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Chemistry: Atoms First

12.4 Free Energy

Chemistry: Atoms First12.4 Free Energy
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Table of contents
  1. Preface
  2. 1 Essential Ideas
    1. Introduction
    2. 1.1 Chemistry in Context
    3. 1.2 Phases and Classification of Matter
    4. 1.3 Physical and Chemical Properties
    5. 1.4 Measurements
    6. 1.5 Measurement Uncertainty, Accuracy, and Precision
    7. 1.6 Mathematical Treatment of Measurement Results
    8. Key Terms
    9. Key Equations
    10. Summary
    11. Exercises
  3. 2 Atoms, Molecules, and Ions
    1. Introduction
    2. 2.1 Early Ideas in Atomic Theory
    3. 2.2 Evolution of Atomic Theory
    4. 2.3 Atomic Structure and Symbolism
    5. 2.4 Chemical Formulas
    6. Key Terms
    7. Key Equations
    8. Summary
    9. Exercises
  4. 3 Electronic Structure and Periodic Properties of Elements
    1. Introduction
    2. 3.1 Electromagnetic Energy
    3. 3.2 The Bohr Model
    4. 3.3 Development of Quantum Theory
    5. 3.4 Electronic Structure of Atoms (Electron Configurations)
    6. 3.5 Periodic Variations in Element Properties
    7. 3.6 The Periodic Table
    8. 3.7 Molecular and Ionic Compounds
    9. Key Terms
    10. Key Equations
    11. Summary
    12. Exercises
  5. 4 Chemical Bonding and Molecular Geometry
    1. Introduction
    2. 4.1 Ionic Bonding
    3. 4.2 Covalent Bonding
    4. 4.3 Chemical Nomenclature
    5. 4.4 Lewis Symbols and Structures
    6. 4.5 Formal Charges and Resonance
    7. 4.6 Molecular Structure and Polarity
    8. Key Terms
    9. Key Equations
    10. Summary
    11. Exercises
  6. 5 Advanced Theories of Bonding
    1. Introduction
    2. 5.1 Valence Bond Theory
    3. 5.2 Hybrid Atomic Orbitals
    4. 5.3 Multiple Bonds
    5. 5.4 Molecular Orbital Theory
    6. Key Terms
    7. Key Equations
    8. Summary
    9. Exercises
  7. 6 Composition of Substances and Solutions
    1. Introduction
    2. 6.1 Formula Mass
    3. 6.2 Determining Empirical and Molecular Formulas
    4. 6.3 Molarity
    5. 6.4 Other Units for Solution Concentrations
    6. Key Terms
    7. Key Equations
    8. Summary
    9. Exercises
  8. 7 Stoichiometry of Chemical Reactions
    1. Introduction
    2. 7.1 Writing and Balancing Chemical Equations
    3. 7.2 Classifying Chemical Reactions
    4. 7.3 Reaction Stoichiometry
    5. 7.4 Reaction Yields
    6. 7.5 Quantitative Chemical Analysis
    7. Key Terms
    8. Key Equations
    9. Summary
    10. Exercises
  9. 8 Gases
    1. Introduction
    2. 8.1 Gas Pressure
    3. 8.2 Relating Pressure, Volume, Amount, and Temperature: The Ideal Gas Law
    4. 8.3 Stoichiometry of Gaseous Substances, Mixtures, and Reactions
    5. 8.4 Effusion and Diffusion of Gases
    6. 8.5 The Kinetic-Molecular Theory
    7. 8.6 Non-Ideal Gas Behavior
    8. Key Terms
    9. Key Equations
    10. Summary
    11. Exercises
  10. 9 Thermochemistry
    1. Introduction
    2. 9.1 Energy Basics
    3. 9.2 Calorimetry
    4. 9.3 Enthalpy
    5. 9.4 Strengths of Ionic and Covalent Bonds
    6. Key Terms
    7. Key Equations
    8. Summary
    9. Exercises
  11. 10 Liquids and Solids
    1. Introduction
    2. 10.1 Intermolecular Forces
    3. 10.2 Properties of Liquids
    4. 10.3 Phase Transitions
    5. 10.4 Phase Diagrams
    6. 10.5 The Solid State of Matter
    7. 10.6 Lattice Structures in Crystalline Solids
    8. Key Terms
    9. Key Equations
    10. Summary
    11. Exercises
  12. 11 Solutions and Colloids
    1. Introduction
    2. 11.1 The Dissolution Process
    3. 11.2 Electrolytes
    4. 11.3 Solubility
    5. 11.4 Colligative Properties
    6. 11.5 Colloids
    7. Key Terms
    8. Key Equations
    9. Summary
    10. Exercises
  13. 12 Thermodynamics
    1. Introduction
    2. 12.1 Spontaneity
    3. 12.2 Entropy
    4. 12.3 The Second and Third Laws of Thermodynamics
    5. 12.4 Free Energy
    6. Key Terms
    7. Key Equations
    8. Summary
    9. Exercises
  14. 13 Fundamental Equilibrium Concepts
    1. Introduction
    2. 13.1 Chemical Equilibria
