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Chemistry: Atoms First

16.4 The Nernst Equation

Chemistry: Atoms First16.4 The Nernst Equation
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:
  • Relate cell potentials to free energy changes
  • Use the Nernst equation to determine cell potentials at nonstandard conditions
  • Perform calculations that involve converting between cell potentials, free energy changes, and equilibrium constants

We will now extend electrochemistry by determining the relationship between Ecell°Ecell° and the thermodynamics quantities such as ΔG° (Gibbs free energy) and K (the equilibrium constant). In galvanic cells, chemical energy is converted into electrical energy, which can do work. The electrical work is the product of the charge transferred multiplied by the potential difference (voltage):

electrical work=volts×(charge in coulombs)=Jelectrical work=volts×(charge in coulombs)=J
16.76

The charge on 1 mole of electrons is given by Faraday’s constant (F)

F=6.022×1023e−mol×1.602×10−19Ce−=9.648×104Cmol=9.648×104JV·molF=6.022×1023e−mol×1.602×10−19Ce−=9.648×104Cmol=9.648×104JV·mol
16.77
total charge=(number of moles of e−)×F=nFtotal charge=(number of moles of e−)×F=nF
16.78

In this equation, n is the number of moles of electrons for the balanced oxidation-reduction reaction. The measured cell potential is the maximum potential the cell can produce and is related to the electrical work (wele) by

Ecell=−welenForwele=−nFEcellEcell=−welenForwele=−nFEcell
16.79

The negative sign for the work indicates that the electrical work is done by the system (the galvanic cell) on the surroundings. In an earlier chapter, the free energy was defined as the energy that was available to do work. In particular, the change in free energy was defined in terms of the maximum work (wmax), which, for electrochemical systems, is wele.

ΔG=wmax=weleΔG=wmax=wele
16.80
ΔG=−nFEcellΔG=−nFEcell
16.81

We can verify the signs are correct when we realize that n and F are positive constants and that galvanic cells, which have positive cell potentials, involve spontaneous reactions. Thus, spontaneous reactions, which have ΔG < 0, must have Ecell > 0. If all the reactants and products are in their standard states, this becomes

ΔG°=−nFEcell°ΔG°=−nFEcell°
16.82

This provides a way to relate standard cell potentials to equilibrium constants, since

ΔG°=−RTlnKΔG°=−RTlnK
16.83
−nFEcell°=−RTlnKorEcell°=RTnFlnK−nFEcell°=−RTlnKorEcell°=RTnFlnK
16.84

Most of the time, the electrochemical reactions are run at standard temperature (298.15 K). Collecting terms at this temperature yields

Ecell°=RTnFlnK=(8.314JK·mol)(298.15K)n×96,485 C/V·mollnK=0.0257 VnlnKEcell°=RTnFlnK=(8.314JK·mol)(298.15K)n×96,485 C/V·mollnK=0.0257 VnlnK
16.85

where n is the number of moles of electrons. For historical reasons, the logarithm in equations involving cell potentials is often expressed using base 10 logarithms (log), which changes the constant by a factor of 2.303:

Ecell°=0.0592 VnlogKEcell°=0.0592 VnlogK
16.86

Thus, if ΔG°, K, or Ecell°Ecell° is known or can be calculated, the other two quantities can be readily determined. The relationships are shown graphically in Figure 16.9.

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 16.9 The relationships between ΔG°, K, and Ecell°.Ecell°. Given any one of the three quantities, the other two can be calculated, so any of the quantities could be used to determine whether a process was spontaneous.

Given any one of the quantities, the other two can be calculated.

Example 16.5

Equilibrium Constants, Standard Cell Potentials, and Standard Free Energy Changes

What is the standard free energy change and equilibrium constant for the following reaction at 25 °C?
2Ag+(aq)+Fe(s)⇌2Ag(s)+Fe2+(aq)2Ag+(aq)+Fe(s)⇌2Ag(s)+Fe2+(aq)
16.87

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)+2e−EFe2+/Fe°=−0.447 Vcathode (reduction):2×(Ag+(aq)+e−⟶Ag(s))EAg+/Ag°=0.7996 VEcell°=Ecathode°−Eanode°=EAg+/Ag°−EFe2+/Fe°=+1.247 Vanode (oxidation):Fe(s)⟶Fe2+(aq)+2e−EFe2+/Fe°=−0.447 Vcathode (reduction):2×(Ag+(aq)+e−⟶Ag(s))EAg+/Ag°=0.7996 VEcell°=Ecathode°−Eanode°=EAg+/Ag°−EFe2+/Fe°=+1.247 V
16.88

Remember that the cell potential for the cathode is not multiplied by two when determining the standard cell potential. With n = 2, the equilibrium constant is then

Ecell°=0.0592 VnlogKEcell°=0.0592 VnlogK
16.89
K=10n×Ecell°/0.0592 VK=10n×Ecell°/0.0592 V
16.90
K=102×1.247 V/0.0592 VK=102×1.247 V/0.0592 V
16.91
K=1042.128K=1042.128
16.92
K=1.3×1042K=1.3×1042
16.93

The standard free energy is then

ΔG°=−nFEcell°ΔG°=−nFEcell°
16.94
ΔG°=−2×96,485JV·mol×1.247 V=−240.6kJmolΔG°=−2×96,485JV·mol×1.247 V=−240.6kJmol
16.95

Check your answer: A positive standard cell potential means a spontaneous reaction, so the standard free energy change should be negative, and an equilibrium constant should be >1.

