Raffi Hovasapian

Entropy Example Problems II

Slide Duration:Table of Contents

46m 5s

- Intro0:00
- Course Overview0:16
- Thermodynamics & Classical Thermodynamics0:17
- Structure of the Course1:30
- The Ideal Gas Law3:06
- Ideal Gas Law: PV=nRT3:07
- Units of Pressure4:51
- Manipulating Units5:52
- Atmosphere : atm8:15
- Millimeter of Mercury: mm Hg8:48
- SI Unit of Volume9:32
- SI Unit of Temperature10:32
- Value of R (Gas Constant): Pv = nRT10:51
- Extensive and Intensive Variables (Properties)15:23
- Intensive Property15:52
- Extensive Property16:30
- Example: Extensive and Intensive Variables18:20
- Ideal Gas Law19:24
- Ideal Gas Law with Intensive Variables19:25
- Graphing Equations23:51
- Hold T Constant & Graph P vs. V23:52
- Hold P Constant & Graph V vs. T31:08
- Hold V Constant & Graph P vs. T34:38
- Isochores or Isometrics37:08
- More on the V vs. T Graph39:46
- More on the P vs. V Graph42:06
- Ideal Gas Law at Low Pressure & High Temperature44:26
- Ideal Gas Law at High Pressure & Low Temperature45:16

46m 2s

- Intro0:00
- Math Lesson 1: Partial Differentiation0:38
- Overview0:39
- Example I3:00
- Example II6:33
- Example III9:52
- Example IV17:26
- Differential & Derivative21:44
- What Does It Mean?21:45
- Total Differential (or Total Derivative)30:16
- Net Change in Pressure (P)33:58
- General Equation for Total Differential38:12
- Example 5: Total Differential39:28

1h 6m 45s

- Intro0:00
- Properties of Thermodynamic State1:38
- Big Picture: 3 Properties of Thermodynamic State1:39
- Enthalpy & Free Energy3:30
- Associated Law4:40
- Energy & the First Law of Thermodynamics7:13
- System & Its Surrounding Separated by a Boundary7:14
- In Other Cases the Boundary is Less Clear10:47
- State of a System12:37
- State of a System12:38
- Change in State14:00
- Path for a Change in State14:57
- Example: State of a System15:46
- Open, Close, and Isolated System18:26
- Open System18:27
- Closed System19:02
- Isolated System19:22
- Important Questions20:38
- Important Questions20:39
- Work & Heat22:50
- Definition of Work23:33
- Properties of Work25:34
- Definition of Heat32:16
- Properties of Heat34:49
- Experiment #142:23
- Experiment #247:00
- More on Work & Heat54:50
- More on Work & Heat54:51
- Conventions for Heat & Work1:00:50
- Convention for Heat1:02:40
- Convention for Work1:04:24
- Schematic Representation1:05:00

1h 6m 33s

- Intro0:00
- The First Law of Thermodynamics0:53
- The First Law of Thermodynamics0:54
- 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 Ways11:56
- Systems Possess Energy, Not Heat or Work12:45
- Scenario 116:00
- Scenario 216:46
- State Property, Path Properties, and Path Functions18:10
- Pressure-Volume Work22:36
- When a System Changes22:37
- Gas Expands24:06
- Gas is Compressed25:13
- Pressure Volume Diagram: Analyzing Expansion27:17
- What if We do the Same Expansion in Two Stages?35:22
- Multistage Expansion43:58
- General Expression for the Pressure-Volume Work46:59
- Upper Limit of Isothermal Expansion50:00
- Expression for the Work Done in an Isothermal Expansion52:45
- Example 2: Find an Expression for the Maximum Work Done by an Ideal Gas upon Isothermal Expansion56:18
- Example 3: Calculate the External Pressure and Work Done58:50

1h 2m 17s

- Intro0:00
- Compression0:20
- Compression Overview0:34
- Single-stage compression vs. 2-stage Compression2:16
- Multi-stage Compression8:40
- Example I: Compression14:47
- Example 1: Single-stage Compression14:47
- Example 1: 2-stage Compression20:07
- Example 1: Absolute Minimum26:37
- More on Compression32:55
- Isothermal Expansion & Compression32:56
- External & Internal Pressure of the System35:18
- Reversible & Irreversible Processes37:32
- Process 1: Overview38:57
- Process 2: Overview39:36
- Process 1: Analysis40:42
- Process 2: Analysis45:29
- Reversible Process50:03
- Isothermal Expansion and Compression54:31
- Example II: Reversible Isothermal Compression of a Van der Waals Gas58:10
- Example 2: Reversible Isothermal Compression of a Van der Waals Gas58:11

1h 4m 39s

- Intro0:00
- Recall0:37
- State Function & Path Function0:38
- First Law2:11
- Exact & Inexact Differential2:12
- Where Does (∆U = Q - W) or dU = dQ - dU Come from?8:54
- Cyclic Integrals of Path and State Functions8:55
- Our Empirical Experience of the First Law12:31
- ∆U = Q - W18:42
- Relations between Changes in Properties and Energy22:24
- Relations between Changes in Properties and Energy22:25
- Rate of Change of Energy per Unit Change in Temperature29:54
- Rate of Change of Energy per Unit Change in Volume at Constant Temperature32:39
- Total Differential Equation34:38
- Constant Volume41:08
- If Volume Remains Constant, then dV = 041:09
- Constant Volume Heat Capacity45:22
- Constant Volume Integrated48:14
- Increase & Decrease in Energy of the System54:19
- Example 1: ∆U and Qv57:43
- Important Equations1:02:06

16m 50s

- Intro0:00
- Joule's Experiment0:09
- Joule's Experiment1:20
- Interpretation of the Result4:42
- The Gas Expands Against No External Pressure4:43
- Temperature of the Surrounding Does Not Change6:20
- System & Surrounding7:04
- Joule's Law10:44
- More on Joule's Experiment11:08
- Later Experiment12:38
- Dealing with the 2nd Law & Its Mathematical Consequences13:52

43m 40s

- Intro0:00
- Changes in Energy & State: Constant Pressure0:20
- Integrating with Constant Pressure0:35
- Defining the New State Function6:24
- Heat & Enthalpy of the System at Constant Pressure8:54
- Finding ∆U12:10
- dH15:28
- Constant Pressure Heat Capacity18:08
- Important Equations25:44
- Important Equations25:45
- Important Equations at Constant Pressure27:32
- Example I: Change in Enthalpy (∆H)28:53
- Example II: Change in Internal Energy (∆U)34:19

32m 23s

- Intro0:00
- The Relationship Between Cp & Cv0:21
- For a Constant Volume Process No Work is Done0:22
- For a Constant Pressure Process ∆V ≠ 0, so Work is Done1:16
- The Relationship Between Cp & Cv: For an Ideal Gas3:26
- The Relationship Between Cp & Cv: In Terms of Molar heat Capacities5:44
- Heat Capacity Can Have an Infinite # of Values7:14
- The Relationship Between Cp & Cv11:20
- When Cp is Greater than Cv17:13
- 2nd Term18:10
- 1st Term19:20
- Constant P Process: 3 Parts22:36
- Part 123:45
- Part 224:10
- Part 324:46
- Define : γ = (Cp/Cv)28:06
- For Gases28:36
- For Liquids29:04
- For an Ideal Gas30:46

39m 15s

- Intro0:00
- General Equations0:13
- Recall0:14
- How Does Enthalpy of a System Change Upon a Unit Change in Pressure?2:58
- For Liquids & Solids12:11
- For Ideal Gases14:08
- For Real Gases16:58
- The Joule Thompson Experiment18:37
- The Joule Thompson Experiment Setup18:38
- The Flow in 2 Stages22:54
- Work Equation for the Joule Thompson Experiment24:14
- Insulated Pipe26:33
- Joule-Thompson Coefficient29:50
- Changing Temperature & Pressure in Such a Way that Enthalpy Remains Constant31:44
- Joule Thompson Inversion Temperature36:26
- Positive & Negative Joule-Thompson Coefficient36:27
- Joule Thompson Inversion Temperature37:22
- Inversion Temperature of Hydrogen Gas37:59

