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- Herausgeber: Wiley-VCH
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Comprehensive resource covering fundamental principles of electrochemical energy conversion and storage technologies including fuel cells, batteries, and capacitors
Starting with the importance and background of electrochemical foundations, Principles of Electrochemical Conversion and Storage Devices explains the working principles and electrochemistry of electrochemical cells. After a summary of thermodynamic and kinetics, different types of fuel cells as well as batteries and capacitors are covered.
This book is written in the style of a textbook, providing illustrative examples and inspiring problems to facilitate the understanding of essential principles of electrochemical cells while offering practical insights for research pursuits. Various application examples are provided at the end of each chapter to strengthen reader understanding of energy storage from a practical point of view.
Written by a highly qualified and awarded academic and based on a culmination of his two decades of personal teaching and research experience in the field, Principles of Electrochemical Conversion and Storage Devices includes information on:
- Common reference electrodes and potentials, standard electrode potentials in aqueous solutions, and current functions for the charge transfer process
- Standard Gibbs free energy of formation of selected compounds, standard heat of combustion of common fuels, and commonly used physical constants
- Latest developments in the field, especially surrounding clean energy technologies, and various experimental methods essential for conducting rigorous electrochemical research
- Characterizing methods, key materials, and governing principles behind all of the covered devices
Providing comprehensive coverage of the subject, Principles of Electrochemical Conversion and Storage Devices is an excellent resource tailored for researchers and students from all technical and natural science disciplines seeking to understand more about the most promising energy-related devices and the potential they hold to change the world.
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Seitenzahl: 454
Veröffentlichungsjahr: 2024
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Table of Contents
Cover
Table of Contents
Title Page
Copyright
Preface
1 Introduction
1.1 Brief History of Electrochemical Cells
1.2 Configuration of Electrochemical Cells
1.3 Half-Reactions in Electrochemical Cells
1.4 Faradaic and Non-Faradaic Reactions
1.5 Nernst Equation
1.6 Overpotential and Reaction Rate
1.7 Several Important Features of Electrochemical Cells
References
Problems
2 Thermodynamics of Electrochemical Cells
2.1 Open Electrochemical Cell Systems
2.2 Closed Electrochemical Cell Systems
2.3 Temperature Dependence of
E
n
and
E
tn
2.4 Pressure Dependence of
E
n
and
E
tn
2.5 Thermal and Chemical Expansion Coefficients
2.6 Heat Production and Consumption in Electrochemical Cells
2.7 Gibbs Phase Rule in Electrochemical Cells
References
Problems
3 Kinetics of Electrochemical Cells
3.1 Bulk Ionic Transport in Solid Inorganic Electrolytes (SIEs)
3.2 Ionic Transport in Solid Amorphous Electrolytes
3.3 Ionic Transport in Aqueous Solution Electrolytes
3.4 Comparison of Aqueous and Non-Aqueous Electrolytes
3.5 Kinetics of Electrode Reactions
References
Problems
4 Fuel Cells and Electrolytic Cells
4.1 Fuel Cells/Electrolytic Cells Basics
4.2 Voltage Losses in FCs and ECs
4.3 Efficiencies of Fuel Cells and Electrolytic Cells
4.4 Fuel Cells with Acidic Electrolytes
4.5 Fuel Cells with Alkaline Electrolytes
4.6 Fuel Cells with Molten Carbonate Electrolytes
4.7 Fuel Cells with Solid Oxide Electrolytes
4.8 Electrolytic Cells
References
Problems
5 Batteries
5.1 Battery Basics
5.2 Rechargeable Batteries with Aqueous Electrolytes
5.3 Rechargeable Batteries with Organic Electrolytes
5.4 Rechargeable Batteries with Solid Electrolytes
5.5 Primary Batteries
References
Problems
6 Capacitors
6.1 Capacitor Basics
6.2 Parallel-Plate Capacitors (PPCs)
6.3 Electrochemical Double-layer Capacitors (EDLCs)
6.4 Electrochemical Pseudocapacitors (ECPCs)
References
Problems
7 Basic Electrochemical Methods
7.1 Controlled Potential Methods
7.2 Controlled Current Methods
7.3 Current Transient Method
7.4 Electrochemical Impedance Spectroscopy
7.5 Electrical Conductivity
7.6 Electrical Conductivity Relaxation (ECR) Method
7.7 Ion Transport Number of Electrolyte
References
Problems
Appendix A: Common Reference Electrodes and Potentials
A.1 Calomel Electrodes
A.2 Silver/Silver Chloride Electrodes
A.3 Converting Potentials Between Reference Electrodes
Appendix B: Standard Electrode Potentials in Aqueous Solutions
Appendix C: Current Functions for Charge Transfer Process
Appendix D: Standard Gibbs Free Energy of Formation of Selected Compounds
Reference
Appendix E: Standard Heat of Combustion of Common Fuels
Appendix F: Commonly Used Physical Constants
Nomenclature
Abbreviations
Greek Symbols
Roman Symbols
Index
End User License Agreement
List of Tables
Chapter 4
Table 4.1 Different types of fuel cells and electrolytic cells.
Table 4.2 Possible pathways for electrochemical reduction of CO
2
.
Chapter 5
Table 5.1 Equations for quantifying the key metrics of the SOMARB.
Chapter 6
Table 6.1 Dielectric constants of some common dielectric materials.
Chapter 7
Table 7.1 Correction factor
C
for the measurement of sheet resistance with t...
Appendix C
Table C.1 Current functions for reversible charge transfer.
a),b)
Appendix E
Table E.1 Heat of combustion and constant-pressure molar heat capacities of ...
Table E.2 Temperature-dependent thermodynamic properties of oxidation reacti...
List of Illustrations
Chapter 1
Figure 1.1 Six forms of energy and their interchangeable transformations.
Figure 1.2 Volta’s voltaic pile [1].
Figure 1.3 A typical three‐electrode setup for standard electrochemical char...
Chapter 2
Figure 2.1 versus
T
of electro-oxidations of various fuels in SOFC under
P
Figure 2.2 Variations of
E
n
(versus air) of SOFC using H
2
O–H
2
fuel with H
2
O/...
Figure 2.3 Energy profiles versus
T
of reactions of H
2
O = H
2
+ 1/2O
2
and CO
2
Figure 2.4 A schematic illustration of
V
–
I
curves in SOFC and SOEC domains....
Figure 2.5 T-dependent thermodynamic efficiency of SOFCs operated on differe...
