An Introduction to Electrochemical Engineering 1527501949, 9781527501942

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An Introduction to Electrochemical Engineering

An Introduction to Electrochemical Engineering By

Carlos M. Marschoff and Pablo D. Giunta

An Introduction to Electrochemical Engineering By Carlos M. Marschoff and Pablo D. Giunta This book first published 2023 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright © 2023 by Editorial Universitaria de Buenos Aires Translation published with permission from Editorial Universitaria de Buenos Aires All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN (10): 1-5275-0194-9 ISBN (13): 978-1-5275-0194-2

TABLE OF CONTENTS

Foreword .................................................................................................... x Part 1: Fundamentals Chapter One ................................................................................................ 2 Introduction Chapter Two ............................................................................................... 8 Electrolyte Resistance 2.1. Introduction 2.2. Electrical conductivity of solutions. Basic concepts. 2.3. Equivalent conductance 2.4. Ionic mobility and transport numbers 2.5. Equivalent conductance dependence on ionic concentration 2.6. Anomalous conductance in aqueous media. Conductance in nonaqueous media and molten salts Chapter Three ........................................................................................... 22 Thermodynamics of Electrode Reactions 3.1. Gibbs’ fundamental equation 3.2. Thermodynamic potentials 3,3, Electrochemical equilibrium conditions 3.4. Galvanic cells 3.5. Electrode potentials 3.6. Galvanic cells in non-equilibrium condition. The case of water decomposition reaction Chapter Four ............................................................................................. 43 The Electrode – Solution Interphase 4.1. Introduction 4.2. The electrical double layer 4.3. Reference electrodes 4.4. Electrode potential control and polarization curves

vi

Table of Contents

Chapter Five ............................................................................................. 53 Electrochemical Kinetics. I. Charge Transfer 5.1. The Butler – Volmer equation 5.2. Tafel equation and linear polarization 5.3. Exchange current 5.4. Consecutive reactions. Reaction mechanisms 5.5. Parallel reactions at electrodes Chapter Six ............................................................................................... 70 Electrochemical kinetics. II. Mass transfer 6.1. Mass transfer overpotential 6.2. The purely diffusional case 6.3. Migration effects 6.4. Convective effects 6.4.1. The semi – infinite plane electrode 6.4.2. The rotating disk electrode 6.5. The mass transfer coefficient 6.6. Current – potential curves Chapter Seven........................................................................................... 96 The Perfectly Stirred Electrochemical Tank Reactor 7.1. Analysis of the perfectly stirred electrochemical tank reactor (PSETR) 7.2. Cascades of PSETR 7.3. The PSETR operating in batch conditions 7.4. The semi – batch PSETR 7.5. PSETR with electrolyte recycling 7.6. PSETR with simultaneous chemical reaction 7.7. Parallel reactions on a PSETR Chapter Eight .......................................................................................... 109 The Plug – Flow Electrochemical Reactor 8.1. Mass balance in a plug – flow electrochemical reactor (PFER) 8.2. The case of a PFER with parallel plate electrodes 8.3. PFER with electrolyte recycling 8.4. PFER with gas evolution at the counter-electrode 8.5. Pressure drops in a PFER

An Introduction to Electrochemical Engineering

vii

Chapter Nine........................................................................................... 120 Thermal Effects in Electrochemical Reactors 9.1. Thermal balance in electrochemical reactors 9.2. The PSETR case 9.3. The PFER case Chapter Ten ............................................................................................ 129 Current and Electric Potential Distribution 10.1. Introduction 10.2. Theoretical foundations 10.3. Primary distribution 10.4. Secondary distribution 10.5. Tertiary distribution Second Part: Areas of Practical Interest Chapter Eleven ....................................................................................... 150 Corrosion 11.1. Importance and impact of corrosion 11.2. Corrosion thermodynamics 11.2.1. General equations 11.2.2. Pourbaix diagrams 11.3. Corrosion kinetics 11.4. Oxides and passivation 11.5. Corrosion types 11.5.1. Uniform corrosion 11.5.2. Localised corrosion 11.5.2.1. Differential aeration corrosion 11.5.2.2. Galvanic corrosion 11.5.2.3. Dealloying 11.5.2.4. Intergranular corrosion 11.5.2.5. Pitting corrosion 11.5.2.6. Corrosion under mechanical requirements 11.6. Protection methods Chapter Twelve ...................................................................................... 206 Industrial Production 12.1. Introduction 12.2. The chlor – alkali process 12.3. Hydrogen production through water electrolysis 12.3.1. Alkaline electrolysis