35m 52s

- Intro0:00
- Adiabatic Changes of State0:10
- Adiabatic Changes of State0:18
- Work & Energy in an Adiabatic Process3:44
- Pressure-Volume Work7:43
- Adiabatic Changes for an Ideal Gas9:23
- Adiabatic Changes for an Ideal Gas9:24
- Equation for a Fixed Change in Volume11:20
- Maximum & Minimum Values of Temperature14:20
- Adiabatic Path18:08
- Adiabatic Path Diagram18:09
- Reversible Adiabatic Expansion21:54
- Reversible Adiabatic Compression22:34
- Fundamental Relationship Equation for an Ideal Gas Under Adiabatic Expansion25:00
- More on the Equation28:20
- Important Equations32:16
- Important Adiabatic Equation32:17
- Reversible Adiabatic Change of State Equation33:02

42m 40s

- Intro0:00
- Fundamental Equations0:56
- Work2:40
- Energy (1st Law)3:10
- Definition of Enthalpy3:44
- Heat capacity Definitions4:06
- The Mathematics6:35
- Fundamental Concepts8:13
- Isothermal8:20
- Adiabatic8:54
- Isobaric9:25
- Isometric9:48
- Ideal Gases10:14
- Example I12:08
- Example I: Conventions12:44
- Example I: Part A15:30
- Example I: Part B18:24
- Example I: Part C19:53
- Example II: What is the Heat Capacity of the System?21:49
- Example III: Find Q, W, ∆U & ∆H for this Change of State24:15
- Example IV: Find Q, W, ∆U & ∆H31:37
- Example V: Find Q, W, ∆U & ∆H38:20

1h 23s

- Intro0:00
- Example I0:11
- Example I: Finding ∆U1:49
- Example I: Finding W6:22
- Example I: Finding Q11:23
- Example I: Finding ∆H16:09
- Example I: Summary17:07
- Example II21:16
- Example II: Finding W22:42
- Example II: Finding ∆H27:48
- Example II: Finding Q30:58
- Example II: Finding ∆U31:30
- Example III33:33
- Example III: Finding ∆U, Q & W33:34
- Example III: Finding ∆H38:07
- Example IV41:50
- Example IV: Finding ∆U41:51
- Example IV: Finding ∆H45:42
- Example V49:31
- Example V: Finding W49:32
- Example V: Finding ∆U55:26
- Example V: Finding Q56:26
- Example V: Finding ∆H56:55

44m 34s

- Intro0:00
- Example I0:15
- Example I: Finding the Final Temperature3:40
- Example I: Finding Q8:04
- Example I: Finding ∆U8:25
- Example I: Finding W9:08
- Example I: Finding ∆H9:51
- Example II11:27
- Example II: Finding the Final Temperature11:28
- Example II: Finding ∆U21:25
- Example II: Finding W & Q22:14
- Example II: Finding ∆H23:03
- Example III24:38
- Example III: Finding the Final Temperature24:39
- Example III: Finding W, ∆U, and Q27:43
- Example III: Finding ∆H28:04
- Example IV29:23
- Example IV: Finding ∆U, W, and Q25:36
- Example IV: Finding ∆H31:33
- Example V32:24
- Example V: Finding the Final Temperature33:32
- Example V: Finding ∆U39:31
- Example V: Finding W40:17
- Example V: First Way of Finding ∆H41:10
- Example V: Second Way of Finding ∆H42:10

59m 7s

- Intro0:00
- Example I: Find ∆H° for the Following Reaction0:42
- Example II: Calculate the ∆U° for the Reaction in Example I5:33
- Example III: Calculate the Heat of Formation of NH₃ at 298 K14:23
- Example IV32:15
- Part A: Calculate the Heat of Vaporization of Water at 25°C33:49
- Part B: Calculate the Work Done in Vaporizing 2 Mols of Water at 25°C Under a Constant Pressure of 1 atm35:26
- Part C: Find ∆U for the Vaporization of Water at 25°C41:00
- Part D: Find the Enthalpy of Vaporization of Water at 100°C43:12
- Example V49:24
- Part A: Constant Temperature & Increasing Pressure50:25
- Part B: Increasing temperature & Constant Pressure56:20

49m 16s

- Intro0:00
- Entropy, Part 10:16
- Coefficient of Thermal Expansion (Isobaric)0:38
- Coefficient of Compressibility (Isothermal)1:25
- Relative Increase & Relative Decrease2:16
- More on α4:40
- More on κ8:38
- Entropy, Part 211:04
- Definition of Entropy12:54
- Differential Change in Entropy & the Reversible Path20:08
- State Property of the System28:26
- Entropy Changes Under Isothermal Conditions35:00
- Recall: Heating Curve41:05
- Some Phase Changes Take Place Under Constant Pressure44:07
- Example I: Finding ∆S for a Phase Change46:05

33m 59s

- Intro0:00
- Math Lesson II0:46
- Let F(x,y) = x²y³0:47
- Total Differential3:34
- Total Differential Expression6:06
- Example 19:24
- More on Math Expression13:26
- Exact Total Differential Expression13:27
- Exact Differentials19:50
- Inexact Differentials20:20
- The Cyclic Rule21:06
- The Cyclic Rule21:07
- Example 227:58

54m 37s

- Intro0:00
- Entropy As a Function of Temperature & Volume0:14
- Fundamental Equation of Thermodynamics1:16
- Things to Notice9:10
- Entropy As a Function of Temperature & Volume14:47
- Temperature-dependence of Entropy24:00
- Example I26:19
- Entropy As a Function of Temperature & Volume, Cont.31:55
- Volume-dependence of Entropy at Constant Temperature31:56
- Differentiate with Respect to Temperature, Holding Volume Constant36:16
- Recall the Cyclic Rule45:15
- Summary & Recap46:47
- Fundamental Equation of Thermodynamics46:48
- For Entropy as a Function of Temperature & Volume47:18
- The Volume-dependence of Entropy for Liquids & Solids52:52

31m 18s

- Intro0:00
- Entropy as a Function of Temperature & Pressure0:17
- Entropy as a Function of Temperature & Pressure0:18
- Rewrite the Total Differential5:54
- Temperature-dependence7:08
- Pressure-dependence9:04
- Differentiate with Respect to Pressure & Holding Temperature Constant9:54
- Differentiate with Respect to Temperature & Holding Pressure Constant11:28
- Pressure-Dependence of Entropy for Liquids & Solids18:45
- Pressure-Dependence of Entropy for Liquids & Solids18:46
- Example I: ∆S of Transformation26:20

23m 6s

- Intro0:00
- Summary of Entropy So Far0:43
- Defining dS1:04
- Fundamental Equation of Thermodynamics3:51
- Temperature & Volume6:04
- Temperature & Pressure9:10
- Two Important Equations for How Entropy Behaves13:38
- State of a System & Heat Capacity15:34
- Temperature-dependence of Entropy19:49

25m 42s

- Intro0:00
- Entropy Changes for an Ideal Gas1:10
- General Equation1:22
- The Fundamental Theorem of Thermodynamics2:37
- Recall the Basic Total Differential Expression for S = S (T,V)5:36
- For a Finite Change in State7:58
- If Cv is Constant Over the Particular Temperature Range9:05
- Change in Entropy of an Ideal Gas as a Function of Temperature & Pressure11:35
- Change in Entropy of an Ideal Gas as a Function of Temperature & Pressure11:36
- Recall the Basic Total Differential expression for S = S (T, P)15:13
- For a Finite Change18:06
- Example 1: Calculate the ∆S of Transformation22:02

