the temperature.

### Entropy Example Problems III

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

• Intro 0:00
• Example I: Isothermal Expansion 0:09
• Example I: Calculate W
• Example I: Calculate ∆U
• Example I: Calculate Q
• Example I: Calculate ∆H
• Example I: Calculate ∆S
• Example II: Adiabatic and Reversible Expansion 6:10
• Example II: Calculate Q
• Example II: Basic Equation for the Reversible Adiabatic Expansion of an Ideal Gas
• Example II: Finding Volume
• Example II: Finding Temperature
• Example II: Calculate ∆U
• Example II: Calculate W
• Example II: Calculate ∆H
• Example II: Calculate ∆S
• 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
• Part B: Calculate ∆H for the Transformation of Solid Lead from 25°C to Liquid Lead at 525°C

### Transcription: Entropy Example Problems III

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

Today, we are going to continue our entropy example problems, let us jump right on in.0004

Our first example for today says that 1 mol of an ideal gas with a constant volume heat capacity of 3/2 Rn is in the initial following state 350°K and 1.5 atm.0011

This is just a continuation of the example problems from the previous section.0025

We have the same initial state, we are just subjecting this initial state to various transformations or transformations and the various circumstances.0030

This time the gas expands isothermally, we have a nice isothermal expansion against 0 external pressure.0038

This time it is going to be a free expansion.0046

A free expansion is when there is nothing on the other end keeping it from expanding.0048

0 external pressure of the gas is 0.75 atm, calculate Q, W, δ U, δ H, δ S and again we want compare the δ S with Q/ T.0054

0 external pressure free expansion until the pressure goes from 1.5 atm to 0.75 atm.0066

Let me go ahead and do this in red since the problems are concerned.0074

The definition of work is DW = P external × DV.0083

The P external is 0 external pressure so this is 0.0090

Therefore, DW =0 which means that our work for this process = 0.0095

In a free expansion no work is done because we are not pushing against anything.0100

There is no force resisting our push, our expansion.0104

This is an ideal gas which makes our job a little bit easier.0110

We are dealing with something that is an isothermal process.0115

Isothermal implies that δ U=0.0119

The whole bunch of things might actually end up being 0 here.0123

δ U = Q - W so Q = δ U + W.0129

Q = 0 + 0 so it looks like the heat in this is also going to be 0.0133

Let us take a look at δ H so DH = let us go ahead and use our equation just so we can actually practice writing it down.0148

CP DT + DH DP DP, this is an isothermal process so this is 0.0159

This is an ideal gas so this is 0 so we have δ H = 0.0170

It looks like everything is going to be 0 here.0177

Let us see what happens with δ S so DS = CV.0179

It is going to be CV we are using pressure here so this is going to be CP/ T DT - nR/ P DP.0194

This is an isothermal process so the change in temperature is 0 so this term goes to 0.0215

Therefore, we have just DS = - nR/ P DP and then when we integrate this differential equation0221

we are going to get the equation δ S = -n × R × log of P2/ P1 the nat log.0233

Let us go ahead and put some numbers in so we get - there is 1 mol that is going to be 8.314 nat log that we have0250

0.75 is the second temperature and 1.50 is the initial temperature.0260

We get δ S = 5.76 J/°K.0267

We end up with 5.76, the initial state the 350°K and 1.5 atm is the final state 350°K, 0.75 atm.0278

You remember the previous lesson, the last two examples, these were the initial and0290

final states because entropy is a state variable, the state function, state property.0297

The path does not matter how you do it.0304

It does not matter what you do, a reversible expansion against a particular pressure.0307

A constant pressure expansion or a free expansion, the entropy ends up being the same because the initial final states are the same.0313

Let us go ahead and do our comparison.0323

We have got δ S = 5.0327

We are slowing down in writing the numbers out properly 5.76 J/°K and then we have Q/ T is going to equal 0 J/ 350°K = 0 J/°K.0336

In this particular case because that Q was 0, our Q/ T = 0 and they do not match.0358

Let us go ahead and take a look at the next example.0370

1 mol of an ideal gas with this particular constant volume heat capacity is initially in the following state.0372

Again, same initial state this time the gas expands adiabatically and reversibly.0380