Figure 2.6 Temperature dependence of of SOFC operated on different fuels....
Figure 2.7 The standard entropy changes of fuel oxidation reactions in SOFC ...
Figure 2.8 (a) Phase diagram of Nb-O; (b) the measured
E
n
(versus O
2
)-SOC pr...
Figure 2.9 (a) A hypothetical ternary phase diagram of Li–M–X; (b)
E
n
(versu...
Chapter 3
Figure 3.1 Schematic illustration of saddle-point plane in fluoride structur...
Figure 3.2 Schematic illustration of saddle-point position in perovskite ABO
Figure 3.3 O
2−
-energies (
black dots
) and oxygen-vacancy energies for o...
Figure 3.4 Ionic transport models for SPEs [6, 7].
Chapter 4
Figure 4.1 A schematic of (a) fuel cell and (b) electrolytic cell configurat...
Figure 4.2 A schematic illustration of elementary steps in the activation po...
Figure 4.3 Plot of activation overpotential as a function of current density...
Figure 4.4 Concentration polarization loss as a function of of the air-ele...
Figure 4.5 The porosity effect of a planar fuel-electrode on
R
conc
at 900 °C...
Figure 4.6
η
tot
as a function of
i
at 950 °C of an SOFC. Dots are exper...
Figure 4.7 Capillary pressure equilibrium in MC components.
Figure 4.8 Components and configurations of SOFCs based on O
2−
- and H
+
...
Figure 4.9 Arrhenius plots of total conductivity in air of several oxide-ion...
Figure 4.10 Schematics of two H
+
-conducting mechanisms in a BaZrO
3
-based...
Figure 4.11 Schematic illustration of electrochemical CO
2
conversion cells w...
Chapter 5
Figure 5.1 Basic shapes of discharge curves experienced in batteries.
Figure 5.2 Binary phase diagram of Li–Bi system.
Figure 5.3 Titration cell of Li|Li
+
-electrolyte|Bi battery at 380 °C....
Figure 5.4 Gibbs triangle of Li–Cl–Cu ternary system.
Figure 5.5
E
n
and MTED of each Gibbs triangle of Li–Cl–Cu ternary system.
Figure 5.6 Schematic illustration of ESW with the HOMO and LUMO levels. A st...
Figure 5.7 The configuration of lead-acid battery.
Figure 5.8 Configuration of an all-vanadium redox flow battery.
Figure 5.9 Standard electrode potentials of various redox couples in aqueous...
Figure 5.10 Configuration of aqueous alkaline batteries.
Figure 5.11 Illustration of Ni cathode chemistry during charge and discharge...
Figure 5.12 Configuration of alkaline metal–air batteries.
Figure 5.13 Schematic illustration of ZIB chemistry.
Figure 5.14 Rechargeable LIBs with organic electrolytes.
Figure 5.15 Schematic illustration of the “rocking-chair” chemistry concept....
Figure 5.16 Schematic illustration of the “induction” effect (use Co
3+
/C...
Figure 5.17 LiMO
2
layered structure hosting foreign ions.
Figure 5.18 Open framework structures of Li
3
V
2
(PO
4
)
3
formed by connecting po...
Figure 5.19 The structure of graphite. (a) Three-dimensional view; (b) two-d...
Figure 5.20 Schematic illustration of the staged Li intercalation scheme in ...
Figure 5.21 A “blow up” view of rechargeable solid-state batteries.
Figure 5.22 The crystal structure of garnet Li
7
La
3
Zr
2
O
12
.
Figure 5.23 The crystal structure of a Li-containing perovskite ABO
3
.
Figure 5.24 Configuration and working principle of rechargeable SOMARB.
Figure 5.25 Zn/MnO
2
primary battery with aqueous alkaline electrolytes.
Figure 5.26 Primary batteries with organic electrolytes and Li metal anode....
Chapter 6
Figure 6.1 A schematic illustration of polar dielectrics.
Figure 6.2 Schematic illustration of non-polar dielectrics.
Figure 6.3 Charge and discharge curves of an
R
–
C
circuit under a constant vo...
Figure 6.4 A parallel
R
–
C
circuit under AC voltage and the corresponding imp...
Figure 6.5 Schematic illustrations of (a) CV and (b) GCD methods to extract ...
Figure 6.6
V
–
Q
characteristics of different types of capacitors.
Figure 6.7 The configuration of a PPC.
Figure 6.8 Schematic of a concentric cylindrical capacitor.
Figure 6.9 Schematic of a concentric spherical capacitor.
Figure 6.10 Illustration of Classical models of EDL.
Figure 6.11 Schematic illustration of “three-dimension” bulk of the electrod...
Figure 6.12 The characteristic CV curves of the three types of ECPCs.
Chapter 7
Figure 7.1 Illustration of controlled potential experiment over a three-elec...
Figure 7.2 Illustration of potential sweep method (a) and (b) the resulting ...
Figure 7.3 Illustration of controlled current experiment over a three-electr...
Figure 7.4 Block diagram of GCI method with a three-electrode configuration....
Figure 7.5
V
–
t
curve from GCI measurement.
Figure 7.6 A typical GITT curve with two rounds of charge and discharge cycl...
Figure 7.7 Block diagram illustrating working principle of the modern EIS sy...
Figure 7.8 A typical EIS spectrum representing a symmetrical solid-state ele...
Figure 7.9 An equivalent electrical circuit model corresponding to Figure 7....
Figure 7.10 A graphical representation of CPE on the complex plane.
Figure 7.11 Schematic illustration of symmetric three-electrode cell (STEC) ...
Figure 7.12 A typical experimental setup for an EIS measurement of an electr...
Figure 7.13 A schematic illustration of 4-probe DC conductivity measurement....
Figure 7.14 A schematic illustration of 4-probe sheet resistance measurement...
Figure 7.15 Sample configurations for ECR measurement. (a) Bulk sample; (b) ...
Figure 7.16 A typical Hittorf cell setup.
Figure 7.17 Illustration of moving boundary method.
Figure 7.18 Concentration cells for
t
i
measurement of (a) Na
+
and (b) O
2
...
Figure 7.19 Configuration of QCM electrochemical cell.
Appendix A
Figure A.1 A schematic showing the configuration of a saturated calomel elec...
Figure A.2 A schematic of the configuration of Ag/AgCl RE.
Figure A.3 Relative positions of Fe
3+
/Fe
2+
redox couple to different...
Appendix C
Figure C.1 Variation of quasireversible current function,
Ψ
(
E
), for dif...