43m 39s

- Intro0:00
- Entropy Example Problems I0:24
- Fundamental Equation of Thermodynamics1:10
- Entropy as a Function of Temperature & Volume2:04
- Entropy as a Function of Temperature & Pressure2:59
- Entropy For Phase Changes4:47
- Entropy For an Ideal Gas6:14
- Third Law Entropies8:25
- Statement of the Third Law9:17
- Entropy of the Liquid State of a Substance Above Its Melting Point10:23
- Entropy For the Gas Above Its Boiling Temperature13:02
- Entropy Changes in Chemical Reactions15:26
- Entropy Change at a Temperature Other than 25°C16:32
- Example I19:31
- Part A: Calculate ∆S for the Transformation Under Constant Volume20:34
- Part B: Calculate ∆S for the Transformation Under Constant Pressure25:04
- Example II: Calculate ∆S fir the Transformation Under Isobaric Conditions27:53
- Example III30:14
- Part A: Calculate ∆S if 1 Mol of Aluminum is taken from 25°C to 255°C31:14
- Part B: If S°₂₉₈ = 28.4 J/mol-K, Calculate S° for Aluminum at 498 K33:23
- Example IV: Calculate Entropy Change of Vaporization for CCl₄34:19
- Example V35:41
- Part A: Calculate ∆S of Transformation37:36
- Part B: Calculate ∆S of Transformation39:10

56m 44s

- Intro0:00
- Example I0:09
- Example I: Calculate ∆U1:28
- Example I: Calculate Q3:29
- Example I: Calculate Cp4:54
- Example I: Calculate ∆S6:14
- Example II7:13
- Example II: Calculate W8:14
- Example II: Calculate ∆U8:56
- Example II: Calculate Q10:18
- Example II: Calculate ∆H11:00
- Example II: Calculate ∆S12:36
- Example III18:47
- Example III: Calculate ∆H19:38
- Example III: Calculate Q21:14
- Example III: Calculate ∆U21:44
- Example III: Calculate W23:59
- Example III: Calculate ∆S24:55
- Example IV27:57
- Example IV: Diagram29:32
- Example IV: Calculate W32:27
- Example IV: Calculate ∆U36:36
- Example IV: Calculate Q38:32
- Example IV: Calculate ∆H39:00
- Example IV: Calculate ∆S40:27
- Example IV: Summary43:41
- Example V48:25
- Example V: Diagram49:05
- Example V: Calculate W50:58
- Example V: Calculate ∆U53:29
- Example V: Calculate Q53:44
- Example V: Calculate ∆H54:34
- Example V: Calculate ∆S55:01

57m 6s

- Intro0:00
- Example I: Isothermal Expansion0:09
- Example I: Calculate W1:19
- Example I: Calculate ∆U1:48
- Example I: Calculate Q2:06
- Example I: Calculate ∆H2:26
- Example I: Calculate ∆S3:02
- Example II: Adiabatic and Reversible Expansion6:10
- Example II: Calculate Q6:48
- Example II: Basic Equation for the Reversible Adiabatic Expansion of an Ideal Gas8:12
- Example II: Finding Volume12:40
- Example II: Finding Temperature17:58
- Example II: Calculate ∆U19:53
- Example II: Calculate W20:59
- Example II: Calculate ∆H21:42
- Example II: Calculate ∆S23:42
- Example III: Calculate the Entropy of Water Vapor25:20
- Example IV: Calculate the Molar ∆S for the Transformation34:32
- Example V44:19
- Part A: Calculate the Standard Entropy of Liquid Lead at 525°C46:17
- Part B: Calculate ∆H for the Transformation of Solid Lead from 25°C to Liquid Lead at 525°C52:23

54m 35s

- Intro0:00
- Entropy & Probability0:11
- Structural Model3:05
- Recall the Fundamental Equation of Thermodynamics9:11
- Two Independent Ways of Affecting the Entropy of a System10:05
- Boltzmann Definition12:10
- Omega16:24
- Definition of Omega16:25
- Energy Distribution19:43
- The Energy Distribution19:44
- In How Many Ways can N Particles be Distributed According to the Energy Distribution23:05
- Example I: In How Many Ways can the Following Distribution be Achieved32:51
- Example II: In How Many Ways can the Following Distribution be Achieved33:51
- Example III: In How Many Ways can the Following Distribution be Achieved34:45
- Example IV: In How Many Ways can the Following Distribution be Achieved38:50
- Entropy & Probability, cont.40:57
- More on Distribution40:58
- Example I Summary41:43
- Example II Summary42:12
- Distribution that Maximizes Omega42:26
- If Omega is Large, then S is Large44:22
- Two Constraints for a System to Achieve the Highest Entropy Possible47:07
- What Happened When the Energy of a System is Increased?49:00

35m 5s

- Intro0:00
- Volume Distribution0:08
- Distributing 2 Balls in 3 Spaces1:43
- Distributing 2 Balls in 4 Spaces3:44
- Distributing 3 Balls in 10 Spaces5:30
- Number of Ways to Distribute P Particles over N Spaces6:05
- When N is Much Larger than the Number of Particles P7:56
- Energy Distribution25:04
- Volume Distribution25:58
- Entropy, Total Entropy, & Total Omega Equations27:34
- Entropy, Total Entropy, & Total Omega Equations27:35

28m 42s

- Intro0:00
- Reversible & Irreversible0:24
- Reversible vs. Irreversible0:58
- Defining Equation for Equilibrium2:11
- Defining Equation for Irreversibility (Spontaneity)3:11
- TdS ≥ dQ5:15
- Transformation in an Isolated System11:22
- Transformation in an Isolated System11:29
- Transformation at Constant Temperature14:50
- Transformation at Constant Temperature14:51
- Helmholtz Free Energy17:26
- Define: A = U - TS17:27
- Spontaneous Isothermal Process & Helmholtz Energy20:20
- Pressure-volume Work22:02

34m 38s

- Intro0:00
- Transformation under Constant Temperature & Pressure0:08
- Transformation under Constant Temperature & Pressure0:36
- Define: G = U + PV - TS3:32
- Gibbs Energy5:14
- What Does This Say?6:44
- Spontaneous Process & a Decrease in G14:12
- Computing ∆G18:54
- Summary of Conditions21:32
- Constraint & Condition for Spontaneity21:36
- Constraint & Condition for Equilibrium24:54
- A Few Words About the Word Spontaneous26:24
- Spontaneous Does Not Mean Fast26:25
- Putting Hydrogen & Oxygen Together in a Flask26:59
- Spontaneous Vs. Not Spontaneous28:14
- Thermodynamically Favorable29:03
- Example: Making a Process Thermodynamically Favorable29:34
- Driving Forces for Spontaneity31:35
- Equation: ∆G = ∆H - T∆S31:36
- Always Spontaneous Process32:39
- Never Spontaneous Process33:06
- A Process That is Endothermic Can Still be Spontaneous34:00

30m 50s

- Intro0:00
- The Fundamental Equations of Thermodynamics0:44
- Mechanical Properties of a System0:45
- Fundamental Properties of a System1:16
- Composite Properties of a System1:44
- General Condition of Equilibrium3:16
- Composite Functions & Their Differentiations6:11
- dH = TdS + VdP7:53
- dA = -SdT - PdV9:26
- dG = -SdT + VdP10:22
- Summary of Equations12:10
- Equation #114:33
- Equation #215:15
- Equation #315:58
- Equation #416:42
- Maxwell's Relations20:20
- Maxwell's Relations20:21
- Isothermal Volume-Dependence of Entropy & Isothermal Pressure-Dependence of Entropy26:21

34m 6s

- Intro0:00
- The General Thermodynamic Equations of State0:10
- Equations of State for Liquids & Solids0:52
- More General Condition for Equilibrium4:02
- General Conditions: Equation that Relates P to Functions of T & V6:20
- The Second Fundamental Equation of Thermodynamics11:10
- Equation 117:34
- Equation 221:58
- Recall the General Expression for Cp - Cv28:11
- For the Joule-Thomson Coefficient30:44
- Joule-Thomson Inversion Temperature32:12