It is no longer isothermal so now are expanding it adiabatically and we are expanding it reversibly until the pressure of the gas is 0.75 atm.0385

That part stays the same.0395

Calculate all of these variables and compare δ S with Q/ T.0397

This time we are doing it adiabatically and reversibly.0403

Let us see here, let us see what this looks like if you remember this is the P and this is the V.0409

We have this which is an isotherm and we have this which is an adiabat.0419

We might have an initial state here, state 2.0427

If we follow the isotherm keeping the temperature constant, it is going to be this.0433

If we follow the adiabat, in other words an adiabatic condition is where there is no heat allowed to transfer anywhere.0438

You are going to end up with a huge temperature drop and reversibly just means we are actually following that path.0446

We are not going this way to that state.0455

We have an isotherm and we have an adiabat.0459

Adiabat that just means that Q = 0, DQ = 0.0463

Adiabatic we automatically know what Q is.0474

Adiabatic implies that Q = 0 so we have taken care of Q.0479

If we look back at reversible adiabatic expansion of an ideal gas, we find the following.0487

This is going to be a little bit of a review of adiabatic processes.0497

We have DU = DQ - DW this is the basic equation.0502

Since Q was 0, adiabatic DQ is 0.0509

What you end up with is DU = - DW.0513

For an ideal gas DU = CV DT and the definition of work DU is just P external × the change in volume.0519

What you end up with is CV DT = - P external × DV so this is the equation.0543

They also said that it is going reversibly and we know what reversible means.0563

Reversible means that the external pressure is actually equal to the external pressure on the system = the pressure in the system.0568

They are always in equilibrium, that is what reversible means.0578

That means we are following this path closely in terms of stair stepping.0581

Therefore, we can replace P external with P and what we get is CV DT =- P DV.0587

This is an ideal gas, P = nRT/ V and we did this derivation back when we discussed adiabatic processes but I thought it would be nice to do it again.0601

We have CV DT =- nRT/ V DV.0617

Let us go ahead and divide both sides by this variable T and we would end up with CV/ T DT = -nR/ V DV.0628

When we go ahead and integrate this equation, we are going to end up with the following.0644

We are going to end up with CV × the nat log of T2/ T1 = - nR × the nat log of V2/ V1.0652

This is our fundamental equation for an adiabatic reversible expansion or a compression.0669

This is the relationship between the constant volume heat capacity are the volume and the temperature for an ideal gas.0678

This is the basic equation.0689

This was the equation that we derived when we discuss this, when we talked about the energies.0694

This is the basic equation for the reversible adiabatic expansion of an ideal gas.0700

Let us go ahead and continue a little bit more.0721

Let me write the equation here on this page so we have it.0722

We have CV × nat log of T2/ T1 is going to equal - n × R × the nat log of V2/ V1.0726

We are going to use this thing.0746

Do you remember the ratio of the constant pressure heat capacity divided by the constant volume heat capacity we called it γ.0749

We are going to go ahead and use this.0757

Let me make this V a little more clear and we are going to do a little bit more mathematical manipulation on this using PV = nRT to derive some equations here.0758

I’m not going to go ahead and go through the mathematical manipulation.0771

If you want you can check your books, it is usually in every single thermodynamics book or you can go back to the previous lessons.0774

I think that I did it there.0782

We got the following relations, we got T1 V1 ⁺γ = T2 V2 ⁺γ.0783

I will go ahead and call this equation 1.0795

It expresses a relationship between temperature and volume.0797

This was also a relationship between temperature and pressure, it is T1 ⁺γ P1¹ - γ = T2 ⁺γ × P2¹ - γ.0801

I will call this equation 2 and there is one for relationship between pressure and volume.0815

We have P1 V1, I’m sorry this is γ -1.0822

The temperature and volume is γ -1, P1 V1 γ = P2 V2 γ.0829

I will go ahead and call this equation 3.0837

Using this CP/ CV calling it γ and then using the ideal gas PV = nRT, I can actually manipulate this fundamental equation0840

to derive these three relationships temperature volume, temperature pressure, and pressure volume, based on the heat capacity.0851

This is for an adiabatic process.0858

We are going to start, manipulating this.0862

The first thing I'm going to do is I’m going to start with equation number 3.0864

It just happens to be the one that I picked, so let us just jump right on in.0869