viii

Table of Contents

12.3.2. Solid polymer electrolysis 12.3.3. Solid electrolyte reactors 12.4. Copper production 12.5. Electrolysis in molten-salt electrolytes. The case of aluminium Chapter Thirteen ..................................................................................... 246 Energy Production 13.1. Introduction 13.2. General considerations on galvanic cells and batteries 13.3. Critical parameters of batteries 13.3.1. Electric potential difference 13.3.2. Battery capacity 13.3.3. Energy characteristics 13.3.4. Shelf life 13.4. Primary batteries 13.4.1. Primary batteries with zinc anode 13.4.2. Primary batteries wit lithium anode 13.5. Secondary batteries 13.5.1. General considerations 13.5.2. Lead – acid batteries 13.5.3. Secondary lithium batteries 13.6. Fuel cells 13.6.1. Alkaline cells 13.6.2. Acid cells 13.6.3. Polymer membrane cells 13.6.4. Molten-carbonate cells 13.6.5. Solid oxide cells 13.7. Energy characteristics Chapter Fourteen .................................................................................... 306 Electroplating 14.1. Introduction 14.2. Electrochemical production of metallic coatings 14.3. Additives 14.4. Electroless production of metallic coatings 14.5. Case analysis of the electrochemical production of copper coatings 14.6. Case analysis of the electroless production of nickel coatings 14.7. Metal coating in the electronics industry

An Introduction to Electrochemical Engineering

ix

Chapter Fifteen ....................................................................................... 324 Electrochemical Machining 15.1. Introduction 15.2. General features of the electrochemical machining process 15.3. The anode reaction 15.4. The cathode reaction 15.5. Techniques for precision enhancement 15.6. Defining the product shape and designing the tool 15.7. Applications 15.7.1. Drilling 15.7.2. Deburring 15.7.3. Production of complicated pieces Appendix 1 ............................................................................................. 344 A.5.1. General Analysis of Electrode Reaction Mechanisms A.5.2. Electrical Modelling of Electrode Reactions Appendix 2 ............................................................................................. 349 A2.1.- Electrical representation of the electrode A2.2.- Electrical representation of an electrochemical cell A.2.3.- Electrical description of an electrode with parallel reactions Problems ................................................................................................. 353

FOREWORD

Since the early days of the 20th century, when chlor-alkali industry and metal electrowinning were, with lead-acid and Leclanché batteries, almost the only economically significant electrochemical processes, a dramatic change has taken place. In fact, the impact and span of electrochemistry in modern industry has evolved since then at an astonishing pace. New inorganic and organic products have been synthesized, surface treatments by electrochemical techniques are a keystone in the production of electronic devices, the mechanisms of corrosion have been understood and consequently the development of techniques for its prevention and control are now available, the introduction of new techniques for producing mechanical pieces that were not amenable to traditional methods and new techniques for energy production by electrochemical methods are some of the present day industrial activities under the scope of electrochemistry. This impressive development has posed unexpected challenges for young engineers that, when entering their first job, feel that they have landed on terra incognita, where the introduction of an electric field in the realm of chemical reactions shakes their foundations. University curricula on industrial chemistry and chemical engineering have traditionally considered that electrochemical engineering is just limited to the analysis of reactors similar to those studied for chemical processes, on which reaction kinetics is governed by the Tafel equation. This is clearly an oversimplification whose consequences, that stem from the fact that undergraduate physical – chemistry courses do not afford knowledge on the fundamentals of electrode processes, are apparent when terra incognita opens its unknown grounds to the newcomer. After more than 25 years teaching electrochemical engineering at the undergraduate and graduate level in the Chemical Engineering Department of the Buenos Aires University, and following many discussions lead with R&D and plant managers of different companies on the expected profile of young engineers, we decided that it was worthwhile to write an introductory

An Introduction to Electrochemical Engineering

xi

book on Electrochemical Engineering that, in the first place, offers an adequate physical – chemistry background on electrochemistry and, on this basis, considers five broad areas of professional activity for engineers. We hope that the present book, whose content has evolved from the quite traditional view our course had in the late 90’s to its present form, will help to improve undergraduate curricula in chemical engineering and, also, be a valuable asset for graduate courses in other engineering branches.