39m 18s

- Intro0:00
- Properties of the Helmholtz & Gibbs Energies0:10
- Equating the Differential Coefficients1:34
- An Increase in T; a Decrease in A3:25
- An Increase in V; a Decrease in A6:04
- We Do the Same Thing for G8:33
- Increase in T; Decrease in G10:50
- Increase in P; Decrease in G11:36
- 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 Constant18:57
- For an Ideal Gas22:18
- Special Note24:56
- Temperature Dependence of Gibbs Energy27:02
- Temperature Dependence of Gibbs Energy #127:52
- Temperature Dependence of Gibbs Energy #229:01
- Temperature Dependence of Gibbs Energy #329:50
- Temperature Dependence of Gibbs Energy #434:50

19m 40s

- Intro0:00
- Entropy of the Universe & the Surroundings0:08
- Equation: ∆G = ∆H - T∆S0:20
- Conditions of Constant Temperature & Pressure1:14
- Reversible Process3:14
- Spontaneous Process & the Entropy of the Universe5:20
- Tips for Remembering Everything12:40
- Verify Using Known Spontaneous Process14:51

54m 16s

- Intro0:00
- Example I0:11
- Example I: Deriving a Function for Entropy (S)2:06
- Example I: Deriving a Function for V5:55
- Example I: Deriving a Function for H8:06
- Example I: Deriving a Function for U12:06
- Example II15:18
- Example III21:52
- Example IV26:12
- Example IV: Part A26:55
- Example IV: Part B28:30
- Example IV: Part C30:25
- Example V33:45
- Example VI40:46
- Example VII43:43
- Example VII: Part A44:46
- Example VII: Part B50:52
- Example VII: Part C51:56

31m 17s

- Intro0:00
- Example I0:09
- Example II5:18
- Example III8:22
- Example IV12:32
- Example V17:14
- Example VI20:34
- Example VI: Part A21:04
- Example VI: Part B23:56
- Example VI: Part C27:56

45m

- Intro0:00
- Example I0:10
- Example II15:03
- Example III21:47
- Example IV28:37
- Example IV: Part A29:33
- Example IV: Part B36:09
- Example IV: Part C40:34

58m 5s

- Intro0:00
- Example I0:41
- Part A: Calculating ∆H3:55
- Part B: Calculating ∆S15:13
- Example II24:39
- Part A: Final Temperature of the System26:25
- Part B: Calculating ∆S36:57
- Example III46:49

25m 20s

- Intro0:00
- Work, Heat, and Energy0:18
- Definition of Work, Energy, Enthalpy, and Heat Capacities0:23
- Heat Capacities for an Ideal Gas3:40
- Path Property & State Property3:56
- Energy Differential5:04
- Enthalpy Differential5:40
- Joule's Law & Joule-Thomson Coefficient6:23
- Coefficient of Thermal Expansion & Coefficient of Compressibility7:01
- Enthalpy of a Substance at Any Other Temperature7:29
- Enthalpy of a Reaction at Any Other Temperature8:01
- Entropy8:53
- Definition of Entropy8:54
- Clausius Inequality9:11
- Entropy Changes in Isothermal Systems9:44
- The Fundamental Equation of Thermodynamics10:12
- Expressing Entropy Changes in Terms of Properties of the System10:42
- Entropy Changes in the Ideal Gas11:22
- Third Law Entropies11:38
- Entropy Changes in Chemical Reactions14:02
- Statistical Definition of Entropy14:34
- Omega for the Spatial & Energy Distribution14:47
- Spontaneity and Equilibrium15:43
- Helmholtz Energy & Gibbs Energy15:44
- Condition for Spontaneity & Equilibrium16:24
- Condition for Spontaneity with Respect to Entropy17:58
- The Fundamental Equations18:30
- Maxwell's Relations19:04
- The Thermodynamic Equations of State20:07
- Energy & Enthalpy Differentials21:08
- Joule's Law & Joule-Thomson Coefficient21:59
- Relationship Between Constant Pressure & Constant Volume Heat Capacities23:14
- One Final Equation - Just for Fun24:04

34m 25s

- Intro0:00
- Complex Numbers0:11
- Representing Complex Numbers in the 2-Dimmensional Plane0:56
- Addition of Complex Numbers2:35
- Subtraction of Complex Numbers3:17
- Multiplication of Complex Numbers3:47
- Division of Complex Numbers6:04
- r & θ8:04
- Euler's Formula11:00
- Polar Exponential Representation of the Complex Numbers11:22
- Example I14:25
- Example II15:21
- Example III16:58
- Example IV18:35
- Example V20:40
- Example VI21:32
- Example VII25:22

59m 57s

- Intro0:00
- Probability & Statistics1:51
- Normalization Condition1:52
- Define the Mean or Average of x11:04
- Example I: Calculate the Mean of x14:57
- Example II: Calculate the Second Moment of the Data in Example I22:39
- Define the Second Central Moment or Variance25:26
- Define the Second Central Moment or Variance25:27
- 1st Term32:16
- 2nd Term32:40
- 3rd Term34:07
- Continuous Distributions35:47
- Continuous Distributions35:48
- Probability Density39:30
- Probability Density39:31
- Normalization Condition46:51
- Example III50:13
- Part A - Show that P(x) is Normalized51:40
- Part B - Calculate the Average Position of the Particle Along the Interval54:31
- Important Things to Remember58:24

42m 5s

- Intro0:00
- Schrӧdinger Equation & Operators0:16
- Relation Between a Photon's Momentum & Its Wavelength0:17
- Louis de Broglie: Wavelength for Matter0:39
- Schrӧdinger Equation1:19
- Definition of Ψ(x)3:31
- Quantum Mechanics5:02
- Operators7:51
- Example I10:10
- Example II11:53
- Example III14:24
- Example IV17:35
- Example V19:59
- Example VI22:39
- Operators Can Be Linear or Non Linear27:58
- Operators Can Be Linear or Non Linear28:34
- Example VII32:47
- Example VIII36:55
- Example IX39:29

30m 26s

- Intro0:00
- Schrӧdinger Equation as an Eigenvalue Problem0:10
- Operator: Multiplying the Original Function by Some Scalar0:11
- Operator, Eigenfunction, & Eigenvalue4:42
- Example: Eigenvalue Problem8:00
- Schrӧdinger Equation as an Eigenvalue Problem9:24
- Hamiltonian Operator15:09
- Quantum Mechanical Operators16:46
- Kinetic Energy Operator19:16
- Potential Energy Operator20:02
- Total Energy Operator21:12
- Classical Point of View21:48
- Linear Momentum Operator24:02
- Example I26:01

21m 34s

- Intro0:00
- The Plausibility of the Schrӧdinger Equation1:16
- The Plausibility of the Schrӧdinger Equation, Part 11:17
- The Plausibility of the Schrӧdinger Equation, Part 28:24
- The Plausibility of the Schrӧdinger Equation, Part 313:45

56m 22s

- Intro0:00
- Free Particle in a Box0:28
- Definition of a Free Particle in a Box0:29
- Amplitude of the Matter Wave6:22
- Intensity of the Wave6:53
- Probability Density9:39
- Probability that the Particle is Located Between x & dx10:54
- Probability that the Particle will be Found Between o & a12:35
- Wave Function & the Particle14:59
- Boundary Conditions19:22
- What Happened When There is No Constraint on the Particle27:54
- Diagrams34:12
- More on Probability Density40:53
- The Correspondence Principle46:45
- The Correspondence Principle46:46
- Normalizing the Wave Function47:46
- Normalizing the Wave Function47:47
- Normalized Wave Function & Normalization Constant52:24