    3. 13.2 Equilibrium Constants
    4. 13.3 Shifting Equilibria: Le Châtelier’s Principle
    5. 13.4 Equilibrium Calculations
    6. Key Terms
    7. Key Equations
    8. Summary
    9. Exercises
  15. 14 Acid-Base Equilibria
    1. Introduction
    2. 14.1 Brønsted-Lowry Acids and Bases
    3. 14.2 pH and pOH
    4. 14.3 Relative Strengths of Acids and Bases
    5. 14.4 Hydrolysis of Salt Solutions
    6. 14.5 Polyprotic Acids
    7. 14.6 Buffers
    8. 14.7 Acid-Base Titrations
    9. Key Terms
    10. Key Equations
    11. Summary
    12. Exercises
  16. 15 Equilibria of Other Reaction Classes
    1. Introduction
    2. 15.1 Precipitation and Dissolution
    3. 15.2 Lewis Acids and Bases
    4. 15.3 Multiple Equilibria
    5. Key Terms
    6. Key Equations
    7. Summary
    8. Exercises
  17. 16 Electrochemistry
    1. Introduction
    2. 16.1 Balancing Oxidation-Reduction Reactions
    3. 16.2 Galvanic Cells
    4. 16.3 Standard Reduction Potentials
    5. 16.4 The Nernst Equation
    6. 16.5 Batteries and Fuel Cells
    7. 16.6 Corrosion
    8. 16.7 Electrolysis
    9. Key Terms
    10. Key Equations
    11. Summary
    12. Exercises
  18. 17 Kinetics
    1. Introduction
    2. 17.1 Chemical Reaction Rates
    3. 17.2 Factors Affecting Reaction Rates
    4. 17.3 Rate Laws
    5. 17.4 Integrated Rate Laws
    6. 17.5 Collision Theory
    7. 17.6 Reaction Mechanisms
    8. 17.7 Catalysis
    9. Key Terms
    10. Key Equations
    11. Summary
    12. Exercises
  19. 18 Representative Metals, Metalloids, and Nonmetals
    1. Introduction
    2. 18.1 Periodicity
    3. 18.2 Occurrence and Preparation of the Representative Metals
    4. 18.3 Structure and General Properties of the Metalloids
    5. 18.4 Structure and General Properties of the Nonmetals
    6. 18.5 Occurrence, Preparation, and Compounds of Hydrogen
    7. 18.6 Occurrence, Preparation, and Properties of Carbonates
    8. 18.7 Occurrence, Preparation, and Properties of Nitrogen
    9. 18.8 Occurrence, Preparation, and Properties of Phosphorus
    10. 18.9 Occurrence, Preparation, and Compounds of Oxygen
    11. 18.10 Occurrence, Preparation, and Properties of Sulfur
    12. 18.11 Occurrence, Preparation, and Properties of Halogens
    13. 18.12 Occurrence, Preparation, and Properties of the Noble Gases
    14. Key Terms
    15. Summary
    16. Exercises
  20. 19 Transition Metals and Coordination Chemistry
    1. Introduction
    2. 19.1 Occurrence, Preparation, and Properties of Transition Metals and Their Compounds
    3. 19.2 Coordination Chemistry of Transition Metals
    4. 19.3 Spectroscopic and Magnetic Properties of Coordination Compounds
    5. Key Terms
    6. Summary
    7. Exercises
  21. 20 Nuclear Chemistry
    1. Introduction
    2. 20.1 Nuclear Structure and Stability
    3. 20.2 Nuclear Equations
    4. 20.3 Radioactive Decay
    5. 20.4 Transmutation and Nuclear Energy
    6. 20.5 Uses of Radioisotopes
    7. 20.6 Biological Effects of Radiation
    8. Key Terms
    9. Key Equations
    10. Summary
    11. Exercises
  22. 21 Organic Chemistry
    1. Introduction
    2. 21.1 Hydrocarbons
    3. 21.2 Alcohols and Ethers
    4. 21.3 Aldehydes, Ketones, Carboxylic Acids, and Esters
    5. 21.4 Amines and Amides
    6. Key Terms
    7. Summary
    8. Exercises
  23. A | The Periodic Table
  24. B | Essential Mathematics
  25. C | Units and Conversion Factors
  26. D | Fundamental Physical Constants
  27. E | Water Properties
  28. F | Composition of Commercial Acids and Bases
  29. G | Standard Thermodynamic Properties for Selected Substances
  30. H | Ionization Constants of Weak Acids
  31. I | Ionization Constants of Weak Bases
  32. J | Solubility Products
  33. K | Formation Constants for Complex Ions
  34. L | Standard Electrode (Half-Cell) Potentials
  35. M | Half-Lives for Several Radioactive Isotopes
  36. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
    14. Chapter 14
    15. Chapter 15
    16. Chapter 16
    17. Chapter 17
    18. Chapter 18
    19. Chapter 19
    20. Chapter 20
    21. Chapter 21
  37. Index