Let us go ahead and do this.0875

First of all, let us go ahead and see what γ is, so CP/ CV = γ so CP =5/2 Rn and CV is going to be 3/2 Rn.0877

γ is going to equal 5/3.0897

Let us go ahead and list what it is that we actually have.0905

We know our first pressure is 1.5 atm, we have our first volume which we can get from nR T1/ P1.0907

Our first volume = nR T1/ P1.0933

I’m just trying to get as many of the variables as possible.0944

I have the first pressure and I have the second pressure.0947

I can calculate the first volume but I want to get the second volume.0950

I have the first temperature I want to get the second temperature.0954

In other words, for this thing I have temperature 1 which was 350°K.0960

We are going to be able to find what the temperature 2 is.0965

I want to be able to find this and that so I have some variables to work with this that is why I'm doing this.0968

I have 1 mol, in this case R, we are using the ideal gas law so I have to use 0.08206.0976

Be very careful with that.0985

Temperature is going to be 350°K and the initial pressure is going to be 1.5 atm.0988

V1 is actually equal to 19.15 L.0997

I know what P2 is, P2 is 0.75 atm.1005

I want to know what volume 2 is so I have P1 V1, I have P2, I want to know what volume 2 is.1013

I’m going to go ahead and go to this equation right here and I know what γ is 5/3 so I get the following.1021

Pressure 1 is 1.5 atm and volume 1 is 19.15 L, 5/3 = 0.75 atm and I'm looking for volume 2 is going to be 5/3.1029

When I solve for volume 2, I get volume 2 = 29.03 L is my volume 2.1056

Let us see what I can do with that.1068

Since I have volume 2, now I want to find temperature 2.1072

Let us find temperature 2.1080

Temperature 2 is just nR T2.1084

Again, PV = nRT therefore T2 = P2 V2/ n × R.1096

The pressure 2 = 0.75 atm, volume 2 = 29.03 L, n is 1, and R is 0.08206.1114

If you use the ideal gas make sure you use the proper form of R and we end up with temperature 2 equal to 265.3°K.1130

This is cooler, we started at 350 and the temperature 1 =350°K.1143

Temperature 2 = 265.3, the gas is expanding.1154

It is expanding adiabatically, we want it to be cooler so our number matches, it is in the proper direction.1159

Cooler which makes sense, an adiabatic expansion gives you the largest temperature drop.1168

We can go ahead and actually work out the problem.1179

We just needed to find what these other values were.1182

We needed to find what V2 was, we need to find what T2 was, so we can use these in our equations.1185

That is what all of this led to.1192

Let us go ahead and start doing this.1194

DU = CV DT which means that δ U = CV × δ T which means 3/2 Rn δ T.1198

Therefore, δ U = 3/2 × 8.314 × 1 and δ T that is going to be the final temperature which is going to be 265.3 -350 because this is the final - the initial.1220

You end up with δ U = -1056 J.1242

In the process of expanding adiabatically and reversibly, the system ends up losing 1056 J of energy.1252

This is an adiabatic process so we said that δ U = - work = - δ U.1260

Let me go ahead and keep writing this – δ U, - - so 1056 J.1279

The gas is expanding therefore, it is doing work on the surroundings.1287

Work is positive from the surroundings point of view.1292

It is expanding so the expanding gas does 1056 J of work on the surroundings.1295

Let us go ahead and take a look at DH, DH = CP DT.1303

Therefore, δ H = CP × δ T this is what we want, we want δ T.1311

We wanted to find that first temperature so we can actually do the problems that we needed to do because DU is CV DT.1318

CV δ T this is CP δ T, that is why we went through the initial process.1329

Remember, we were only given the initial temperature, the initial and final pressures.1334

I use those relationships in order to find the second volume and the second temperature so that I can do these problems.1339

That = 5/2 Rn δ T and again CP if the 3/2 Rn is the constant volume capacity, this is an ideal gas at constant pressure heat capacity of 5/2 Rn.1348

This is just the 3/2 + 1.1364

Based on this relation, from ideal gas the difference between the CP and the CV= R.1373

If I’m given this I can find this.1379

Therefore, we have δ H = 5/2 0.314 × 1 × 265.3 -350.1383

Therefore, our δ H = - 1760 J.1400

Let us go ahead and do δ S.1419

What shall we do?1423

δ S for an ideal gas, I’m not going to go through the entire equation, I’m going to go through the final version of the equation.1426

