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Chemistry: Atoms First 2e

16.4 Potential, Free Energy, and Equilibrium

Chemistry: Atoms First 2e16.4 Potential, Free Energy, and Equilibrium
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  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 Salts
    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 Coupled Equilibria
    5. Key Terms
    6. Key Equations
    7. Summary
    8. Exercises
  17. 16 Electrochemistry
    1. Introduction
    2. 16.1 Review of Redox Chemistry
    3. 16.2 Galvanic Cells
    4. 16.3 Electrode and Cell Potentials
    5. 16.4 Potential, Free Energy, and Equilibrium
    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
By the end of this section, you will be able to:
  • Explain the relations between potential, free energy change, and equilibrium constants
  • Perform calculations involving the relations between cell potentials, free energy changes, and equilibrium
  • Use the Nernst equation to determine cell potentials under nonstandard conditions

So far in this chapter, the relationship between the cell potential and reaction spontaneity has been described, suggesting a link to the free energy change for the reaction (see chapter on thermodynamics). The interpretation of potentials as measures of oxidant strength was presented, bringing to mind similar measures of acid-base strength as reflected in equilibrium constants (see the chapter on acid-base equilibria). This section provides a summary of the relationships between potential and the related thermodynamic properties ΔG and K.

E° and ΔG°

The standard free energy change of a process, ΔG°, was defined in a previous chapter as the maximum work that could be performed by a system, wmax. In the case of a redox reaction taking place within a galvanic cell under standard state conditions, essentially all the work is associated with transferring the electrons from reductant-to-oxidant, welec:

ΔG°=wmax=welecΔG°=wmax=welec

The work associated with transferring electrons is determined by the total amount of charge (coulombs) transferred and the cell potential:

ΔG°=welec=nFEcell°ΔG°=nFEcell°ΔG°=welec=nFEcell°ΔG°=nFEcell°

where n is the number of moles of electrons transferred, F is Faraday’s constant, and E°cell is the standard cell potential. The relation between free energy change and standard cell potential confirms the sign conventions and spontaneity criteria previously discussed for both of these properties: spontaneous redox reactions exhibit positive potentials and negative free energy changes.

E° and K

Combining a previously derived relation between ΔG° and K (see the chapter on thermodynamics) and the equation above relating ΔG° and E°cell yields the following:

ΔG°=RTlnK=nFEcell°Ecell°=(RTnF)lnKΔG°=RTlnK=nFEcell°Ecell°=(RTnF)lnK

This equation indicates redox reactions with large (positive) standard cell potentials will proceed far towards completion, reaching equilibrium when the majority of reactant has been converted to product. A summary of the relations between E°, ΔG° and K is depicted in Figure 16.7, and a table correlating reaction spontaneity to values of these properties is provided in Table 16.2.

A diagram is shown that involves three double headed arrows positioned in the shape of an equilateral triangle. The vertices are labeled in red. The top vertex is labeled “K.“ The vertex at the lower left is labeled “delta G superscript degree symbol.” The vertex at the lower right is labeled “E superscript degree symbol subscript cell.” The right side of the triangle is labeled “E superscript degree symbol subscript cell equals ( R T divided by n  F ) l n K.” The lower side of the triangle is labeled “delta G superscript degree symbol equals negative n F E superscript degree symbol subscript cell.” The left side of the triangle is labeled “delta G superscript degree symbol equals negative R T l n K.”
Figure 16.7 Graphic depicting the relation between three important thermodynamic properties.
K ΔG° E°cell  
> 1 < 0 > 0

Reaction is spontaneous under standard conditions

Products more abundant at equilibrium

< 1 > 0 < 0

Reaction is non-spontaneous under standard conditions

Reactants more abundant at equilibrium

= 1 = 0 = 0

Reaction is at equilibrium under standard conditions

Reactants and products equally abundant

Table 16.2

Example 16.6

Equilibrium Constants, Standard Cell Potentials, and Standard Free Energy Changes Use data from Appendix L to calculate the standard cell potential, standard free energy change, and equilibrium constant for the following reaction at 25 °C. Comment on the spontaneity of the forward reaction and the composition of an equilibrium mixture of reactants and products.