45m 24s

- Intro0:00
- Free Particle in a Box0:08
- Free Particle in a 1-dimensional Box0:09
- For a Particle in a Box3:57
- Calculating Average Values & Standard Deviations5:42
- Average Value for the Position of a Particle6:32
- Standard Deviations for the Position of a Particle10:51
- Recall: Energy & Momentum are Represented by Operators13:33
- Recall: Schrӧdinger Equation in Operator Form15:57
- Average Value of a Physical Quantity that is Associated with an Operator18:16
- Average Momentum of a Free Particle in a Box20:48
- The Uncertainty Principle24:42
- Finding the Standard Deviation of the Momentum25:08
- Expression for the Uncertainty Principle35:02
- Summary of the Uncertainty Principle41:28

48m 43s

- Intro0:00
- 2-Dimension0:12
- Dimension 20:31
- Boundary Conditions1:52
- Partial Derivatives4:27
- Example I6:08
- The Particle in a Box, cont.11:28
- Operator Notation12:04
- Symbol for the Laplacian13:50
- The Equation Becomes…14:30
- Boundary Conditions14:54
- Separation of Variables15:33
- Solution to the 1-dimensional Case16:31
- Normalization Constant22:32
- 3-Dimension28:30
- Particle in a 3-dimensional Box28:31
- In Del Notation32:22
- The Solutions34:51
- Expressing the State of the System for a Particle in a 3D Box39:10
- Energy Level & Degeneracy43:35

46m 18s

- Intro0:00
- Postulate I0:31
- Probability That The Particle Will Be Found in a Differential Volume Element0:32
- Example I: Normalize This Wave Function11:30
- Postulate II18:20
- Postulate II18:21
- Quantum Mechanical Operators: Position20:48
- Quantum Mechanical Operators: Kinetic Energy21:57
- Quantum Mechanical Operators: Potential Energy22:42
- Quantum Mechanical Operators: Total Energy22:57
- Quantum Mechanical Operators: Momentum23:22
- Quantum Mechanical Operators: Angular Momentum23:48
- More On The Kinetic Energy Operator24:48
- Angular Momentum28:08
- Angular Momentum Overview28:09
- Angular Momentum Operator in Quantum Mechanic31:34
- The Classical Mechanical Observable32:56
- Quantum Mechanical Operator37:01
- Getting the Quantum Mechanical Operator from the Classical Mechanical Observable40:16
- Postulate II, cont.43:40
- Quantum Mechanical Operators are Both Linear & Hermetical43:41

39m 28s

- Intro0:00
- Postulate III0:09
- Postulate III: Part I0:10
- Postulate III: Part II5:56
- Postulate III: Part III12:43
- Postulate III: Part IV18:28
- Postulate IV23:57
- Postulate IV23:58
- Postulate V27:02
- Postulate V27:03
- Average Value36:38
- Average Value36:39

35m 32s

- Intro0:00
- The Postulates & Principles of Quantum Mechanics, Part III0:10
- Equations: Linear & Hermitian0:11
- Introduction to Hermitian Property3:36
- Eigenfunctions are Orthogonal9:55
- The Sequence of Wave Functions for the Particle in a Box forms an Orthonormal Set14:34
- Definition of Orthogonality16:42
- Definition of Hermiticity17:26
- Hermiticity: The Left Integral23:04
- Hermiticity: The Right Integral28:47
- Hermiticity: Summary34:06

29m 55s

- Intro0:00
- The Postulates & Principles of Quantum Mechanics, Part IV0:09
- Operators can be Applied Sequentially0:10
- Sample Calculation 12:41
- Sample Calculation 25:18
- Commutator of Two Operators8:16
- The Uncertainty Principle19:01
- In the Case of Linear Momentum and Position Operator23:14
- When the Commutator of Two Operators Equals to Zero26:31

54m 25s

- Intro0:00
- Example I: Three Dimensional Box & Eigenfunction of The Laplacian Operator0:37
- Example II: Positions of a Particle in a 1-dimensional Box15:46
- Example III: Transition State & Frequency29:29
- Example IV: Finding a Particle in a 1-dimensional Box35:03
- Example V: Degeneracy & Energy Levels of a Particle in a Box44:59

46m 58s

- Intro0:00
- Review0:25
- Wave Function0:26
- Normalization Condition2:28
- Observable in Classical Mechanics & Linear/Hermitian Operator in Quantum Mechanics3:36
- Hermitian6:11
- Eigenfunctions & Eigenvalue8:20
- Normalized Wave Functions12:00
- Average Value13:42
- If Ψ is Written as a Linear Combination15:44
- Commutator16:45
- Example I: Normalize The Wave Function19:18
- Example II: Probability of Finding of a Particle22:27
- Example III: Orthogonal26:00
- Example IV: Average Value of the Kinetic Energy Operator30:22
- Example V: Evaluate These Commutators39:02

44m 11s

- Intro0:00
- Example I: Good Candidate for a Wave Function0:08
- Example II: Variance of the Energy7:00
- Example III: Evaluate the Angular Momentum Operators15:00
- Example IV: Real Eigenvalues Imposes the Hermitian Property on Operators28:44
- Example V: A Demonstration of Why the Eigenfunctions of Hermitian Operators are Orthogonal35:33

35m 33s

- Intro0:00
- The Harmonic Oscillator0:10
- Harmonic Motion0:11
- Classical Harmonic Oscillator4:38
- Hooke's Law8:18
- Classical Harmonic Oscillator, cont.10:33
- General Solution for the Differential Equation15:16
- Initial Position & Velocity16:05
- Period & Amplitude20:42
- Potential Energy of the Harmonic Oscillator23:20
- Kinetic Energy of the Harmonic Oscillator26:37
- Total Energy of the Harmonic Oscillator27:23
- Conservative System34:37

43m 4s

- Intro0:00
- The Harmonic Oscillator II0:08
- Diatomic Molecule0:10
- Notion of Reduced Mass5:27
- Harmonic Oscillator Potential & The Intermolecular Potential of a Vibrating Molecule7:33
- The Schrӧdinger Equation for the 1-dimensional Quantum Mechanic Oscillator14:14
- Quantized Values for the Energy Level15:46
- Ground State & the Zero-Point Energy21:50
- Vibrational Energy Levels25:18
- Transition from One Energy Level to the Next26:42
- Fundamental Vibrational Frequency for Diatomic Molecule34:57
- Example: Calculate k38:01

26m 30s

- Intro0:00
- The Harmonic Oscillator III0:09
- The Wave Functions Corresponding to the Energies0:10
- Normalization Constant2:34
- Hermite Polynomials3:22
- First Few Hermite Polynomials4:56
- First Few Wave-Functions6:37
- Plotting the Probability Density of the Wave-Functions8:37
- Probability Density for Large Values of r14:24
- Recall: Odd Function & Even Function19:05
- More on the Hermite Polynomials20:07
- Recall: If f(x) is Odd20:36
- Average Value of x22:31
- Average Value of Momentum23:56

41m 10s

- Intro0:00
- Possible Confusion from the Previous Discussion0:07
- Possible Confusion from the Previous Discussion0:08
- Rotation of a Single Mass Around a Fixed Center8:17
- Rotation of a Single Mass Around a Fixed Center8:18
- Angular Velocity12:07
- Rotational Inertia13:24
- Rotational Frequency15:24
- Kinetic Energy for a Linear System16:38
- Kinetic Energy for a Rotational System17:42
- Rotating Diatomic Molecule19:40
- Rotating Diatomic Molecule: Part 119:41
- Rotating Diatomic Molecule: Part 224:56
- Rotating Diatomic Molecule: Part 330:04
- Hamiltonian of the Rigid Rotor36:48
- Hamiltonian of the Rigid Rotor36:49