PART 1 FUNDAMENTALS

CHAPTER ONE INTRODUCTION

In many cases processes, called reactions, occur in which a well-defined number of substances, generally named reagents, interact transferring atoms between molecules, thus producing a set of new substances which are called products. If the transferred atom shows no change in its oxidation state the process is called a chemical reaction; on the other hand, if the atom transfer includes a change in its oxidation state, because electrons are also exchanged, the process is called an electrochemical reaction. Electrochemical reactions can be classified in two groups according to the way in which electrons are transferred. Thus, the electrochemical reaction is called homogeneous when electron transfer occurs between two species that are dissolved in the same phase. Many reactions between ions in water correspond to this group such as: Fe+3 + Cu+ =

Fe+2 + Cu+2

(1.1)

Co+3 + Cr+2

Co+2 + Cr+3

(1.2)

=

Fe(CN)63- + MnO42- =

Fe(CN)64- + MnO4-

(1.3)

These reactions might take place in several steps but in all cases, as shown in Figure 1.1, electron transfer is produced when the involved species collide with enough energy to overcome the activation barrier.

Introduction

3

Figure 1.1: Schematic representation of the reduction of an Fe+3 ion and the oxidation of Cu+ ion.

An electrochemical reaction is heterogeneous when electron transfer does not occur by direct contact of the reactants but takes place, separately, at two independent interphases in which an electronic conductor, the electrode, is in contact with an ionic conductor, the electrolyte, and an electric potential difference exists between them. When the reaction proceeds spontaneously such arrangement is called an electrochemical cell and the electric potential difference is generated by the reaction. When the process is not spontaneous, an electric potential difference must be applied to the electrodes in order that the reaction can occur, in which case the arrangement is called an electrochemical reactor. Two reactions that occur heterogeneously are: ‫ݑܥ‬ଶା ൅ ܼ݊ ՞ ‫ ݑܥ‬൅ ܼ݊ଶା

(1.4)

2‫ܪ‬ଶ ܱ ՞ 2‫ܪ‬ଶ ൅ ܱଶ

(1.5)

Figures 1.2a and 1.2b schematically show the way in which electron transfer between reagents takes place in both cases.

4

Chapter One

Figure 1.2.a: Spontaneous reduction of Cu+2 to Cu and oxidation of Zn to Zn+2 in an electrochemical cell.

Figure 1.2.b: Water decomposition into hydrogen and oxygen in an electrochemical reactor.

Introduction

5

It is to be pointed out, hence, that heterogeneous electrochemical reactions imply that reduction and oxidation processes occur separately on the electrodes. The electrode on which the reduction reaction takes place is the cathode and the electrode on which the oxidation reaction occurs is the anode. Taking into consideration the electrochemical reaction (1.4) the processes occurring at each electrode are: Cathode

‫ݑܥ‬ଶା ൅ 2݁ ି ՜ ‫ݑܥ‬ሺ‫ݏ‬ሻ

(1.4a)

Anode:

Zn (s) ĺ Zn2 + + 2 e-

(1.4b)

while for reaction (1.5) the electrode processes are Cathode:

2 H2O + 2 e- ĺ H2 + 2 OH-

(1.5a)

Anode:

2 OH- ĺ H2O + ½ O2 + 2 e-

(1.5b)

Considering now the general electrode reaction: A + ȣ*e- ĺ Aȣ*-

(1.6)

Faraday´s law indicates that the number of electrodes that flow through the circuit per unit time is proportional to the reaction rate and, hence, we have: ௗ௡ಲ ௗ௧

ൌ

ூ జ‫࣠ כ‬

(1.7)

where ݊஺ is the number of moles of A, I is the electric current, ȣ* is the number of electrons exchanged and ࣠ is Faraday´s constant (࣠ ൌ 96487 Coulomb). If the electric current I is to be maintained, an electric potential difference ǻȌ between both electrodes must be established and the product |I.ǻȌ| indicates the electric power that must be exchanged. When the reaction occurs spontaneously this electric power is obtained from the reaction energy itself while, if the reaction is not spontaneous, electric power must be provided by an external source. In the first case, the transformation of reactants into products results in the production of electrical energy: Zn + Cu2+ = Zn2+ + Cu + I. ǻȌ