2Ag+(aq)+Fe(s)2Ag(s)+Fe2+(aq)2Ag+(aq)+Fe(s)2Ag(s)+Fe2+(aq)

Solution The reaction involves an oxidation-reduction reaction, so the standard cell potential can be calculated using the data in Appendix L.

anode (oxidation):Fe(s)Fe2+(aq)+2eEFe2+/Fe°=−0.447 Vcathode (reduction):2×(Ag+(aq)+eAg(s))EAg+/Ag°=0.7996 VEcell°=Ecathode°Eanode°=EAg+/Ag°EFe2+/Fe°=+1.247 Vanode (oxidation):Fe(s)Fe2+(aq)+2eEFe2+/Fe°=−0.447 Vcathode (reduction):2×(Ag+(aq)+eAg(s))EAg+/Ag°=0.7996 VEcell°=Ecathode°Eanode°=EAg+/Ag°EFe2+/Fe°=+1.247 V

With n = 2, the equilibrium constant is then

Ecell°=0.0592 VnlogK K=10n×Ecell°/0.0592 V K=102×1.247 V/0.0592 V K=1042.128 K=1.3×1042Ecell°=0.0592 VnlogK K=10n×Ecell°/0.0592 V K=102×1.247 V/0.0592 V K=1042.128 K=1.3×1042

The standard free energy is then

ΔG°=nFEcell° ΔG°=−2×96,485Cmol×1.247JC=−240.6kJmolΔG°=nFEcell° ΔG°=−2×96,485Cmol×1.247JC=−240.6kJmol

The reaction is spontaneous, as indicated by a negative free energy change and a positive cell potential. The K value is very large, indicating the reaction proceeds to near completion to yield an equilibrium mixture containing mostly products.

Check Your Learning What is the standard free energy change and the equilibrium constant for the following reaction at room temperature? Is the reaction spontaneous?

Sn(s)+2Cu2+(aq)Sn2+(aq)+2Cu+(aq)Sn(s)+2Cu2+(aq)Sn2+(aq)+2Cu+(aq)

Answer:

Spontaneous; n = 2; Ecell°=+0.291 V;Ecell°=+0.291 V; ΔG°=−56.2kJmol;ΔG°=−56.2kJmol; K = 6.8 ×× 109.

Potentials at Nonstandard Conditions: The Nernst Equation

Most of the redox processes that interest science and society do not occur under standard state conditions, and so the potentials of these systems under nonstandard conditions are a property worthy of attention. Having established the relationship between potential and free energy change in this section, the previously discussed relation between free energy change and reaction mixture composition can be used for this purpose.

ΔG=ΔG°+RTlnQΔG=ΔG°+RTlnQ

Notice the reaction quotient, Q, appears in this equation, making the free energy change dependent upon the composition of the reaction mixture. Substituting the equation relating free energy change to cell potential yields the Nernst equation:

nFEcell=nFEcell°+RTlnQnFEcell=nFEcell°+RTlnQ
Ecell=Ecell°RTnFlnQEcell=Ecell°RTnFlnQ

This equation describes how the potential of a redox system (such as a galvanic cell) varies from its standard state value, specifically, showing it to be a function of the number of electrons transferred, n, the temperature, T, and the reaction mixture composition as reflected in Q. A convenient form of the Nernst equation for most work is one in which values for the fundamental constants (R and F) and a factor converting from natural to base-10 logarithms have been included:

Ecell=Ecell°0.0592VnlnQEcell=Ecell°0.0592VnlnQ

Example 16.7

Predicting Redox Spontaneity Under Nonstandard Conditions Use the Nernst equation to predict the spontaneity of the redox reaction shown below.