Figure C.4 Variation of Δ(
Λ
,
α
) with
Λ α
. Dashed lines s...
Guide
Cover
Table of Contents
Title Page
Copyright
Preface
Begin Reading
Appendix A Common Reference Electrodes and Potentials
Appendix B Standard Electrode Potentials in Aqueous Solutions
Appendix C Current Functions for Charge Transfer Process
Appendix D Standard Gibbs Free Energy of Formation of Selected Compounds
Appendix E Standard Heat of Combustion of Common Fuels
Appendix F Commonly Used Physical Constants
Nomenclature
Index
End User License Agreement
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Principles of Electrochemical Conversion and Storage Devices
Kevin Huang
Author
Prof. Kevin HuangUniversity of South Carolina541 Main StreetColumbiaSouth Carolina 29201USA
Cover Images: © Kevin Huang, University of South Carolina, USA
All books published by WILEY-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.
Library of Congress Card No.: applied for
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Bibliographic information published by the Deutsche NationalbibliothekThe Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at <http://dnb.d-nb.de>.
© 2025 WILEY-VCH GmbH, Boschstraße 12, 69469 Weinheim, Germany
All rights reserved (including those of translation into other languages, text and data mining and training of artificial technologies or similar technologies). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.
Print ISBN: 978-3-527-35060-5ePDF ISBN: 978-3-527-83830-1ePub ISBN: 978-3-527-83829-5oBook ISBN: 978-3-527-85163-8
Preface
In the quest for a sustainable and decarbonized energy future, electrochemical cells have captured significant interest from researchers and practitioners alike due to their unparalleled efficiency in converting and storing energy. This strong interest has fueled a rapid expansion of the electrochemical research landscape, playing a pivotal role in the advancement of clean energy technologies.
This book emerges as a response to the growing demand for a comprehensive and authoritative resource that explains the basic principles at the heart of electrochemical cells. It stands not as just another compilation of edited works, but as a culmination of two decades of my personal teaching and research in the realm of electrochemistry.
Tailored for graduate students and researchers seeking a solid foundation of the subject, this book is written in the style of a textbook, providing illustrative examples and inspiring problems to facilitate the understanding of essential principles of electrochemical cells while offering practical insights for research pursuits. Benefiting from my role as an instructor for the “Energy Storage” course at the University of South Carolina over the past 10 years, this book is a distillation of lecture notes blended with my teaching and research experience. My hope is that it will seamlessly bridge the gap between theoretical understanding and real-world applications.
The structure of the book is designed to be both coherent and comprehensive. It commences with an introduction to electrochemical cells in Chapter 1, after which it embarks on a deep dive into the foundational aspects of thermodynamics and kinetics in Chapters 2 and 3. The book then unfolds with an exploration of materials and designs specific to fuel cells, electrolyzers, batteries, and capacitors in Chapters 4–6. Finally, it is ended in Chapter 7, which introduces various experimental methods essential for conducting rigorous electrochemical research.
In essence, this book is an earnest endeavor to empower the next generation of students and researchers with the knowledge and skills necessary to navigate the intricate realm of electrochemical cells. It is my sincere belief that, equipped with this understanding, they will contribute meaningfully to the ongoing transition toward a cleaner, sustainable energy paradigm.
January 2024
Dr. Kevin Huang
SmartState Endowed ProfessorDirector of South Carolina SmartStateCenter for Solid Oxide Fuel CellsThe University of South CarolinaColumbia, South CarolinaUSA
1Introduction
The Earth, a dynamic and interconnected system, relies fundamentally on the Sun as its primary energy source. All naturally occurring raw energy on our planet, whether directly or indirectly, stems from the radiant power of the Sun. From the growth of plants through photosynthesis to the formation of fossil fuels over millions of years, the Sun’s energy plays a pivotal role in shaping the Earth’s energy landscape. Wind patterns and water movements, too, are consequences of the Sun’s heating of the Earth’s surface. Over the course of evolution, humans have acquired the knowledge to harness and convert these raw natural energies into forms, such as electricity and heat, useful for sustaining daily life. This journey of transformation has resulted in various interchangeable forms of energy, which have become the foundation of our modern existence. In Figure 1.1, we show the six forms of energy that humans have learned to interchangeably transform, underscoring their essential role in supporting and shaping our daily lives.
However, the law of energy conservation requires that energy cannot be created nor destroyed. This means that even though energy changes its form, the total quantity of energy always stays the same. For example, many of the energy converters widely used today involve the transformation of chemical energy through thermal energy into electrical energy. The efficiency of such systems is limited by the fundamental laws of thermodynamics. As of today, the conversion efficiency from naturally occurred fossil fuels to useful electricity is ∼35% on average. Such a low efficiency implies a faster depletion of fossil fuels and more emissions of greenhouse gas (CO2) into the atmosphere. In this aspect, those direct energy-conversion devices, such as fuel cells and electrolyzers that bypass the intermediate step of chemical-to-thermal conversion, are advantageous. Unbound by the thermodynamic law, these direct energy converters can achieve high efficiency, thus less fuel consumption and carbon emissions in producing the same amount of electricity. Electrochemical cells are a representative class of such high-efficiency direct energy converters. They are currently deemed one of the best technologies to address all aspects of energy-related challenges such as emissions/pollutions, efficiency, intermittency, cost, and supply chains.
Figure 1.1 Six forms of energy and their interchangeable transformations.
1.1 Brief History of Electrochemical Cells
The history of electrochemical cells began with Italian physicist Alessandro Volta. In 1800, he demonstrated a so-called “voltaic pile”, see Figure 1.2[1], in which Cu and Zn discs were separated by cardboard or felt spacers soaked in salt water (the electrolyte). This is believed to be the first prototype of the modern battery producing electrical current from chemical reactions. Following Volta’s pioneering work, British scientist Humphry Davy was the first to link the production of electricity with the occurring chemical reactions (precisely Gibbs free energy change of the reactions). His student Michael Faraday took a step further predicting how much product can be produced by passing a certain amount of electric current though a chemical compound, a process that he called “electrolysis” and later known as Faraday’s law. The above two important works laid the foundation for today’s electrochemical cells to produce power and chemicals.
Figure 1.2 Volta’s voltaic pile [1].
Source: acrogame/Adobe Stock.