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)
16.96

Answer:

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

Now that the connection has been made between the free energy and cell potentials, nonstandard concentrations follow. Recall that

ΔG=ΔG°+RTlnQΔG=ΔG°+RTlnQ
16.97

where Q is the reaction quotient (see the chapter on equilibrium fundamentals). Converting to cell potentials:

−nFEcell=−nFEcell°+RTlnQorEcell=Ecell°−RTnFlnQ−nFEcell=−nFEcell°+RTlnQorEcell=Ecell°−RTnFlnQ
16.98

This is the Nernst equation. At standard temperature (298.15 K), it is possible to write the above equations as

Ecell=Ecell°−0.0257 VnlnQorEcell=Ecell°−0.0592 VnlogQEcell=Ecell°−0.0257 VnlnQorEcell=Ecell°−0.0592 VnlogQ
16.99

If the temperature is not 298.15 K, it is necessary to recalculate the value of the constant. With the Nernst equation, it is possible to calculate the cell potential at nonstandard conditions. This adjustment is necessary because potentials determined under different conditions will have different values.

Example 16.6

Cell Potentials at Nonstandard Conditions

Consider the following reaction at room temperature:
Co(s)+Fe2+(aq,1.94M)⟶Co2+(aq, 0.15M)+Fe(s)Co(s)+Fe2+(aq,1.94M)⟶Co2+(aq, 0.15M)+Fe(s)
16.100

Is the process spontaneous?

Solution

There are two ways to solve the problem. If the thermodynamic information in Appendix G were available, you could calculate the free energy change. If the free energy change is negative, the process is spontaneous. The other approach, which we will use, requires information like that given in Appendix L. Using those data, the cell potential can be determined. If the cell potential is positive, the process is spontaneous. Collecting information from Appendix L and the problem,
Anode (oxidation):Co(s)⟶Co2+(aq)+2e−ECo2+/Co°=−0.28 VCathode (reduction):Fe2+(aq)+2e−⟶Fe(s)EFe2+/Fe°=−0.447 VEcell°=Ecathode°−Eanode°=−0.447 V−(−0.28 V)=−0.17 VAnode (oxidation):Co(s)⟶Co2+(aq)+2e−ECo2+/Co°=−0.28 VCathode (reduction):Fe2+(aq)+2e−⟶Fe(s)EFe2+/Fe°=−0.447 VEcell°=Ecathode°−Eanode°=−0.447 V−(−0.28 V)=−0.17 V
16.101

The process is not spontaneous under standard conditions. Using the Nernst equation and the concentrations stated in the problem and n = 2,

Q=[Co2+][Fe2+]=0.15M1.94M=0.077Q=[Co2+][Fe2+]=0.15M1.94M=0.077
16.102
Ecell=Ecell°−0.0592 VnlogQEcell=Ecell°−0.0592 VnlogQ
16.103
Ecell=−0.17 V−0.0592 V2log0.077Ecell=−0.17 V−0.0592 V2log0.077
16.104
Ecell=−0.17 V+0.033 V=−0.014 VEcell=−0.17 V+0.033 V=−0.014 V
16.105

The process is (still) nonspontaneous.

Check Your Learning

What is the cell potential for the following reaction at room temperature?
Al(s)│Al3+(aq,0.15M)║Cu2+(aq,0.025M)│Cu(s)Al(s)│Al3+(aq,0.15M)║Cu2+(aq,0.025M)│Cu(s)
16.106

What are the values of n and Q for the overall reaction? Is the reaction spontaneous under these conditions?

Answer:

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

Finally, we will take a brief look at a special type of cell called a concentration cell. In a concentration cell, the electrodes are the same material and the half-cells differ only in concentration. Since one or both compartments is not standard, the cell potentials will be unequal; therefore, there will be a potential difference, which can be determined with the aid of the Nernst equation.

Example 16.7

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)
16.107

Solution

From the information given:
Anode:Zn(s)⟶Zn2+(aq, 0.10M)+2e−Eanode°=−0.7618 VCathode:Zn2+(aq, 0.50M)+2e−⟶Zn(s)Ecathode°=−0.7618 V¯Overall:Zn2+(aq, 0.50M)⟶Zn2+(aq, 0.10M)Ecell°=0.000 VAnode:Zn(s)⟶Zn2+(aq, 0.10M)+2e−Eanode°=−0.7618 VCathode:Zn2+(aq, 0.50M)+2e−⟶Zn(s)Ecathode°=−0.7618 V¯Overall:Zn2+(aq, 0.50M)⟶Zn2+(aq, 0.10M)Ecell°=0.000 V
16.108

The standard cell potential is zero because the anode and cathode involve the same reaction; only the concentration of Zn2+ changes. Substituting into the Nernst equation,

Ecell=0.000 V−0.0592 V2log0.100.50=+0.021 VEcell=0.000 V−0.0592 V2log0.100.50=+0.021 V
16.109

and the process is spontaneous at these conditions.

Check your answer: In a concentration cell, the standard cell potential will always be zero. To get a positive cell potential (spontaneous process) the reaction quotient Q must be <1. Q < 1 in this case, so the process is spontaneous.

Check Your Learning

What value of Q for the previous concentration cell would result in a voltage of 0.10 V? If the concentration of zinc ion at the cathode was 0.50 M, what was the concentration at the anode?

Answer:

Q = 0.00042; [Zn2+]cat = 2.1 ×× 10−4 M.

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