Learning Objectives

By the end of this section, you will be able to:
  • Define Gibbs free energy, and describe its relation to spontaneity
  • Calculate free energy change for a process using free energies of formation for its reactants and products
  • Calculate free energy change for a process using enthalpies of formation and the entropies for its reactants and products

One of the challenges of using the second law of thermodynamics to determine if a process is spontaneous is that we must determine the entropy change for the system and the entropy change for the surroundings. An alternative approach involving a new thermodynamic property defined in terms of system properties only was introduced in the late nineteenth century by American mathematician Josiah Willard Gibbs. This new property is called the Gibbs free energy change (G) (or simply the free energy), and it is defined in terms of a system’s enthalpy and entropy as the following:

G=H−TSG=H−TS
12.27

Free energy is a state function, and at constant temperature and pressure, the standard free energy change (ΔG°) may be expressed as the following:

ΔG=ΔH−TΔSΔG=ΔH−TΔS
12.28

(For simplicity’s sake, the subscript “sys” will be omitted henceforth.)

We can understand the relationship between this system property and the spontaneity of a process by recalling the previously derived second law expression:

ΔSuniv=ΔS+qsurrTΔSuniv=ΔS+qsurrT
12.29

The first law requires that qsurr = −qsys, and at constant pressure qsys = ΔH, and so this expression may be rewritten as the following:

ΔSuniv=ΔS−ΔHTΔSuniv=ΔS−ΔHT
12.30

ΔH is the enthalpy change of the system. Multiplying both sides of this equation by −T, and rearranging yields the following:

−TΔSuniv=ΔH−TΔS−TΔSuniv=ΔH−TΔS
12.31

Comparing this equation to the previous one for free energy change shows the following relation:

ΔG=−TΔSunivΔG=−TΔSuniv
12.32

The free energy change is therefore a reliable indicator of the spontaneity of a process, being directly related to the previously identified spontaneity indicator, ΔSuniv. Table 12.3 summarizes the relation between the spontaneity of a process and the arithmetic signs of these indicators.

Relation between Process Spontaneity and Signs of Thermodynamic Properties
ΔSuniv > 0 ΔG < 0 spontaneous
ΔSuniv < 0 ΔG > 0 nonspontaneous
ΔSuniv = 0 ΔG = 0 reversible (at equilibrium)
Table 12.3

Calculating Free Energy Change

Free energy is a state function, so its value depends only on the conditions of the initial and final states of the system. A convenient and common approach to the calculation of free energy changes for physical and chemical reactions is by use of widely available compilations of standard state thermodynamic data. One method involves the use of standard enthalpies and entropies to compute standard free energy changes according to the following relation as demonstrated in Example 12.7.