Modern electrochemistry has now become a branch of chemistry studying the interplay of electrical and chemical energy. Governed by Faraday’s law, a large portion of this field deals with the study of changes in chemical energy (or Gibbs free energy) caused by the passage of an electrical current or vice versa, the production of electrical current by chemical reactions [2]. These basic laws of electrochemistry have been successfully applied to a wide range of fields from fundamental phenomena (e.g. electrophoresis and corrosion) and to technologies (batteries, sensors, fuel cells, water electrolyzers, smelters, and metal platers), making significant impacts on every part of our life and economy.
1.2 Configuration of Electrochemical Cells
In electrochemical cells, chemical and electrical energy can be reversibly transformed into each other. An electrochemical cell consists of three basic components: electrolyte, cathode, and anode. Electrolyte is an ionic conductor and electron insulator. Cathode is the electrode where reduction reactions (accepting electrons from external circuit) occur, while anode is the opposite electrode to cathode, where oxidation reactions (releasing electrons to external circuit) take place. Therefore, both cathode and anode are typically electron conductors and catalytically active to the respective reduction and oxidation reactions. As such, they are often labeled as electrocatalysts by electrochemists to differentiate from conventional catalysts in chemical catalysis that do not involve electron transfer from/to external circuit. At the device level (e.g. batteries, electrolyzers, and fuel cells), current collector is also considered as an integral component of the cell. The performance (e.g. power, or chemical production rate) of an electrochemical cell is generally determined by the ohmic resistance of electrolyte and current collector, and polarization resistances (activation and concentration) of the two electrodes.
The modern electrochemical cells primarily comprise fuel cells, electrolyzers, and batteries, but are often extended to include pseudocapacitors (supercapacitors) and photoelectrochemical cells that involve electron transfer. Fuel cells are a type of concentration or galvanic cell operated in an open system for the purpose of producing electrical power. For a continuous power generation, the underlying chemical reactions must be spontaneous, which is characterized by a negative Gibbs free energy change and a positive electromotive force (EMF) or Nernst potential (En). Conversely, electrolyzer is a type of electrolytic cell operated in an open system for the purpose of producing chemicals. Since the underlying chemical reactions are non-spontaneous, which is characterized by positive Gibbs free energy change and negative En, electrical current is needed to drive the reaction. If the produced chemicals can be stored externally for later use, electrolyzers are also viewed as an energy storage device. The capacity of fuel cells and electrolyzers is scaled by the surface area (not by mass) because they are open systems.
In contrast to fuel cells and electrolyzers, batteries operate as a closed (or semi-closed) system with alternating power (En > 0) and chemical production (En < 0) modes. The chemical energy is shuttled between the two electrodes with electrons as the charge regulator through the external circuit. Therefore, the capacity of a battery is influenced by the mass of active electrodes and typically scaled by weight and/or volume.
1.3 Half-Reactions in Electrochemical Cells
The overall reaction of an electrochemical cell is represented by a chemical reaction. This chemical reaction is made up of two independent half-electrode reactions that describe the real chemical changes at the two electrodes. Each half-reaction describes electro-active species involved in electron transfer at the corresponding electrode and defines a fixed potential E (E0 under standard condition); refer to Appendix B for these values. The open circuit voltage OCV of a full cell is, therefore, the difference of the electrode potentials of the two half-electrode reactions,
For the half-reaction of interest, the electrode at which it occurs is called the working electrode (WE). To accurately characterize its behaviors, a three-electrode cell configuration is needed, where a reference electrode (RE) and counter electrode (CE) are included, see Figure 1.3.
Example 1.1
Calculate the OCV of the cell: (−) Cu(s)|Cu2+(aq) ||salt bridge|| Ag+(aq)|Ag(s) (+).
Solution
From Appendix B, E0(Cu/Cu2+) = 0.520 V, E0(Ag/Ag+) = 0.7996 V, OCV = 0.7996 − 0.520 = 0.2796 V.
The functionality of RE is to provide a constant potential to which the potential of WE is referred. Therefore, RE is an electrode made up of phases of a constant composition, which enables it, by Gibbs’s phase rule, to exhibit a fixed potential at a given temperature and pressure, even under the passage of small currents. Thus, they are also termed “nonpolarizable electrode.” The primary reference, chosen by convention, is the normal hydrogen electrode (NHE), aka. the standard hydrogen electrode (SHE), Pt|H2 (a = 1)|H+ (a = 1), represented by the half-reaction of
Its thermodynamic potential has been assigned zero at all temperatures. However, SHE is not convenient to use. In practice, the saturated calomel electrode (SCE), Hg/Hg2Cl2/KCl (saturated in water), is commonly used as a RE, which has a standard potential of 0.242 V versus SHE and the half-reaction of
Figure 1.3 A typical three‐electrode setup for standard electrochemical characterization.
Another commonly used RE is the silver–silver chloride electrode, Ag/AgCl/KCl (saturated in water), with a potential of 0.197 V versus SHE, in which the electrode reaction is
Note that the Ag/AgCl RE should be avoided for use in alkaline electrolytes due to the reaction of Ag with OH−, forming Ag2O that blocks the electrode tip. A table of commonly used reference electrode potentials can be found in Appendix A.
Each half-reaction has a fixed potential (E0) under standard condition (a = 1, P = 1 atm, T = 25 °C). The modern electrochemistry defines the standard potential of the half-reaction accepting electrons from the external circuit as the tabulated values; for easy reference, they are listed in Appendix B. By this definition, when the potential of an electrode is moved from its open-circuit value toward more positive potentials, the active species with the most positive E0 will be reduced first and is deemed as the oxidant. Therefore, strong oxidizing agents are found in anions and/or neutral species with high positive E0. Vice versa for the potential of an electrode is moved from its open-circuit value toward more negative potentials, in which the active species with the most negative E0 will be oxidized first and deemed as the reductant. Thus, strong reducing agents are found in cations and/or neutral species with high negative E0. In discussing electrochemical cells, the electrode at which reductions occur is called the cathode, and the electrode at which oxidations occur is the anode. A current in which electrons cross the interface from the electrode to electrolyte is a cathodic current, while electrons flow from electrolyte into the electrode is an anodic current. Therefore, in electrolytic cells, the cathode is deemed negative (−) with respect to the anode; but in galvanic cells, the cathode is viewed positive (+) with respect to the anode.