30m 32s

- Intro0:00
- The Rigid Rotator II0:08
- Cartesian Coordinates0:09
- Spherical Coordinates1:55
- r6:15
- θ6:28
- φ7:00
- Moving a Distance 'r'8:17
- Moving a Distance 'r' in the Spherical Coordinates11:49
- For a Rigid Rotator, r is Constant13:57
- Hamiltonian Operator15:09
- Square of the Angular Momentum Operator17:34
- Orientation of the Rotation in Space19:44
- Wave Functions for the Rigid Rotator20:40
- The Schrӧdinger Equation for the Quantum Mechanic Rigid Rotator21:24
- Energy Levels for the Rigid Rotator26:58

35m 19s

- Intro0:00
- The Rigid Rotator III0:11
- When a Rotator is Subjected to Electromagnetic Radiation1:24
- Selection Rule2:13
- Frequencies at Which Absorption Transitions Occur6:24
- Energy Absorption & Transition10:54
- Energy of the Individual Levels Overview20:58
- Energy of the Individual Levels: Diagram23:45
- Frequency Required to Go from J to J + 125:53
- Using Separation Between Lines on the Spectrum to Calculate Bond Length28:02
- Example I: Calculating Rotational Inertia & Bond Length29:18
- Example I: Calculating Rotational Inertia29:19
- Example I: Calculating Bond Length32:56

33m 48s

- Intro0:00
- Equations Review0:11
- Energy of the Harmonic Oscillator0:12
- Selection Rule3:02
- Observed Frequency of Radiation3:27
- Harmonic Oscillator Wave Functions5:52
- Rigid Rotator7:26
- Selection Rule for Rigid Rotator9:15
- Frequency of Absorption9:35
- Wave Numbers10:58
- Example I: Calculate the Reduced Mass of the Hydrogen Atom11:44
- Example II: Calculate the Fundamental Vibration Frequency & the Zero-Point Energy of This Molecule13:37
- Example III: Show That the Product of Two Even Functions is even19:35
- Example IV: Harmonic Oscillator24:56

46m 43s

- Intro0:00
- Example I: Harmonic Oscillator0:12
- Example II: Harmonic Oscillator23:26
- Example III: Calculate the RMS Displacement of the Molecules38:12

40m

- Intro0:00
- The Hydrogen Atom I1:31
- Review of the Rigid Rotator1:32
- Hydrogen Atom & the Coulomb Potential2:50
- Using the Spherical Coordinates6:33
- Applying This Last Expression to Equation 110:19
- Angular Component & Radial Component13:26
- Angular Equation15:56
- Solution for F(φ)19:32
- Determine The Normalization Constant20:33
- Differential Equation for T(a)24:44
- Legendre Equation27:20
- Legendre Polynomials31:20
- The Legendre Polynomials are Mutually Orthogonal35:40
- Limits37:17
- Coefficients38:28

35m 58s

- Intro0:00
- Associated Legendre Functions0:07
- Associated Legendre Functions0:08
- First Few Associated Legendre Functions6:39
- s, p, & d Orbital13:24
- The Normalization Condition15:44
- Spherical Harmonics20:03
- Equations We Have Found20:04
- Wave Functions for the Angular Component & Rigid Rotator24:36
- Spherical Harmonics Examples25:40
- Angular Momentum30:09
- Angular Momentum30:10
- Square of the Angular Momentum35:38
- Energies of the Rigid Rotator38:21

36m 18s

- Intro0:00
- The Hydrogen Atom III0:34
- Angular Momentum is a Vector Quantity0:35
- The Operators Corresponding to the Three Components of Angular Momentum Operator: In Cartesian Coordinates1:30
- The Operators Corresponding to the Three Components of Angular Momentum Operator: In Spherical Coordinates3:27
- Z Component of the Angular Momentum Operator & the Spherical Harmonic5:28
- Magnitude of the Angular Momentum Vector20:10
- Classical Interpretation of Angular Momentum25:22
- Projection of the Angular Momentum Vector onto the xy-plane33:24

33m 55s

- Intro0:00
- The Hydrogen Atom IV0:09
- The Equation to Find R( r )0:10
- Relation Between n & l3:50
- The Solutions for the Radial Functions5:08
- Associated Laguerre Polynomials7:58
- 1st Few Associated Laguerre Polynomials8:55
- Complete Wave Function for the Atomic Orbitals of the Hydrogen Atom12:24
- The Normalization Condition15:06
- In Cartesian Coordinates18:10
- Working in Polar Coordinates20:48
- Principal Quantum Number21:58
- Angular Momentum Quantum Number22:35
- Magnetic Quantum Number25:55
- Zeeman Effect30:45

51m 53s

- Intro0:00
- The Hydrogen Atom V: Where We Are0:13
- Review0:14
- Let's Write Out ψ₂₁₁7:32
- Angular Momentum of the Electron14:52
- Representation of the Wave Function19:36
- Radial Component28:02
- Example: 1s Orbital28:34
- Probability for Radial Function33:46
- 1s Orbital: Plotting Probability Densities vs. r35:47
- 2s Orbital: Plotting Probability Densities vs. r37:46
- 3s Orbital: Plotting Probability Densities vs. r38:49
- 4s Orbital: Plotting Probability Densities vs. r39:34
- 2p Orbital: Plotting Probability Densities vs. r40:12
- 3p Orbital: Plotting Probability Densities vs. r41:02
- 4p Orbital: Plotting Probability Densities vs. r41:51
- 3d Orbital: Plotting Probability Densities vs. r43:18
- 4d Orbital: Plotting Probability Densities vs. r43:48
- Example I: Probability of Finding an Electron in the 2s Orbital of the Hydrogen45:40

51m 53s

- Intro0:00
- The Hydrogen Atom VI0:07
- Last Lesson Review0:08
- Spherical Component1:09
- Normalization Condition2:02
- Complete 1s Orbital Wave Function4:08
- 1s Orbital Wave Function4:09
- Normalization Condition6:28
- Spherically Symmetric16:00
- Average Value17:52
- Example I: Calculate the Region of Highest Probability for Finding the Electron21:19
- 2s Orbital Wave Function25:32
- 2s Orbital Wave Function25:33
- Average Value28:56
- General Formula32:24

34m 29s

- Intro0:00
- The Hydrogen Atom VII0:12
- p Orbitals1:30
- Not Spherically Symmetric5:10
- Recall That the Spherical Harmonics are Eigenfunctions of the Hamiltonian Operator6:50
- Any Linear Combination of These Orbitals Also Has The Same Energy9:16
- Functions of Real Variables15:53
- Solving for Px16:50
- Real Spherical Harmonics21:56
- Number of Nodes32:56

43m 49s

- Intro0:00
- Example I: Angular Momentum & Spherical Harmonics0:20
- Example II: Pair-wise Orthogonal Legendre Polynomials16:40
- Example III: General Normalization Condition for the Legendre Polynomials25:06
- Example IV: Associated Legendre Functions32:13

1h 1m 57s

- Intro0:00
- Example I: Normalization & Pair-wise Orthogonal0:13
- Part 1: Normalized0:43
- Part 2: Pair-wise Orthogonal16:53
- Example II: Show Explicitly That the Following Statement is True for Any Integer n27:10
- Example III: Spherical Harmonics29:26
- Angular Momentum Cones56:37
- Angular Momentum Cones56:38
- Physical Interpretation of Orbital Angular Momentum in Quantum mechanics1:00:16

48m 33s

- Intro0:00
- Example I: Show That ψ₂₁₁ is Normalized0:07
- Example II: Show That ψ₂₁₁ is Orthogonal to ψ₃₁₀11:48
- Example III: Probability That a 1s Electron Will Be Found Within 1 Bohr Radius of The Nucleus18:35
- Example IV: Radius of a Sphere26:06
- Example V: Calculate <r> for the 2s Orbital of the Hydrogen-like Atom36:33