ΔG°=ΔH°−TΔS°ΔG°=ΔH°−TΔS°
12.33

Example 12.7

Evaluation of ΔG° from ΔH° and ΔS°

Use standard enthalpy and entropy data from Appendix G to calculate the standard free energy change for the vaporization of water at room temperature (298 K). What does the computed value for ΔG° say about the spontaneity of this process?

Solution

The process of interest is the following:
H2O(l)⟶H2O(g)H2O(l)⟶H2O(g)
12.34

The standard change in free energy may be calculated using the following equation:

ΔG298°=ΔH°−TΔS°ΔG298°=ΔH°−TΔS°
12.35

From Appendix G, here is the data:

Substance ΔHf°(kJ/mol)ΔHf°(kJ/mol) S298°(J/K·mol)S298°(J/K·mol)
H2O(l) −286.83 70.0
H2O(g) −241.82 188.8

Combining at 298 K:

ΔH°=ΔH298°=ΔHf°(H2O(g))−ΔHf°(H2O(l))=[−241.82 kJ−(−286.83)]kJ/mol=45.01 kJ/molΔH°=ΔH298°=ΔHf°(H2O(g))−ΔHf°(H2O(l))=[−241.82 kJ−(−286.83)]kJ/mol=45.01 kJ/mol
12.36
ΔS°=ΔS298°=S298°(H2O(g))−S298°(H2O(l))=188.8J/mol·K−70.0J/K=118.8J/mol·KΔS°=ΔS298°=S298°(H2O(g))−S298°(H2O(l))=188.8J/mol·K−70.0J/K=118.8J/mol·K
12.37
ΔG°=ΔH°−TΔS°ΔG°=ΔH°−TΔS°
12.38

Converting everything into kJ and combining at 298 K:

ΔG298°=ΔH°−TΔS°=45.01 kJ/mol−(298 K×118.8J/mol·K)×1 kJ1000 JΔG298°=ΔH°−TΔS°=45.01 kJ/mol−(298 K×118.8J/mol·K)×1 kJ1000 J
12.39
45.01 kJ/mol−35.4 kJ/mol=9.6 kJ/mol45.01 kJ/mol−35.4 kJ/mol=9.6 kJ/mol
12.40

At 298 K (25 °C) ΔG298°>0,ΔG298°>0, and so boiling is nonspontaneous (not spontaneous).

Check Your Learning

Use standard enthalpy and entropy data from Appendix G to calculate the standard free energy change for the reaction shown here (298 K). What does the computed value for ΔG° say about the spontaneity of this process?
C2H6(g)⟶H2(g)+C2H4(g)C2H6(g)⟶H2(g)+C2H4(g)
12.41

Answer:

ΔG298°=102.0 kJ/mol;ΔG298°=102.0 kJ/mol; the reaction is nonspontaneous (not spontaneous) at 25 °C.

The standard free energy change for a reaction may also be calculated from standard free energy of formation (ΔGf°)(ΔGf°) values of the reactants and products involved in the reaction. The standard free energy of formation is the free energy change that accompanies the formation of one mole of a substance from its elements in their standard states. Similar to the standard enthalpy of formation, ΔGf°ΔGf° is by definition zero for elemental substances under standard state conditions. The approach used to calculate ΔG°ΔG° for a reaction from ΔGf°ΔGf° values is the same as that demonstrated previously for enthalpy and entropy changes. For the reaction

mA+nB⟶xC+yD,mA+nB⟶xC+yD,
12.42

the standard free energy change at room temperature may be calculated as

ΔG298°=ΔG°=∑νΔG298°(products)−∑νΔG298°(reactants)=[xΔGf°(C)+yΔGf°(D)]−[mΔGf°(A)+nΔGf°(B)].ΔG298°=ΔG°=∑νΔG298°(products)−∑νΔG298°(reactants)=[xΔGf°(C)+yΔGf°(D)]−[mΔGf°(A)+nΔGf°(B)].
12.43

Example 12.8

Calculation of ΔG298°ΔG298°

Consider the decomposition of yellow mercury(II) oxide.
HgO(s,yellow)⟶Hg(l)+12O2(g)HgO(s,yellow)⟶Hg(l)+12O2(g)
12.44

Calculate the standard free energy change at room temperature, ΔG298°,ΔG298°, using (a) standard free energies of formation and (b) standard enthalpies of formation and standard entropies. Do the results indicate the reaction to be spontaneous or nonspontaneous under standard conditions?