(1.8)

6

Chapter One

In the second case, electrical energy must be applied in order to obtain the products: 2 H2O + I. ǻȌ = 2 H2 + O2

(1.9)

Heterogeneous electrochemical reactions are of practical interest since it is possible, by controlling the electric current, a direct management of reaction rate. Hence, a central goal for the electrochemical engineer will be to optimise the electric current value: in the case of industrial production of substances the aim is to minimise energy consumption per kilogram of product whereas, in the case of electricity generation by means of electrochemical reactions, the goal is to maximise the amount of energy obtained per kilogram of reactant. The energy balance of the process is, thus, the core problem to be solved by the engineer and, on a first analysis, the following contributions can be identified for a process occurring at a given current value: x The amount of energy that, according to the reaction spontaneity, will be produced or required, which depends on the thermodynamic electric potential difference, ǻȌtherm x The amount of energy required to overcome the activation barriers involved in electrode reaction kinetics, which depends on the “activation overpotential”, ǻȌact. x The amount of energy required to keep reactants flowing to, and products flowing from, the electrodes, which depends on the “mass transfer overpotential” ǻȌmt. x Finally, the amount of energy employed in overcoming the electrical resistance of the medium that separates the electrodes, which depends on the “resistance overpotential”, ǻȌǷ, Consequently, the total potential difference in an electrochemical cell can be described by the equation: ǻȌ = ǻȌtherm - ǻȌa - ǻȌmt - ǻȌȍ

(1.10)

where the value of ǻȌtherm might be positive or negative. If ǻȌ is positive, the cell will be able provide energy while, if ǻȌ is negative, an external energy source will be needed in order that reaction occurs.

Introduction

7

The purpose of this book is to provide the reader with the basic knowledge required to understand the main factors that impact on the different contributions shown in Eqn. (1.10) and apply it to some cases of practical interest.

CHAPTER TWO THE ELECTROLYTE RESISTANCE

2.1. Introduction In order to sustain an electric current through a piece of matter an electric potential difference must be applied. The relationship between electric current and potential is given by Ohm’s law: ȟȲஐ ൌ ‫ܫ‬. ܴ

(2.1)

where R is the electrical resistance and ȟȲஐ is the electric potential difference required to maintain the current flow I. When electrochemical reactions are employed to produce substances or to obtain energy at industrial scale, large currents are needed and, consequently, a critical goal is to reduce the resistance of the electrolyte in the cell as much as possible. In order to give an idea of the importance of this fact, Table 2.1 shows some typical current values for several industrial processes. Process Aluminium production Copper refining Chlorine production Hydrogen production

Production (kg/h) 32 23 185 0.34

Current (A) 100,000 20,000 150,000 10,000

Table 2.1: Typical current values in industrial cells for some important industrial processes. With these current values, applying equation 2.1 and assuming that electric energy is provided at a cost of 0.02 US$/kWh, Table 2.2 shows the cost per kilogram of product, for three electrolyte resistance values.

The Electrolyte Resistance

Process Aluminium production Copper refining Chlorine production Hydrogen production

R=1ȍ 6,250 349 2,432 5,882

9

R = 0,1 ȍ R = 0,01 ȍ 625 62.5 34.9 3.49 243.2 24.32 588.2 58.82

Table 2.2: Energy cost in US$ per kilogram due to electrolyte resistance. As it is clearly seen, minimizing electrolyte resistance is of utmost economic importance and, therefore, it is necessary to understand its causes.

2.2. Electrical conductivity of solutions: Basic concepts The electrical resistance of a homogeneous piece of a given material, with length ࣦ and cross-sectional area ࣛ, is given by: ܴൌߩ

ࣦ

(2.2)

ࣛ

where ȡ is the specific resistance, or resistivity, which is usually expressed in ȍ.cm. The specific conductance, or conductivity, of a material is defined as: ߢ ൌ ߩିଵ ൌ

ࣦ

(2.3)

ࣛୖ

with units ࣭.cm-1 where ࣭ = Siemens = ȍ-1. Table 2.3 shows resistivity and conductivity values for several electronic and ionic conductors. Material Silver Copper Gold Aluminium Tungsten Zinc Steel Lead Titanium Mercury Graphite