48m 33s

- Intro0:00
- Example I: Probability Density vs. Radius Plot0:11
- Example II: Hydrogen Atom & The Coulombic Potential14:16
- Example III: Find a Relation Among <K>, <V>, & <E>25:47
- Example IV: Quantum Mechanical Virial Theorem48:32
- Example V: Find the Variance for the 2s Orbital54:13

48m 33s

- Intro0:00
- Example I: Derive a Formula for the Degeneracy of a Given Level n0:11
- Example II: Using Linear Combinations to Represent the Spherical Harmonics as Functions of the Real Variables θ & φ8:30
- Example III: Using Linear Combinations to Represent the Spherical Harmonics as Functions of the Real Variables θ & φ23:01
- Example IV: Orbital Functions31:51

59m 18s

- Intro0:00
- Quantum Numbers Specify an Orbital0:24
- n1:10
- l1:20
- m1:35
- 4th Quantum Number: s2:02
- Spin Orbitals7:03
- Spin Orbitals7:04
- Multi-electron Atoms11:08
- Term Symbols18:08
- Russell-Saunders Coupling & The Atomic Term Symbol18:09
- Example: Configuration for C27:50
- Configuration for C: 1s²2s²2p²27:51
- Drawing Every Possible Arrangement31:15
- Term Symbols45:24
- Microstate50:54

34m 54s

- Intro0:00
- Microstates0:25
- We Started With 21 Possible Microstates0:26
- ³P State2:05
- Microstates in ³P Level5:10
- ¹D State13:16
- ³P State16:10
- ²P₂ State17:34
- ³P₁ State18:34
- ³P₀ State19:12
- 9 Microstates in ³P are Subdivided19:40
- ¹S State21:44
- Quicker Way to Find the Different Values of J for a Given Basic Term Symbol22:22
- Ground State26:27
- Hund's Empirical Rules for Specifying the Term Symbol for the Ground Electronic State27:29
- Hund's Empirical Rules: 128:24
- Hund's Empirical Rules: 229:22
- Hund's Empirical Rules: 3 - Part A30:22
- Hund's Empirical Rules: 3 - Part B31:18
- Example: 1s²2s²2p²31:54

38m 3s

- Intro0:00
- Spin Quantum Number: Term Symbols III0:14
- Deriving the Term Symbols for the p² Configuration0:15
- Table: MS vs. ML3:57
- ¹D State16:21
- ³P State21:13
- ¹S State24:48
- J Value25:32
- Degeneracy of the Level27:28
- When Given r Electrons to Assign to n Equivalent Spin Orbitals30:18
- p² Configuration32:51
- Complementary Configurations35:12

57m 49s

- Intro0:00
- Lyman Series0:09
- Spectroscopic Term Symbols0:10
- Lyman Series3:04
- Hydrogen Levels8:21
- Hydrogen Levels8:22
- Term Symbols & Atomic Spectra14:17
- Spin-Orbit Coupling14:18
- Selection Rules for Atomic Spectra21:31
- Selection Rules for Possible Transitions23:56
- Wave Numbers for The Transitions28:04
- Example I: Calculate the Frequencies of the Allowed Transitions from (4d) ²D →(2p) ²P32:23
- Helium Levels49:50
- Energy Levels for Helium49:51
- Transitions & Spin Multiplicity52:27
- Transitions & Spin Multiplicity52:28

1h 1m 20s

- Intro0:00
- 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

56m 34s

- Intro0:00
- Example I: Find the Term Symbols for the nd² Configuration0:11
- Example II: Find the Term Symbols for the 1s¹2p¹ Configuration27:02
- Example III: Calculate the Separation Between the Doublets in the Lyman Series for Atomic Hydrogen41:41
- Example IV: Calculate the Frequencies of the Lines for the (4d) ²D → (3p) ²P Transition48:53

18m 24s

- Intro0:00
- Quantum Mechanics Equations0:37
- De Broglie Relation0:38
- Statistical Relations1:00
- The Schrӧdinger Equation1:50
- The Particle in a 1-Dimensional Box of Length a3:09
- The Particle in a 2-Dimensional Box of Area a x b3:48
- The Particle in a 3-Dimensional Box of Area a x b x c4:22
- The Schrӧdinger Equation Postulates4:51
- The Normalization Condition5:40
- The Probability Density6:51
- Linear7:47
- Hermitian8:31
- Eigenvalues & Eigenfunctions8:55
- The Average Value9:29
- Eigenfunctions of Quantum Mechanics Operators are Orthogonal10:53
- Commutator of Two Operators10:56
- The Uncertainty Principle11:41
- The Harmonic Oscillator13:18
- The Rigid Rotator13:52
- Energy of the Hydrogen Atom14:30
- Wavefunctions, Radial Component, and Associated Laguerre Polynomial14:44
- Angular Component or Spherical Harmonic15:16
- Associated Legendre Function15:31
- Principal Quantum Number15:43
- Angular Momentum Quantum Number15:50
- Magnetic Quantum Number16:21
- z-component of the Angular Momentum of the Electron16:53
- Atomic Spectroscopy: Term Symbols17:14
- Atomic Spectroscopy: Selection Rules18:03

50m 2s

- Intro0:00
- Spectroscopic Overview: Which Equation Do I Use & Why1:02
- Lesson Overview1:03
- Rotational & Vibrational Spectroscopy4:01
- Frequency of Absorption/Emission6:04
- Wavenumbers in Spectroscopy8:10
- Starting State vs. Excited State10:10
- Total Energy of a Molecule (Leaving out the Electronic Energy)14:02
- Energy of Rotation: Rigid Rotor15:55
- Energy of Vibration: Harmonic Oscillator19:08
- Equation of the Spectral Lines23:22
- Harmonic Oscillator-Rigid Rotor Approximation (Making Corrections)28:37
- Harmonic Oscillator-Rigid Rotor Approximation (Making Corrections)28:38
- Vibration-Rotation Interaction33:46
- Centrifugal Distortion36:27
- Anharmonicity38:28
- Correcting for All Three Simultaneously41:03
- Spectroscopic Parameters44:26
- Summary47:32
- Harmonic Oscillator-Rigid Rotor Approximation47:33
- Vibration-Rotation Interaction48:14
- Centrifugal Distortion48:20
- Anharmonicity48:28
- Correcting for All Three Simultaneously48:44

59m 47s

- Intro0:00
- Vibration-Rotation0:37
- What is Molecular Spectroscopy?0:38
- Microwave, Infrared Radiation, Visible & Ultraviolet1:53
- Equation for the Frequency of the Absorbed Radiation4:54
- Wavenumbers6:15
- Diatomic Molecules: Energy of the Harmonic Oscillator8:32
- Selection Rules for Vibrational Transitions10:35
- Energy of the Rigid Rotator16:29
- Angular Momentum of the Rotator21:38
- Rotational Term F(J)26:30
- Selection Rules for Rotational Transition29:30
- Vibration Level & Rotational States33:20
- Selection Rules for Vibration-Rotation37:42
- Frequency of Absorption39:32
- Diagram: Energy Transition45:55
- Vibration-Rotation Spectrum: HCl51:27
- Vibration-Rotation Spectrum: Carbon Monoxide54:30

46m 22s

- Intro0:00
- Vibration-Rotation Interaction0:13
- Vibration-Rotation Spectrum: HCl0:14
- Bond Length & Vibrational State4:23
- Vibration Rotation Interaction10:18
- Case 112:06
- Case 217:17
- Example I: HCl Vibration-Rotation Spectrum22:58
- Rotational Constant for the 0 & 1 Vibrational State26:30
- Equilibrium Bond Length for the 1 Vibrational State39:42
- Equilibrium Bond Length for the 0 Vibrational State42:13
- Bₑ & αₑ44:54