1.4 Faradaic and Non-Faradaic Reactions
It is important to recognize that there are two types of processes possibly occurring at electrodes of an electrochemical cell. One is reactions in which electrons are transferred across the electrode/electrolyte interface, causing oxidation or reduction to occur. Since half-reaction follows Faraday’s law, it is often called faradaic process. Good examples of Faradaic cells are fuel cells and batteries. Another kind is processes such as adsorption and desorption of active species on the surface of electrodes. These processes change the electrode’s surface structure, potential or electrolyte composition, but do not transfer electrons across the electrolyte/electrode interface. Therefore, they are referred to as non-faradaic processes. Good examples of non-Faradaic cells are electrochemical double-layer capacitors and pseudocapacitors [3].
An electrode at which no electron transfer can occur across the electrode/electrolyte interface, regardless of the potential applied, is called an ideally polarizable electrode (IPE). While no real electrode can behave as an IPE over the whole potential range available in electrolytes, some electrode–solution systems can approach ideal polarizability over limited potential ranges. An immediate example is Hg/KCl electrode/electrolyte interface.
The IPE features a horizontal line on polarization i-E curve. Since electrons cannot cross the IPE interface when a potential (V) is applied, the behavior of the electrode/electrolyte interface is similar to that of a capacitor. As is commonly known, when V is applied across a capacitor with capacitance C, charge (Q) will accumulate on its metal plates until Q satisfies Q = C * V. The separation of charged species and oriented dipoles at the electrode/electrolyte interface is called the electrical double layer (EDL), which will be discussed in Chapter 6. Therefore, the electrode/electrolyte interfacial impedance of an electrochemical cell is often modeled by a double-layer capacitance of the non-Faradaic process, Cdl, in parallel to charge transfer resistance of the faradaic process, in the equivalent circuit modeling of electrochemical impedance spectroscopy; the latter will be discussed in detail in Chapter 7.
1.5 Nernst Equation
The faradaic half-reactions can be generalized into
If the kinetics of electron transfer of (1.5) are fast and kf = kb, the activities of О (aO) and R (aR) at the electrode surface can be assumed to be at equilibrium, the corresponding potential En of (1.5) can be expressed by
Eq. (1.6) is often referred to as Nernst equation. It provides a linkage between electrode potential E and the concentrations of active species in the electrode process. En is also known as Nernst potential. It is often measured as OCV of a cell. If a system follows the Nernst equation or an equation derived from it, the electrode reaction is often said to be thermodynamically or electrochemically reversible.
Example 1.2
Consider a concentration cell Cu(s)|CuSO4(aq)a1 ||salt bridge||CuSO4(aq)a2|Cu(s). Calculate En of the cell at 25 °C if a1 = 0.75 and a2 = 0.25.
Solution
From Eq. (1.6) and E0 = 0 V, n = 2, En = 8.314 (J/K/mol) × (273 + 25) (K)/2/96,500 (As/mol) × ln(0.75/0.25) = 0.014 V.
1.6 Overpotential and Reaction Rate
When an electrical (faradaic) current is passed through the cell, the electrode equilibrium is disturbed (polarized) and the original electrode potential will change. The departure of the electrode potential from the equilibrium value upon passage of current is termed polarization. The extent of polarization is measured by the overpotential, η,
Note that Eeq = E0 if the cell is under the standard condition.
Since an electrode process is a heterogeneous reaction occurring only at the electrode/electrolyte interface, its rate () is connected to the current i in the following form:
Eq. (1.8) is also known as Faraday’s law. When a steady-state current is reached, the rates of all in-series reaction steps in the electrode process are the same. The magnitude of this rate is often limited by one or more slowest reactions (called rate-determining steps or RDS).
Example 1.3
A galvanic cell consisting of standard concentrations: (−) Cu(s)|Cu2+(aq) ||salt bridge|| Ag+(aq)|Ag(s) (+) is connected to a small light bulb at 0.12 A for 40 minutes. How many grams of copper are dissolved from the anode?
Solution
The total charge passing through the light bulb is 0.12 A × 40 (min) × 60 (s/min) = 288 (As), which is equivalent to 288 (As)/2/96,500 (As/mol) = 1.49 × 10−3 mol. With the atomic weight of Cu, 63.546 g/mol, the total amount of Cu dissolved from the anode is 1.49 × 10−3 (mol) × Cu 63.546 (g/mol) = 0.0948 (g).
1.7 Several Important Features of Electrochemical Cells
Due to its direct reversible electrons-to-molecules conversion, electrochemical cells generally have fast response time, high power/energy density, high conversion efficiency, and low/zero emissions. Specifically for open electrochemical cell systems such as fuel cells and electrolyzers, this is reflected by the high power density or chemical production rate at high conversion efficiency compared to internal combustion engines or chemical reactors. For closed electrochemical cell systems such as batteries and capacitors, this is not only reflected by the high energy density and efficiency, but also in design flexibility of power and energy to meet specific needs. Above all, because of monolithic design, electrochemical cells are usually compact in geometry and can be easily made into modular and scalable systems for volume- or weight-limited applications such as transportation and distributed generations. However, the commercial viability of an electrochemical cell system is almost exclusively determined by service life (years or degradation rate) and cost ($/kW, $/kWh/cycle). In large-scale stationary energy storage systems, these two criteria have become the major considerations for capital investments and commercial development.
References
1
Decker, F. (2005). Volta and the ‘pile’. In:
Electrochemistry Encyclopedia
. Case Western Reserve University.
2
Faraday, M. (1834). On electrical decomposition.
Philos. Trans. R. Soc.
124: 77–122.
https://doi.org/10.1098/rstl.1834.0008.S2CID 116224057
.
3
Bard, A.J. and Faulkner, L.R. (2001).
Electrochemical Methods: Fundamentals and Applications
. New York, USA: John Wiley & Sons, Inc.
Problems
1.1
For a Cu-Zn galvanic cell, use the standard electrode potentials listed in
Appendix B
to calculate
E
n
of the cell.
1.2
The electrolysis of molten MgCl
2
is used to obtain Mg metal. Calculate how many amps are needed to produce 35.6 g of Mg in 2.50 hours?
1.3
How many minutes does it take to produce 10.0 liter of oxygen gas under standard condition by electrolyzing neutral water with a current of 1.3 A? How much H
2
is produced at the same time?