It is equal to the constant pressure heat capacity × LN of T2/ T1 -nR × LN of P2/ P1.1432

That = 5/2 × 8.314 × 1 × the nat log of 265.3/ 350 -1 × 8.314 × the nat log of 0.75/ 1.5.1449

When I actually do that I end up with δ S= 0.1480

If I do Q/ T this is actually equal to Q reversible/ T.1487

Q reversible is adiabatic 0/ the temperature which is 350 so we end up with 0.1496

You see they match.1503

Reversible processes the entropy δ S and Q reversible/ T they will match.1507

Let us see what else have we got.1518

This one, the entropy of water is 69.95 J/°K at 25°C.1522

This is the entropy not that change in entropy, this is S not δ S, 25°C and 1 atm, 1 atm so it is standard.1531

This is S standard 298 for water is 69.95.1546

In other words, the general sense of disorder of liquid water and 25°C 1 atm pressure is 69.95 J/°K.1558

Given the following, calculate the entropy of water vapor at 180°C and 0.75 atm.1569

Let us see what we have got.1580

They give us the constant pressure heat capacity of the liquid water which is 75.29 J/ mol °K.1582

They give us the constant pressure heat capacity of the gas which is 33.58 J/°K.1589

They give us the heat of vaporization of water 40.656 kJ have to be injected into 1 mol of water that much energy has to be put into water to convert it from liquid to gas.1595

That is what this means.1607

Assume ideal behavior for the water vapor accounts, we can assume that it is an ideal gas.1609

Let us see what we can do here.1615

The basic equation that you are going to work with here is the following.1617

Two things that are happening, you are going for 25°C where water is a liquid.1621

You are going to a 180°C where water is a gas.1629

You have to calculate the entropy of the phase change, the entropy of the temperature rise, and there is one other thing going on here.1633

You are going from 1 atm to .75 atm so you have to account for the entropy change for the pressure change.1642

Here is what it looks like.1649

The standard entropy at any temperature is going to equal the standard entropy at the temperature that you know which in this case is 298 +1653

the entropy in going from this initial temperature all the way to the boiling point for water of the liquid state of water/ T DT.1671

This from the third law.1684

The third law says that if you want to calculate the entropy from temperature 1 to temperature 21688

it is going to be the constant pressure heat capacity of the particular phase divided by the temperature DT.1694

The entropy that we start off with a given temperature now we are going to calculate,1703

we are going to add to that the entropy in going from, in this particular case 25 C to 100°C.1709

The entropy change that accompanies the vaporization of the water at the boiling temperature,1719

the conversion of going from liquid to water vapor, we have more temperature rise.1724

The temperature boiling up to the particular temperature that we want and this time, since it is going to be water vapor it is going to be this.1733

That takes care of the temperature part, now we have to adjust for the pressure part - this is an ideal gas nR × the nat log of pressure 2/ pressure 1.1743

I hope that makes sense.1760

Remember that the CP/ T DT - nR/ P DP this accounts for the pressure change in entropy, the rise in entropy.1762

This accounts for the change in temperature.1780

All I have done is I have accounted for the entropy that I start off with.1782

The entropy of the temperature rise going from 25 to 100.1787

The change in entropy in going from liquid to the water vapor.1791

The change in entropy in going from 100°C to 180°C.1795

I account for the entropy decrease in the pressure difference.1799

In this particular case, because the pressure is going down 1 atm to 0.75 atm, that pressure decline is going to actually accompany volume increase.1804

Our volume increase is going to be actually you can end up getting a positive number here.1814

It is going to be slightly more entropy than usual.1818

Let us go ahead and put the numbers in.1821

This is going to be, we said 180°C.1824

It is going to be 453°K.1830

Our S and now it can no longer standard pressure 1 atm.1837

It is going to be 0.7518 atm and 453°K that is going to equal, the entropy of the temperature that we do know so the S at 298 = 69.95 +1845

the integral from 298 to 373 of the constant pressure heat capacity 75.29.1866

This is going to be 75.29/ T DT + 40656 J divided by the boiling temperature which is 373°K + this 1.1878