29m 24s

- Intro0:00
- The Non-Rigid Rotator0:09
- Pure Rotational Spectrum0:54
- The Selection Rules for Rotation3:09
- Spacing in the Spectrum5:04
- Centrifugal Distortion Constant9:00
- Fundamental Vibration Frequency11:46
- Observed Frequencies of Absorption14:14
- Difference between the Rigid Rotator & the Adjusted Rigid Rotator16:51
- Adjusted Rigid Rotator21:31
- Observed Frequencies of Absorption26:26

30m 53s

- Intro0:00
- The Anharmonic Oscillator0:09
- Vibration-Rotation Interaction & Centrifugal Distortion0:10
- Making Corrections to the Harmonic Oscillator4:50
- Selection Rule for the Harmonic Oscillator7:50
- Overtones8:40
- True Oscillator11:46
- Harmonic Oscillator Energies13:16
- Anharmonic Oscillator Energies13:33
- Observed Frequencies of the Overtones15:09
- True Potential17:22
- HCl Vibrational Frequencies: Fundamental & First Few Overtones21:10
- Example I: Vibrational States & Overtones of the Vibrational Spectrum22:42
- Example I: Part A - First 4 Vibrational States23:44
- Example I: Part B - Fundamental & First 3 Overtones25:31
- Important Equations27:45
- Energy of the Q State29:14
- The Difference in Energy between 2 Successive States29:23
- Difference in Energy between 2 Spectral Lines29:40

1h 1m 33s

- Intro0:00
- Electronic Transitions0:16
- Electronic State & Transition0:17
- Total Energy of the Diatomic Molecule3:34
- Vibronic Transitions4:30
- Selection Rule for Vibronic Transitions9:11
- More on Vibronic Transitions10:08
- Frequencies in the Spectrum16:46
- Difference of the Minima of the 2 Potential Curves24:48
- Anharmonic Zero-point Vibrational Energies of the 2 States26:24
- Frequency of the 0 → 0 Vibronic Transition27:54
- Making the Equation More Compact29:34
- Spectroscopic Parameters32:11
- Franck-Condon Principle34:32
- Example I: Find the Values of the Spectroscopic Parameters for the Upper Excited State47:27
- Table of Electronic States and Parameters56:41

33m 47s

- Intro0:00
- Example I: Calculate the Bond Length0:10
- Example II: Calculate the Rotational Constant7:39
- Example III: Calculate the Number of Rotations10:54
- Example IV: What is the Force Constant & Period of Vibration?16:31
- Example V: Part A - Calculate the Fundamental Vibration Frequency21:42
- Example V: Part B - Calculate the Energies of the First Three Vibrational Levels24:12
- Example VI: Calculate the Frequencies of the First 2 Lines of the R & P Branches of the Vib-Rot Spectrum of HBr26:28

1h 1m 5s

- Intro0:00
- Example I: Calculate the Frequencies of the Transitions0:09
- Example II: Specify Which Transitions are Allowed & Calculate the Frequencies of These Transitions22:07
- Example III: Calculate the Vibrational State & Equilibrium Bond Length34:31
- Example IV: Frequencies of the Overtones49:28
- Example V: Vib-Rot Interaction, Centrifugal Distortion, & Anharmonicity54:47

33m 31s

- Intro0:00
- Example I: Part A - Derive an Expression for ∆G( r )0:10
- Example I: Part B - Maximum Vibrational Quantum Number6:10
- Example II: Part A - Derive an Expression for the Dissociation Energy of the Molecule8:29
- Example II: Part B - Equation for ∆G( r )14:00
- Example III: How Many Vibrational States are There for Br₂ before the Molecule Dissociates18:16
- Example IV: Find the Difference between the Two Minima of the Potential Energy Curves20:57
- Example V: Rotational Spectrum30:51

1h 1m 15s

- Intro0:00
- Statistical Thermodynamics: The Big Picture0:10
- Our Big Picture Goal0:11
- Partition Function (Q)2:42
- The Molecular Partition Function (q)4:00
- Consider a System of N Particles6:54
- Ensemble13:22
- Energy Distribution Table15:36
- Probability of Finding a System with Energy16:51
- The Partition Function21:10
- Microstate28:10
- Entropy of the Ensemble30:34
- Entropy of the System31:48
- Expressing the Thermodynamic Functions in Terms of The Partition Function39:21
- The Partition Function39:22
- Pi & U41:20
- Entropy of the System44:14
- Helmholtz Energy48:15
- Pressure of the System49:32
- Enthalpy of the System51:46
- Gibbs Free Energy52:56
- Heat Capacity54:30
- Expressing Q in Terms of the Molecular Partition Function (q)59:31
- Indistinguishable Particles1:02:16
- N is the Number of Particles in the System1:03:27
- The Molecular Partition Function1:05:06
- Quantum States & Degeneracy1:07:46
- Thermo Property in Terms of ln Q1:10:09
- Example: Thermo Property in Terms of ln Q1:13:23

47m 23s

- Intro0:00
- Lesson Overview0:19
- Monatomic Ideal Gases6:40
- Monatomic Ideal Gases Overview6:42
- Finding the Parition Function of Translation8:17
- Finding the Parition Function of Electronics13:29
- Example: Na17:42
- Example: F23:12
- Energy Difference between the Ground State & the 1st Excited State29:27
- The Various Partition Functions for Monatomic Ideal Gases32:20
- Finding P43:16
- Going Back to U = (3/2) RT46:20

54m 9s

- Intro0:00
- Diatomic Gases0:16
- Diatomic Gases0:17
- Zero-Energy Mark for Rotation2:26
- Zero-Energy Mark for Vibration3:21
- Zero-Energy Mark for Electronic5:54
- Vibration Partition Function9:48
- When Temperature is Very Low14:00
- When Temperature is Very High15:22
- Vibrational Component18:48
- Fraction of Molecules in the r Vibration State21:00
- Example: Fraction of Molecules in the r Vib. State23:29
- Rotation Partition Function26:06
- Heteronuclear & Homonuclear Diatomics33:13
- Energy & Heat Capacity36:01
- Fraction of Molecules in the J Rotational Level39:20
- Example: Fraction of Molecules in the J Rotational Level40:32
- Finding the Most Populated Level44:07
- Putting It All Together46:06
- Putting It All Together46:07
- Energy of Translation51:51
- Energy of Rotation52:19
- Energy of Vibration52:42
- Electronic Energy53:35

48m 32s

- Intro0:00
- Example I: Calculate the Fraction of Potassium Atoms in the First Excited Electronic State0:10
- Example II: Show That Each Translational Degree of Freedom Contributes R/2 to the Molar Heat Capacity14:46
- Example III: Calculate the Dissociation Energy21:23
- Example IV: Calculate the Vibrational Contribution to the Molar heat Capacity of Oxygen Gas at 500 K25:46
- Example V: Upper & Lower Quantum State32:55
- Example VI: Calculate the Relative Populations of the J=2 and J=1 Rotational States of the CO Molecule at 25°C42:21

57m 30s

- Intro0:00
- Example I: Make a Plot of the Fraction of CO Molecules in Various Rotational Levels0:10
- Example II: Calculate the Ratio of the Translational Partition Function for Cl₂ and Br₂ at Equal Volume & Temperature8:05
- Example III: Vibrational Degree of Freedom & Vibrational Molar Heat Capacity11:59
- Example IV: Calculate the Characteristic Vibrational & Rotational temperatures for Each DOF45:03

For more information, please see full course syllabus of Physical Chemistry

# Physical Chemistry Entropy Example Problems II

In this lesson our instructor does some example problems on entropy. He works on a total of five complete example problems.

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2 answers

Last reply by: Professor Hovasapian

Sun Nov 25, 2018 5:27 AM

Post by Kimberly Davis on November 24, 2018

For example II, can we just calc deltaU= Cv dT ? I got the same answer as you did but took a different approach.

3 answers

Last reply by: Professor Hovasapian

Tue Oct 2, 2018 5:17 AM

Post by Van Anh Do on February 23, 2016

For example 1, I think you forgot to calculate dS.