Co(s)+Fe2+(aq,1.94M)Co2+(aq, 0.15M)+Fe(s)Co(s)+Fe2+(aq,1.94M)Co2+(aq, 0.15M)+Fe(s)

Solution Collecting information from Appendix L and the problem,

Anode (oxidation):Co(s)Co2+(aq)+2eECo2+/Co°=−0.28 VCathode (reduction):Fe2+(aq)+2eFe(s)EFe2+/Fe°=−0.447 VEcell°=Ecathode°Eanode°=−0.447 V(−0.28 V)=−0.17 VAnode (oxidation):Co(s)Co2+(aq)+2eECo2+/Co°=−0.28 VCathode (reduction):Fe2+(aq)+2eFe(s)EFe2+/Fe°=−0.447 VEcell°=Ecathode°Eanode°=−0.447 V(−0.28 V)=−0.17 V

Notice the negative value of the standard cell potential indicates the process is not spontaneous under standard conditions. Substitution of the Nernst equation terms for the nonstandard conditions yields:

Q=[Co2+][Fe2+]=0.15M1.94M=0.077 Ecell=Ecell°0.0592 VnlogQ Ecell=−0.17 V0.0592 V2log0.077 Ecell=−0.17 V+0.033 V=−0.14 VQ=[Co2+][Fe2+]=0.15M1.94M=0.077 Ecell=Ecell°0.0592 VnlogQ Ecell=−0.17 V0.0592 V2log0.077 Ecell=−0.17 V+0.033 V=−0.14 V

The cell potential remains negative (slightly) under the specified conditions, and so the reaction remains nonspontaneous.

Check Your Learning For the cell schematic below, identify values for n and Q, and calculate the cell potential, Ecell.

Al(s)Al3+(aq,0.15M)Cu2+(aq,0.025M)Cu(s)Al(s)Al3+(aq,0.15M)Cu2+(aq,0.025M)Cu(s)

Answer:

n = 6; Q = 1440; Ecell = +1.97 V, spontaneous.

A concentration cell is constructed by connecting two nearly identical half-cells, each based on the same half-reaction and using the same electrode, varying only in the concentration of one redox species. The potential of a concentration cell, therefore, is determined only by the difference in concentration of the chosen redox species. The example problem below illustrates the use of the Nernst equation in calculations involving concentration cells.

Example 16.8

Concentration Cells What is the cell potential of the concentration cell described by

Zn(s)Zn2+(aq, 0.10M)Zn2+(aq, 0.50M)Zn(s)Zn(s)Zn2+(aq, 0.10M)Zn2+(aq, 0.50M)Zn(s)

Solution From the information given:

Anode:Zn(s)Zn2+(aq, 0.10M)+2eEanode°=−0.7618 VCathode:Zn2+(aq, 0.50M)+2eZn(s)Ecathode°=−0.7618 V¯Overall:Zn2+(aq, 0.50M)Zn2+(aq, 0.10M)Ecell°=0.000 VAnode:Zn(s)Zn2+(aq, 0.10M)+2eEanode°=−0.7618 VCathode:Zn2+(aq, 0.50M)+2eZn(s)Ecathode°=−0.7618 V¯Overall:Zn2+(aq, 0.50M)Zn2+(aq, 0.10M)Ecell°=0.000 V

Substituting into the Nernst equation,

Ecell=0.000 V0.0592 V2log0.100.50=+0.021 VEcell=0.000 V0.0592 V2log0.100.50=+0.021 V

The positive value for cell potential indicates the overall cell reaction (see above) is spontaneous. This spontaneous reaction is one in which the zinc ion concentration in the cathode falls (it is reduced to elemental zinc) while that in the anode rises (it is produced by oxidation of the zinc anode). A greater driving force for zinc reduction is present in the cathode, where the zinc(II) ion concentration is greater (Ecathode > Eanode).

Check Your Learning The concentration cell above was allowed to operate until the cell reaction reached equilibrium. What are the cell potential and the concentrations of zinc(II) in each half-cell for the cell now?

Answer:

Ecell = 0.000 V; [Zn2+]cathode = [Zn2+]anode = 0.30 M

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