This is going to be 373 to 453, this time we are going to use the 33.58 which is the constant pressure heat capacity for water vapor, 33.58/ T DT.1900

We are going to subtract so it was going to be 1 mol and this could be 8.314 and this is going to be the nat log of 0.75 atm1917

which is P2 or the initial which is 1 atm.1936

This is the integral that we solve.1942

Let us go ahead and write out a little bit more.1945

S at 0.75 atm and 453 = 69.95 + 75.29 × the nat log of 373/ 298 + when I do the 4656 divided by the 373 I get 109.1950

I'm going to add to that + 33.58 × the nat log of 453/ 373.1983

It is going to be -2.39 and I get the entropy of water vapor at 180°C which is 453°K and 0.75 atm is going to equal 205 J/ mol/°K.1996

Here we go, just a basic application of the mathematical portion of the third law of thermodynamics.2028

There is nothing strange going on here.2036

You start with a particular entropy that you know and you account for the any temperature change.2038

If there is a phase change you account for the entropy change for the phase change.2044

After the change of phase there is still the temperature increase then you account for that temperature increase.2048

If there is a pressure increase or decrease, you account for that pressure increase or decrease.2054

That is all that you are doing.2059

You are accounting for every single change that takes place in the system.2060

Let us see what we have got here.2068

Silicon dioxide experiences the following transformation.2076

We go from 25°C and 1 atm pressure, we are going to raise the temperature to 225°C and we are going to raise the pressure to 1500 atm.2080

There are two things going on, change in temperature and a change in pressure.2090

Given the following data, calculate the molar δ S for the transformation.2093

The constant pressure heat capacity for the solid is this, notice it is not a constant, it is going to be a function of T.2099

The density of silicon is 2.648 g/ cm³ and the coefficient of thermal expansion is 3.530 × 10⁻⁵.2107

Sorry I forgot the unit here, for thermal expansion this is going to be per °K.2117

Let me do it in black, my apologies.2124

It is important that we get these.2128

Coefficient of the thermal expansion this is α.2132

α = 3.530 × 10⁻⁵ /°K.2137

It represents the percentage change in the volume of something when you heated up, relative to how much you started off with.2142

Let us go ahead and see what we can do.2154

We are changing temperature and we are changing pressure, let me go back to red here.2157

We are going to use this equation, is our fundamental equation so the DS = CP/ T DT – V A DP.2164

We are dealing with a solid here, this is a general system.2181

We are no longer dealing with an ideal gas.2183

For the most general equation, the entropy change when you change temperature and pressure this is it right here.2185

There is nothing strange going on here.2194

δ S when we integrate this so we get δ S = the integral from temperature 1 to temperature 2 of CP/ T DT - V × A × the integral from pressure 1 to pressure 2 of DP.2199

We have our final equation of DS = this stays the same, we cannot pull the CP out because CP is now no longer constant.2229

It is going to be the integral of CP/ T DT from T1 to T2 , this is just DP so - V A and δ P.2241

This is the equation that we are going to use.2256

It looks like we are going to need the sub V.2259

We are going to need the molar volume.2264

We need to know what the volume is of this particular thing.2266

They wanted us to calculate the molar δ S, they gave us the density, they did not give us the molar volume.2271

The first thing we have to do is we have to calculate V.2277

Let me go ahead and do that in blue.2280

We have silicon dioxide and that is going to be 60.09 g/ mol, let us start off with that way.2287

We have 60.09 g/ mol and we have 1 cm³ is 2.648 g.2308

2.648 g that takes care of that so that = 22.69 cm³/ mol so that is the molar volume.2323

This is V but it is in cm³.2339

I want to express it in dm³ which is the same as the L.2342

That is going to equal 22.698 × 10⁻³ cm³.2348

I hope you guys are okay with converting from cm to dm to m, things like that.2355

Dm/ mol which is the same as 22.69 so dm³ is a L.2360

22.69 × 10⁻³ L/ mol this is what I wanted, this is my V right here.2370

In 1 mol of this stuff because we are calculating molar, the volume is 22.69 × 10⁻³ L.2380

1 mol of silicon dioxide has this volume, now I can go ahead and do my problem.2387