2Thermodynamics of Electrochemical Cells
Thermodynamics is the foundation of all electrochemical cells. It governs the relationship of electrode potential and basic thermodynamic quantities, defines the theoretical upper limits of electrochemical performance, and predicts the dependences of electrochemical properties on concentration, temperature, and pressure. It also offers a unique way to attain thermodynamic properties of chemical reactions through accurate electrochemical measurements. In this chapter, the laws of thermodynamics are applied to explain the electrochemical, chemical, and mechanical behaviors of electrodes in electrochemical cells. The electrochemical cells to be discussed are divided into “open” and “closed” systems to distinguish the power/chemical generating fuel cells/electrolyzers from the energy storing batteries/capacitors. One will find that Nernst potential (En) is directly related to Gibbs free energy change (ΔG) of chemical reactions in an electrochemical cell. However, to use the Nernst equation, one should remember that electrochemical cells should be at equilibrium and thermodynamically reversible. Practically, it is impossible to meet this requirement because all actual electrochemical processes occur at certain rates. With a desired accuracy, however, an electrochemical process may be carried out in such a manner that Nernst equations can be applied.
2.1 Open Electrochemical Cell Systems
Open electrochemical cell systems refer to those operating with a continuous supply of reactants and constant generation of products without changing the composition or mass of the electrodes and the electrolyte. Representative examples of open electrochemical cell systems are fuel cells (galvanic cells) and electrolyzers (electrolytic cells). The former is operated under a constant supply of oxidant such as air and chemical fuels such as hydrogen and generation of power and heat, whereas the latter is operated under a constant supply of electricity, heat, and chemically inert feedstocks such as H2O/CO2, and production of active oxidants/fuels such as oxygen and hydrogen/syngas.
2.1.1 Nernst Potential En of Galvanic Cells
The fundamental relationship between En and ΔG in a galvanic cell, regardless of open or closed system, can be understood by and derived from the thermodynamic first law. When a system undergoes a reversible process under isothermal and isobaric conditions, the first law of thermodynamics states that decrease in the Gibbs free energy, G, of the system equals , the maximum work (other than work of expansion) done by the system. For an increment of such a process [1]:
When there is application of Eq. (2.1) to an electrochemical cell, the work is the electric charge transported across a voltage difference created between the two electrodes because of the Gibbs free energy change (ΔGr) of chemical reaction occurring in the cell. If dm mole of charge carrier with charge n are transported through a voltage difference Δφ, then the electrical work (J) is the product of the charge transported, dm × n × F (C), and the electric potential difference, Δφ (volts),
Combining Eqs. (2.1) with (2.2) leads to
For reversibly transporting 1 mole of ions, En = Δφ, Eq. (2.3) then becomes
Eq. (2.4) is the foundation for calculating En of an electrochemical cell from ΔGr of the undergoing chemical reaction. Under a standard state, Eq. (2.4) is rewritten as:
where and refer to Nernst potential and Gibbs free energy change of the reaction under standard state, respectively. As the standard state of a component is simply a reference state to which the component in any other state is compared, it follows that any state can be chosen as the standard state, and the choice is normally made purely based on convenience. The commonly used standard state is the pure substance (a = 1), one atmosphere pressure (Po = 1 atm), and the temperature of the system. Note that all thermodynamic databases compile thermodynamic properties (such as G, H, and S) of substances under standard state as a function of temperature for use.
When the system is not at standard state, En will deviate from by a term. The difference between the two states can be derived from the mass–action law. Consider a global chemical reaction occurring in a galvanic cell:
in which the reactants and products are not in their standard states, ΔGr of reaction (2.6) is given by:
and, from Eq. (2.4), the corresponding En of the cell is given by:
where . Typically, of reaction (2.6) can be obtained from the Gibbs free energy of formation, , of products and reactants by:
where ν is the stoichiometric coefficient of the products or reactants. of a substance is a thermodynamic property available in thermodynamic database. In Appendix D of this book, of common materials are provided. With the known , En at any condition of reactants and products can be calculated by Eq. (2.8). In the following, several examples are provided to illustrate the utility of Eq. (2.8).
Example 2.1
Calculate En of H2–air fuel cells.
Solution
Consider the cathode and anode reactions in a solid oxide fuel cell and overall electrochemical oxidation reactions of H2 as follows:
Applying Eq. (2.8) to (2.10) with n = 2, system pressure Pt = 1 atm, Pi = aiPt (i = H2, O2 and H2O) leads to
Since of chemical 2.10 is equal to (H2O), which is available in Appendix D, of H2 fuel as a function of temperature under Pt = 1atm can be calculated out and the results are shown in Figure 2.1. Similarly, one can also calculate at different T for CO, CH4, and C as the fuel; the results are compared in Figure 2.1. Evidently, H2 and CO oxidations produce an more sensitive to temperature than those from CH4 and C oxidations. In later section, we will discuss in detail on the T-dependence of . Under a practical condition, e.g.,Po2 = 0.21 atm and , a similar graph to Figure 2.1 can be plotted using Eq. (2.11).
Under a non-standard state, for example, when the oxidant is air (Po2 = 0.21 atm), En of a specific fuel oxidation reaction can also be calculated from Eq. (2.11). Figure 2.2 illustrates En versus logarithm of under different T with air and H2 as oxidant and fuel, respectively. The analytical form of relationship derived from the thermodynamic data in Appendix D is given by:
Figure 2.1 versus T of electro-oxidations of various fuels in SOFC under Pt = 1 atm.
Figure 2.2 Variations of En (versus air) of SOFC using H2O–H2 fuel with H2O/H2 ratio at different T. Also included is the En (versus air) for Ni–O2 reaction.
One can also calculate En with CO, CH4, and C as the fuel and air as the oxidant, see Problem 2.1.
Example 2.2
Calculate En of Ni–air fuel cell.
Solution
From the overall Ni oxidation reaction Ni + 1/2O2 (air) = NiO potentially occurring inside the anode and Eq. (2.8), one can derive En associated with the Ni oxidation as follows using ΔGr data in Appendix D:
Note that Ni-oxidation related En is no longer dependent on , which is understandable from the Gibbs phase rule since Ni and NiO are both solids and the degree of freedom or the number of independent variables of the Ni–NiO system is one under an isobaric condition, meaning the phase equilibrium is only T-dependent. Plotting Eq. (2.13) in Figure 2.2 (dash lines) suggests that it would require a mixture of 99.9% H2O–0.1% H2 to oxidize Ni metal at 1000 °C. It also infers that the cell voltage would become invariant at the value predicted by Eq. (2.13) if the oxidation of Ni occurs.