ȡ (ȍ.cm) 1.6 . 10-6 1.7 . 10-6 2.2 . 10-6 2.6 . 10-6 5.1 . 10-6 5.6 . 10-6 11.9 . 10-6 21.9 . 10-6 46.2 . 10-6 95.8 . 10-6 700 . 10-6

ț (࣭.cm-1) 6.3 . 105 5.9 . 105 4.5 . 105 3.8 . 105 1.9 . 105 1.8 . 105 8.4 . 104 4.6 . 104 5.9 . 104 2.1 . 104 1.4 . 103

Chapter Two

10

NaCl (molten at 1000 °C) H2SO4 3.5 M (18 °C) NaCl (saturated at 18 °C) KOH 1 M (18 °C) NaCl 1 M (18 °C) LiCl 1 M (18 °C) Zirconia (85% ZrO2 15% Y2O3) a 1000 °C Water (25 °C) Methanol (25 °C) Benzene (25 °C)

2.43 . 10-3 1.35 . 10-2 4.67 . 10-2 5.43 . 10-2 0.134 0.158 0.20 1.82 . 105 1.43 . 106 2 . 1011

4.2 . 102 73.9 21.4 18.4 7.44 6.34 5.0 5.5 . 10-6 7 . 10-7 5 . 10-12

Table 2.3: Resistivity and conductivity values for several materials. In metals electric current flows because of the movement of electrons in the conduction band, but in an electrolyte solution electrical conductivity is due to the movement of dissolved ions and any attempt to understand the mechanisms which lead to the observed resistance values must be based on this fact. A first intuitive analysis suggests that the conductivity of an electrolyte will depend on the concentration of ionic species: a larger number of ions able to contribute to electricity transport should yield a larger conductivity value. A second intuitive assumption is that, since the kinetic energy of dissolved ions increases with temperature, the electrolyte conductivity should be enhanced at higher temperature values. In Table 2.4 results obtained in highly precise measurements on KCl solutions are shown which confirm both points. Molality g KCl/1000 g solution 71.1352 7.41913 0.745263

ț273.15K झ.cm-1

ț291.15K झ.cm-1

ț298.15K झ.cm-1

0.065176 0.007137 0.0007736

0.097838 0.011167 0.0012205

0.111342 0.012857 0.0014087

Table 2.4: Conductivity of KCl solutions at three temperature values. A careful look at data of Table 2.4 indicates that conductivity increases with concentration and therefore one might assume that in a first, coarse, approximation the conductivity of a mixture of ionic compounds should be the sum of the conductivity of each component of the mixture. Thus, if we

The Electrolyte Resistance

11

have measured the conductivity of several solutions, say CuCl2 at concentration c1; NaNO3 at concentration c2 and K2SO4 at concentration c3 whose values are, respectively, ț1, ț2 and ț3 the conductivity of a mixture of the three compounds at these concentrations could be assumed to be given by: ߢ ൎ ߢଵ ൅ ߢଶ ൅ ߢଷ

(2.4)

This approach, however, is only valid at highly dilute solutions and, if applied to solutions with component concentrations of practical interest, significant errors will be committed. Such errors can be particularly serious, as shown in Figure 2.1, where it is seen that when ionic concentrations increase the slope of the conductivity vs concentration curves diminishes and, eventually, reaches a maximum.

Figure 2.1 Conductivity vs concentration curves for several ionic compounds.

2.3. Equivalent conductance If the resistance of a given solution is to be directly obtained from experimental conductivity values it will be necessary to have specific data for each compound and, moreover, for each mixture over a given concentration range. Thus, a large number of costly and time-consuming measurements should be performed. However, and as suggested by the additivity rule of conductivity at highly dilute solutions, we might conclude that each ionic species makes a well defined contribution to the conductivity

Chapter Two

12

of a mixture. Hence, the idea is to look for a method that allow to express conductivity of a solution in such a way that its dependence with concentration be described by means of general equations in which the individual contribution of each ion can be identified. With this goal in mind the molar conductance, Ȧ௠ , of a solution of an electrolyte with concentration c is defined as: Ȧ௠ ൌ

఑ ௖

(2.5)