Let us go ahead and rewrite what I need.2396

I need to go back to red.2401

There we go so δ S = the integral from T1 to T2 of CP/ T DT – V A δ P.2406

I got my δ S = the integral from 298 to 498 and now I write 46.94 + 34.31 × 10⁻³ × T -11.30 × 10⁵ T⁻²2421

that is the constant pressure heat capacity of silica/ T that is the integral part and it is going to be –volume.2455

The volume is 22.69 × 10⁻³ L/ mol and A which is the coefficient of thermal expansion they give us at 3.530 × 10⁻⁵/°K.2467

The change in pressure δ P is going to be 1500 – 1.2494

I’ m going to keep multiplying this by, notice this is going to end up being in L atm.2501

I need to put this into J.2512

I’m going to multiply it by 8.314 J = .08206 L atm.2515

It is important if you are going to be working in L atm, you want to convert this to J.2531

This is the conversion factor, it is just the ratio of the 2 R.2537

8.314 J = .08206 L atm they are both units of energy.2542

When I do that, atm cancels atm, L cancels L , you end up with J/ mol °K.2548

Everything works out right.2563

When you do all this, let us actually calculate this.2564

The integral comes out to be 26.88 J / mol °K and this ends up being 0.12 J/ mol °K.2567

Our final δ S is 26.76 J/ mol °K.2590

Clearly the change in temperature accounts for 26.88 J/ mol °K change in entropy.2600

The change in the pressure from 1 to 1500 atm, 1500 atm is massive, it is huge.2607

1500 atm, all that pressure change.2614

This is solid and a solid is not going to contract all that much under that much pressure, even that much pressure.2618

Its only difference is 0.12.2627

Clearly, the change in entropy of the system especially of the solid or liquid, under a variation of pressure is virtually negligible.2630

You are not losing anything by just ignoring this 0.12 but we want you to see it because we want to be able to solve the problem.2640

This is how you solve the problem, your basic equation.2646

Let us see what else have we got here.2657

The standard entropy of lead at 25°C is 64.8 J/ mol °K.2663

Standard means 1 atm it does not necessarily mean 25°C.2671

In general chemistry, when you see this little degree sign on top, it generally means 25°C and 1 atm.2679

The degree sign ° really just means the temperature.2686

We specify the temperature because in these problems now we are a little bit more sophisticated than we were in General Chemistry.2689

We can calculate S, δ H, δ S at different temperatures.2696

This degree sign ° represents the temperature, that is what standard means.2702

Standard pressure is 1 atm.2708

The constant pressure heat capacity is this, you notice it is not at constant, it is a function of the temperature.2712

The constant heat capacity to liquid at 32.51 again it is not a constant.2720

The melting temperature of lead is 327.4°C, the δ H of fusion of melting is 4.770 kJ/ mol.2728

They want us to calculate the standard entropy of liquid lead at 525°C.2738

Here it is going to be solid lead at 25, what is the entropy of liquid lead and calculate the δ H2744

for this transformation of solid lead from 25°C to liquid lead at 525°C.2749

This is going to be exactly like the problems that we did a little bit earlier when we calculated the entropy change for water,2758

from liquid water to water vapor.2763

Let us see what we can do.2768

We have got 25 and 525.2770

Let us see what we have.2774

We have S at 298 = 64.80 J/°K and we want S.2780

Still standard so there is no change in pressure, all we are doing is actually changing the temperature, it is actually melting the lead.2799

It is going to be at 823 this is what we want.2805

Let us see what we have got.2812

We are going to write S of 823 = S of 298 + the integral from 298 to the melting temperature of the solid heat capacity/ T DT.2814

The entropy at a given temperature + the new entropy at the new temperature which is melting and we are going to account2839

for the entropy of the phase change δ H of fusion/ the temperature of melting, boiling from going to solid to liquid.2848

This is the melting temperature.2857

We are going to heat that liquid some more.2860

We are going to go from the melting temperature all the way to this 823, except this time we are going to use the constant pressure heat capacity of liquid.2862

That is the integral that we want.2875

Let us go ahead and put in our numbers so that S 823 = 64.80 + the integral from 298 to the temperature of melting2877

which is 300 and something, they said would actually ends up being 600.4.2891

It is going to be the 22.13 + 0.01172 T + 0.96 × 10⁵ T⁻² / T DT.2900

We are going to add to that the 4770 J divided by the melting temperature which was 4850.4 °K, that is going to be the transition.2924