2.1.2 Thermoneutral Potential Etn of Electrolytic Cells
In the last section, we have discussed that Nernst potential En is determined by the Gibbs free energy change (ΔGr) of the global chemical reaction occurring in the galvanic cell. For electrolytic cells, while En is still applicable, a new term called “thermoneutral potential (Etn)” is better used to describe the performance of electrolytic cells due to the nature of electrical-to-chemical energy conversion. Etn is defined by:
where ΔHr (>0) is the enthalpy change of the chemical reaction occurred in the electrolytic cell, e.g., H2O = H2 + 1/2O2 and CO2 = CO + 1/2O2. The use of Etn is important to high-temperature electrolytic cells since TΔSr = ΔHr − ΔGr becomes significant. In contrast, for low-temperature electrolytic cells (e.g. proton-exchange membrane cells and liquid alkaline cells), it is insignificant compared to ΔHr and ΔGr such that ΔHr ≈ ΔGr. We here use open-system solid oxide electrolytic cells (SOECs) to discuss some important thermodynamic aspects of electrolytic cells.
For open-system SOECs, the energy inputs are electricity, heat, and chemical feedstock such as H2O and CO2, and the products are value-added chemicals such as H2 and CO. Since ΔGr representing electrical energy input is TΔSr less than ΔHr stored in chemicals produced, additional thermal energy TΔSr is needed for electrolytic cell operation, particularly for high-temperature SOECs when TΔSr is significant. From the thermodynamic data in Appendix D, the relative energy position among ΔHr, ΔGr, and TΔSr using H2O = H2 + 1/2O2 and CO2 = CO + 1/2O2 as two examples is illustrated in Figure 2.3.
Figure 2.3 Energy profiles versus T of reactions of H2O = H2 + 1/2O2 and CO2 = CO + 1/2O2.
Using Eqs. (2.12, 2.14) and 2.14, the relative positions of Etn and En versus the cell voltage Vc on a hypothetical V–I curve for illustration purpose are shown in Figure 2.4. From the hypothetical V–I curve, three regimes can be distinguished:
Figure 2.4 A schematic illustration of V–I curves in SOFC and SOEC domains.
SOFC regime (
V
c
<
E
n
): electricity and heat generated are
P
e
=
V
c
×
I
and , respectively.
SOEC regime (
E
n
<
V
c
<
E
tn
): electricity and heat needed are
P
e
=
V
c
×
I
and , respectively.
Extended SOEC regime (
V
c
>
E
tn
): electricity needed, and heat generated are
P
e
=
V
c
×
I
and , respectively.
Note that the SOFC current is considered negative for the oxygen reduction reaction (cathodic) occurring at the air electrode and the SOEC current is considered positive for the oxygen evolution reaction (anodic) occurring at the air electrode. Clearly, at Vc ≥ Etn, TΔSr ≤ 0, meaning that there is no heat needed to balance the energy requirement. The Joule heating at high Vc is sufficient to sustain the operating temperature. The above thermodynamic insights are important to SOEC operation. At Vc < Etn, cold spots are expected within the stack due to the additional heat drawn from the cell. At Vc > Etn, the cell temperature is expected to be at or higher than the normal setpoint. Therefore, to expect cooler cell temperature during initial loading process (Vc < Eth), the SOEC stack temperature can be intentionally set to slightly higher than the wanted temperature to minimize the cold spots.
2.1.3 Thermodynamic Efficiency of Electrochemical Cells
Thermodynamic efficiency (ɛo) represents the highest conversion (chemical-to-electricity and vice versa) efficiency that an electrochemical cell can achieve. It is analogous to the highest Carnot efficiency for all internal combustion engines. For a galvanic cell under the standard state, ɛo is defined by the ratio of Gibbs free energy change (electrical energy produced) to enthalpy change (chemical energy input) of the global reaction undergoing in the cell:
Clearly, ɛo of a fuel cell is both fuel and T dependent. For example, fuel cells with H2 fuel, ɛo = 76% at 800 °C. For carbon fuel, it approaches ∼100% (see below example).
For electrolytic cells, ɛo can be defined as:
Therefore, the highest ɛo can approach to 100% when the applied voltage Vc equals Eth, where , meaning all electrons are converted into molecules.
Example 2.3
Calculate thermodynamic efficiency of SOFC with H2, CO, CH4, and C as fuels.
Solution
Using Eq. (2.15) and the thermodynamic data in Appendix D, the thermodynamic efficiency of SOFCs operating on full oxidation of fuels can be calculated as a function of T. Figure 2.5 shows the results.
Figure 2.5 T-dependent thermodynamic efficiency of SOFCs operated on different fuels.
2.2 Closed Electrochemical Cell Systems
Closed electrochemical systems refer to electrochemical cells with no external mass supply of chemical reactants nor net mass output of chemical products; electrical current and/or heat are the only input and/or output of the system. Therefore, they are typically reversible electrical–chemical conversion devices suited for storing electricity and chemicals. Good examples of closed electrochemical cell systems are batteries (rechargeable and non-rechargeable) and capacitors. Strictly speaking, metal–air batteries are semi-closed systems since the cathodes of these systems are constantly open to air.
2.2.1 En of Batteries
Like any electrochemical cell, the Nernst potential (the maximum voltage) of a battery is determined by ΔGr of the global chemical reaction occurring inside the battery. Therefore, Eq. (2.8) is also applicable to batteries.
Example 2.4
Calculate En of Li/I2 battery.
Solution
The global reaction of Li/I2 battery is:
From Appendix D, ; MLiI = 133.85 g/mol.
From Eq. (2.4), En = −(−2 69 670 (J))/1/96 500 (C/mol) = 2.795 (V).
2.2.2 Theoretical Energy Density of Battery Cells
Since the maximum electrical energy of a battery equals the Gibbs free energy change of the global chemical reaction (summation of the two half-electrode reactions), the maximum theoretical energy density (MTED) on mass basis of a battery cell can be calculated by [2]:
Here n is the number of electrons transferred during the electrode reaction and M is the molecular weight of the products or reactants. If there are multiple products, M is the sum of each individual product’s molar weight Mi multiplied by the corresponding stoichiometric coefficients νi in the chemical reaction, M = ∑ νiMi.
If the densities of the products are known, the mass-based MTED can be easily converted into volume-based MTED. The readers can exercise this conversion with the following example.
Example 2.5
Calculate MTED of Li/I2 battery.
Solution
From Eq. (2.17) and Example 2.4,
MTED = 26 805 × 1 × 2.795/(133.85) = 559.77 (Wh/kg).