Clearly, molar conductance units are ࣭.cm2mol-1. When studying a mixture containing ions of different electrical charge, the direct use of molar conductance has some drawbacks and, for this reason, the use of the concept of equivalent conductance, Ȧ, defined as: Ȧൌ

఑ ௖.௡೐೜

(2.6)

has been customary where, for a solution of the ionic compound ‫ܣ‬௡ಲ ‫ܤ‬௡ಳ , ݊௘௤ ൌ ݊஺ ‫ݖ‬஺ ൌ െ݊஻ ‫ݖ‬஻ , is the number of equivalents in a mole of electrolyte, ݊஺ and ݊஻ indicate the stoichiometric number of cations and anions in the electrolyte molecule, and ‫ݖ‬஺ and ‫ݖ‬஻ their charge. Ȧ units are, then, ࣭.cm2.eq1 . By the end of the 19th century Kohlrausch (Kohlrausch, 1874) performed a systematic experimental study on solution conductivity variation with concentration for a large number of 1 – 1 electrolytes and found that, if concentrations were kept below 0.001 M, in all cases it was verified that: ߉ ൌ ߉° െ ‫ܣ‬. ܿ ଵ/ଶ

(2.7)

which is known as Kohlrausch’s equation, where ߉° is the equivalent conductance at infinite dilution of the electrolyte and A is a positive parameter. Considering (2.6) it comes that: ߢ ൌ ߉°. ܿ െ ‫ܣ‬. ܿ ଷ/ଶ

(2.8)

From the obtained experimental data Kohlrausch showed that, for a binary ionic compound, the electrolyte equivalent conductance at infinite dilution could be described as the sum of two independent contributions, one from the cation and other from the anion:

The Electrolyte Resistance

߉° ൌ ߣ௢ା ൅ ߣ௢ି

13

(2.9)

and thus, recording the ߣ௢ values for the different ions, it is possible to obtain the equivalent conductance at infinite dilution of any ionic compound. Table 2.5 shows the equivalent conductance at infinite dilution for some of the most usual ions in aqueous solution. Ion H+ OHLi+ Na+ K+ Rb+ Cs+ Ag+ NH4+ FClBrINO3ClO4Mg2+ Ca2+ Sr2+ Ba2+ Cu2+ Zn2+ Pb2+ Co2+ Al3+ La3+ SO42CO32-

Ȝ°/( झ.cm2.eq-1) 349.8 199.1 38.6 50.1 73.5 77.8 77.2 61.9 73.5 55.4 76.4 78.1 76.8 71.5 67.3 53.0 59.5 59.4 63.6 53.6 52.8 69.5 55.0 69.8 69.7 80.0 69.3

Table 2.5.: Equivalent conductance at infinite dilution of several ions at 298.15 K.

Chapter Two

14

2.4. Ionic mobility and transport numbers In the absence of an electric field ions move randomly in an electrolytic solution and, if concentration gradients are nil, the net ionic flow at any point is zero. However, if an electric field is applied, a force will operate on each ion whose value depends on the electric field intensity and the ion electric charge, ‫ݖ‬௝ . If the electric field is not extraordinarily high, this force will be almost instantaneously balanced by the solution viscosity and ions will attain a limiting velocity v ሬԦ୨ that, according to Stoke’s law, is: ሬvԦ୨ ൌ

ሬԦ ࡲ

(2.10)

଺గ௥ೕ ఔ

ሬԦ is the applied force, ‫ݎ‬௝ the effective ionic radius and ߥ the solution where ࡲ viscosity. Defining ionic mobility, ‫ݑ‬௝ , as the velocity attained by the ion under unit force: ‫ݑ‬௝ =

ሬሬሬሬԦห ห௩ ണ ሬԦห หࡲ

(2.11)

and since the applied force on an ion is: ሬሬሬԦ ൌ ‫ݖ‬௝ . ࡱ ሬሬሬԦ ൌ ‫ݖ‬௝ ൫െࢺࢸ ሬሬሬሬሬሬԦ൯ ࡲ

(2.12)

ሬሬሬԦ is the electric field, the electrical mobility of an ion, ‫ݑ‬௝ᇱ , is defined where ࡱ as the velocity attained by the ion under a unit electric field: ‫ݑ‬௝ᇱ ൌ