We are going to add to that the entropy change in going from 600.4 to 823.2937

This is going to be the 32.51 -0.00301 T/ T DT.2947

I can go ahead and let you work out the arithmetic here.2963

I’m not going to give you the answer, this is the answer right here.2967

You can work out the integral and whatever number you get that is the answer.2970

I just decided that I did not feel working this particular thing out, this is what is important.2974

You are just taking the entropy at any other temperature is the entropy at some temperature that you know.2983

For all practical purposes, the entropy of the temperature you know comes from the table of thermodynamic data in the back of your books.2990

Just like in the back of your General Chemistry books, there was the enthalpy, the free energy, and the entropy.2997

Those are standard third law entropy at 25°C and 1 atm pressure, that is going to be your starting point.3004

If you want to calculate the entropy of any other temperature at any other pressure, you have to account for the change.3016

25°C this is the entropy, you add to it the entropy that you get from the temperature rise from 298 to 600.3024

This is the entropy rising going from the solid to liquid state and this is going to be entropy change in going from 600.4 to 823°K.3033

That is all you are doing and it is going to be the same thing for δ H.3045

You remember when we did that earlier for energy, we are going to be doing it again in just a moment.3049

If we want the standard entropy at still higher temperatures, at temperatures above boiling,3061

let us say we took this solid lead to liquid lead, that is why we boiled off that lead to turn it into lead vapor3086

and we raise the temperatures too, then we just add more terms.3092

Temperatures above boiling then we just add the appropriate terms.3096

In this particular case, they would add the δ H of vaporization divided by the temperature of boiling, that is the entropy in going from liquid to vapor.3113

We add to that if we are going to still raise it above the boiling temperature, temperature of boiling to whatever final temperature and3124

this time we are going to use the constant pressure heat capacity of the gas / T DT.3131

That is all nice and simple.3138

Let us do part B, this time it asks for the δ H.3145

I’m going to go ahead and write δ H = CP DT + DH DP T DP.3157

The pressure is constant in this particular case so this is 0.3166

δ H is just equal to the integral from T1 to T2 of CP DT.3171

If there are any phase changes we simply add the δ H for the phase change, same as before.3181

Our general equation becomes the δ H at any temperature T that I want is going to equal T0 to T melting of the CP solid DT.3189

In this particular case, lead going from 298 to its melting temperature + the δ H of fusion and we are talking about δ H here not δ S.3207

Once I have actually melted it, I raise the temperature some more, it is going to be the melting temperatures3220

to whatever final temperature that I want.3227

This time it is going to be the constant pressure heat capacity of the liquid DT.3230

We have δ H at 823 = 298 to 600.4 of the 22.13 + 0.01172 T + 0.96 × 10⁵ T⁻² DT.3239

I'm going to add to that the 4770 J and I'm going to add to that the change in enthalpy for going from the melting temperature of 823.3274

This time I’m going to use the heat capacity of liquid lead which is 32.51 -0.00301 T DT.3291

I'm going to let you work out whenever this number is.3305

This is the answer, the rest is just arithmetic and integration and stuff like that.3312

If I needed to know the δ H at higher temperature still, let us say I want to take solid lead turn it into liquid lead like I did here and3320

then take the liquid lead and turn it into a gaseous lead and still raise the temperature, I would add the two following terms.3329

I have to account for the δ H that comes from the vaporization and then if I raise the temperature beyond that,3336

I would go from the boiling temperature to whatever temperature I wanted and this time I would use the constant pressure heat capacity of the gas.3345

It is solid to liquid, transition from solid to liquid temperature change during the liquid phase, if I needed to,3366

I would include the δ H of the transition from liquid to gas and then any other temperature change during the gas.3385

This is just normal heat.3392

This is the solid phase, the liquid phase, the gas phase, these are constant temperature processes.3399

These are the phase changes so these are accounted for by the δ H.3406

This is a vaporization, this is the δ H of fusion.3411

Here I have to do them in terms of integration.3416

That is all that is happening here.3420

Thank you so much for joining us here at www.educator.com.3424

We will see you next time, bye.3426

OR

### Start Learning Now

Our free lessons will get you started (Adobe Flash® required).