2.2.3 Maximum Theoretical Charge Capacity of a Battery Cell
The mass-based maximum theoretical charge capacity (MTCD) of a battery cell can be calculated by
Note that MTCD is a material’s property. It describes how many charges are stored in the redox active materials, i.e., electrodes. It is different from MTED, which is also dependent on the Nernst potential, which is determined by the Gibbs free energy change (ΔGr) of the global chemical reaction that a battery undergoes.
Example 2.6
Calculate MTCD of Zn/Zn2+ redox couple.
Solution
The atomic weight of Zn is 65.38 g/mol. The number of charges transferred in Zn/Zn2+ redox couple is n = 2.
Using Eq. (2.18)(2.18), the MTCD of Zn/Zn2+ is
MTCD = 26 805 × 2/65.38 (g/mol) = 820 (mAh/g)
The corresponding volume-based MTCD is
MTCD = 820 (mAh/g) × 7.133 (g/cm3) = 5849 (mAh/cm3).
2.2.4 Round-Trip Efficiency of Battery Cell
The round-trip efficiency (RTE) describes the reversibility of a battery cell to transform energy between chemical and electrical forms during charge and discharge cycles. It is conceivable that the higher the RTE, the more reversible the cell. It is generically expressed by the ratio of energy output Eout and energy input Ein cycled through the battery cell:
where q, V, I, and t are charge, voltage, current, and time (cycle duration), respectively. The subscripts “d” and “c” represent “discharge” and “charge” cycles, respectively. Under galvanic cycling, Eq. (2.19) can be simplified into
If Id = Ic and td = tc, Eq. (2.20) can be further simplified into
Under this circumstance, the voltages of discharge and charge cycles determine the RTE of the battery.
Example 2.7
Calculate RTE of an all-vanadium redox flow battery. Consider an all-vanadium redox flow battery undergoing galvanic charge and discharge cycles. For the discharge cycle, the battery operates on a galvanic current of 50 mA/cm2 for 4 hours and exhibits a voltage of 900 mV. For the charge cycle, the battery operates on a galvanic current of 50 mA/cm2 for 4 hours and exhibits a voltage of 1200 mV.
Solution
Using Eq. (2.20),
RTE = (50 (mA/cm2) × 4 (h) ×0.9 (V))/(50 (mA/cm2) × 4 (h) × 1.2 (V)) = 75%.
2.3 Temperature Dependence of En and Etn
The temperature (T) dependence of En and Etn is an important thermodynamic information for electrochemical cells, particularly for those operating at high temperatures. Regardless of the system configuration (open or closed), it is solely determined by the dependence of ΔGr and ΔHr on T under isobaric condition. For a known global chemical reaction undergoing within an electrochemical cell, the following En versus T relationship can be derived from:
In other words, the temperature dependence of En is determined by the entropy change of the reaction. If ΔSr > 0, En increases with T and vice versa. The above equations are also applicable to standard state.
Example 2.8
The standard En of the cell Zn(s)|Zn2+(aq) || salt bridge||Fe2+(aq), Fe3+(aq)|Pt(s) is +1.534 V at 25 °C and 1.576 V at 65 °C. Write the cell reactions. Calculate , and K at 25 °C.
Solution
At Zn-electrode: Zn(s) → Zn2+(aq) + 2e−
At Fe2+/Fe3+ electrode: 2Fe3+(aq) + 2e− → 2Fe2+(aq)
Overall reaction: Zn(s) + 2 Fe3+(aq) → Zn2+(aq) + 2Fe2+(aq)
According to Eq. (2.5) and n = 2,
According to Eq. (2.23),
From , K = 7 × 1051 (mol/l).
Example 2.9
Calculate the T-coefficient of of SOFCs with air oxidant and different fuels.
Solution
With the thermodynamic data in Appendix D, we can calculate how of SOFC operating on different fuels varies with T using Eq. (2.23). Figure 2.6 shows that of H2 and CO electrochemical oxidations in SOFC have negative dependence on T, whereas those for CH4 and C are almost T-independent or slightly positive. Although of all fuels appear to be less sensitive to T, the magnitude of the is clearly different for different fuels. Higher absolute values of for CO and H2 infer more heat being released and less chemical energy being converted into electrical power. On the other hand, the slightly positive for the C electrochemical oxidation reaction can be used an indicative of coking for a specific hydrocarbon fuel under different SOFC operating conditions.
Figure 2.6 Temperature dependence of of SOFC operated on different fuels.
Note that Eq. (2.23) also provides a way to obtain ΔHr via the following equation provided that the E = E(T) relationship is known
For high-temperature electrolyzers, the T-dependence of Etn under isobaric condition can be derived from Eq. (2.24) in the following form:
Since ΔHr > 0 under electrolytic mode is a weak function of T. Thus, Etn does not change with T significantly. This is why low-T and high-T SOECs often use the same Etn in the literature.
2.4 Pressure Dependence of En and Etn
The effect of system pressure (P) on En can be predicted by differentiation of Eq. (2.4) with respect to pressure P under a constant T[3]:
Rearrangement of Eq. (2.26) yields
This relationship implies that En of a cell increases with P if the global reaction is volume decreasing. An immediate example is a global reaction H2 + 0.5O2 = H2O occurring in SOFC operated on H2 fuel.
Example 2.10
Derive the pressure enhancement factor of En in H2–O2 SOFC.
Solution
In 2.10, assuming ideal gas behavior for all gases involved, the partial molar volume of component A (e.g. A = H2, H2O, and O2) in the mixture is given by
where XA = mA/∑ mi; mA and PA are the moles and partial pressure of component A, respectively. The overall change of molar volume of the 2.10 is, therefore, given by:
Substitution of Eq. (2.29) into Eq. (2.27) and performing integration of from atmospheric Po = 1 atm to an elevated pressure Pt leads to the increase ΔEn in En
Clearly, higher Pt favor a higher En. The complete equation of En as a function of T, concentration, and P for 2.10 can then be written from Eqs. (2.11, 2.30) and 2.30 for H2 electro-oxidation in SOFCs as
For CO fuel, the pressure enhancement term of E is the same as Eq. (2.30), and therefore the complete equation of E with T, , and Pt as variables is given by:
For direct electrochemical oxidations of C and CH4 fuels, the system pressure Pt should have no effect on the E as the volumes of the reactants and products involved in the reactions remain unchanged. However, if the primary fuels are reformed into simple fuels such as H2 and CO, the Pt effect on the simple fuels remains. Therefore, Expressions 2.31 and 2.32 are the two most general equations for calculating En of fuels under all circumstances.
2.5 Thermal and Chemical Expansion Coefficients
It is obvious that both thermal and chemical