Solution

The required data are available in Appendix G and are shown here.
Compound ΔGf°(kJ/mol)ΔGf°(kJ/mol) ΔHf°(kJ/mol)ΔHf°(kJ/mol) S298°(J/K·mol)S298°(J/K·mol)
HgO (s, yellow) −58.43 −90.46 71.13
Hg(l) 0 0 75.9
O2(g) 0 0 205.2

(a) Using free energies of formation:

ΔG298°=∑νGSf°(products)−∑νΔGf°(reactants)ΔG298°=∑νGSf°(products)−∑νΔGf°(reactants)
12.45
=[1ΔGf°Hg(l)+12ΔGf°O2(g)]−1ΔGf°HgO(s,yellow)=[1ΔGf°Hg(l)+12ΔGf°O2(g)]−1ΔGf°HgO(s,yellow)
12.46
=[1mol(0 kJ/mol)+12mol(0 kJ/mol)]−1 mol(−58.43 kJ/mol)=58.43 kJ/mol=[1mol(0 kJ/mol)+12mol(0 kJ/mol)]−1 mol(−58.43 kJ/mol)=58.43 kJ/mol
12.47

(b) Using enthalpies and entropies of formation:

ΔH298°=∑νΔHf°(products)−∑νΔHf°(reactants)ΔH298°=∑νΔHf°(products)−∑νΔHf°(reactants)
12.48
=[1ΔHf°Hg(l)+12ΔHf°O2(g)]−1ΔHf°HgO(s,yellow)=[1ΔHf°Hg(l)+12ΔHf°O2(g)]−1ΔHf°HgO(s,yellow)
12.49
=[1 mol(0 kJ/mol)+12mol(0 kJ/mol)]−1 mol(−90.46 kJ/mol)=90.46 kJ/mol=[1 mol(0 kJ/mol)+12mol(0 kJ/mol)]−1 mol(−90.46 kJ/mol)=90.46 kJ/mol
12.50
ΔS298°=∑νΔS298°(products)−∑νΔS298°(reactants)ΔS298°=∑νΔS298°(products)−∑νΔS298°(reactants)
12.51
=[1ΔS298°Hg(l)+12ΔS298°O2(g)]−1ΔS298°HgO(s,yellow)=[1ΔS298°Hg(l)+12ΔS298°O2(g)]−1ΔS298°HgO(s,yellow)
12.52
=[1 mol(75.9 J/mol K)+12mol(205.2 J/mol K)]−1 mol(71.13 J/mol K)=107.4 J/mol K=[1 mol(75.9 J/mol K)+12mol(205.2 J/mol K)]−1 mol(71.13 J/mol K)=107.4 J/mol K
12.53
ΔG°=ΔH°−TΔS°=90.46 kJ−298.15 K×107.4 J/K·mol×1 kJ1000 JΔG°=ΔH°−TΔS°=90.46 kJ−298.15 K×107.4 J/K·mol×1 kJ1000 J
12.54
ΔG°=(90.46−32.01)kJ/mol=58.45 kJ/molΔG°=(90.46−32.01)kJ/mol=58.45 kJ/mol
12.55

Both ways to calculate the standard free energy change at 25 °C give the same numerical value (to three significant figures), and both predict that the process is nonspontaneous (not spontaneous) at room temperature.

Check Your Learning

Calculate ΔG° using (a) free energies of formation and (b) enthalpies of formation and entropies (Appendix G). Do the results indicate the reaction to be spontaneous or nonspontaneous at 25 °C?
C2H4(g)⟶H2(g)+C2H2(g)C2H4(g)⟶H2(g)+C2H2(g)
12.56

Answer:

141.5 kJ/mol, nonspontaneous

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