ሬሬሬሬԦห ห௩ ണ ሬሬԦ ห หࡱ

ൌ

ሬሬሬሬԦห ห௩ ണ ሬሬሬሬሬሬԦ ห ห ࢺࢸ

(2.13)

Ionic electrical mobility clearly has a direct link with ionic equivalent conductance as can be seen if a volume element with unit length and crosssectional area, containing a solution of the ionic compound ‫ܣ‬௔ ‫ܤ‬௕ with concentration c is considered. In this volume the ionic total charge for each species is: |࣫஺ | ൌ |࣫஻ | ൌ ࣠. |‫ݖ‬஺ ܽ|. ܿ ൌ ࣠. |‫ݖ‬஻ ܾ|. ܿ

(2.14)

Applying a unit potential difference (|߂ߖ| ൌ 1ܸ) on this volume the electrical current I is:

The Electrolyte Resistance

‫ ܫ‬ൌ ߢ. ߂ߖ ൌ ߢ. 1ܸ

15

(2.15)

Now, since conductivity can be expressed in terms of ionic electrical mobility by: ߢ ൌ ࣠ሺ‫ݖ‬஺ . ‫ݑ‬஺ᇱ . ܽ. ܿ ൅ ‫ݖ‬஻ . ‫ݑ‬஻ᇱ . ܾ. ܿሻ

(2.16)

and since |‫ݖ‬஺ ܽ| ൌ |‫ݖ‬஻ ܾ| ൌ ݊௘௤ , from (2.6): ߢ ൌ Ȧ. ݊௘௤ . ܿ ൌ ሺߣ஺ ൅ ߣ஻ ሻ. ݊௘௤ . ܿ

(2.17)

and it comes that: ࣠.‫ݑ‬஺ᇱ = ߣ஺

(2.18)

࣠.‫ݑ‬஻ᇱ = ߣ஻

(2.19)

and

Under no concentration gradients, the current fraction carried by each ion is: ߬஺ =

, ௨ಲ , , ௨ಲ ା௨ಳ

(2.20)

for species A and ߬஻ =

, ௨ಳ , , ௨ಲ ା௨ಳ

(2.21)

for species B. ߬஺ and ߬஻ are the transport numbers of A and B. In the general case it can be shown that in a solution with several ionic components with concentrations and charges ci and zi, the transport number for each species is: ,

߬௝ = σ

ห௭ೕ ห.௖ೕ .௨ೕ , ೔ሺ|௭೔ |.௖೔ .௨೔ ሻ

where the sum is performed over all the ionic species.

(2.22)

16

Chapter Two

2.5. Equivalent conductance dependence on ionic concentration When the equivalent conductance of an electrolyte solution is measured at very low concentrations, there is a linear dependence with c1/2, as shown in Eqn. (2.7). However, as concentration increases the ߉ vs c1/2 curve deviates from linearity and this deviation is larger when ionic charges are higher and, also, if the electrolyte is only partially dissociated. Figure 2.2 shows experimental results for some specific cases.

Figure 2.2: Equivalent conductance dependence on concentration for several electrolytes at concentrations below 1 M.

In order to understand this behaviour, it is necessary to consider the fact that ions in solution are not free but associated with solvent molecules through charge – dipole and chemical interactions. In the case of aqueous solutions, hydrated ions are the entities that move under an electric field and, consequently, their mobility is related to the hydration sphere radius. In turn, hydration sphere radii depend on the hydration number, i.e., the number of water molecules attached to the particular ion, which is a function of both charge and size of the ion: for two ions with the same charge value the polarizing effect on water dipoles will be more intense for species with lower ionic radius. Thus, hydration number of alkali ions increases as K+ < Na+ < Li+ and, correspondingly, equivalent ionic conductance at infinite dilution shows the reverse behaviour ߣ௢௄శ ൐ ߣ௢ே௔శ ൐ ߣ௢௅௜ శ .

The Electrolyte Resistance

17

On a first approximation an electrolytic solution might be described as a dielectric continuum in which charged spheres, whose radii depend on the hydration number, interact through electrostatic forces. Hence, if an observer is installed on a hydrated positive ion, the charge distribution around it will not be homogeneous, because the ion density of negative hydrated ions will exceed that of positive ions, and a negative ionic atmosphere is built. From this description, the behaviour of the equivalent conductance vs concentration curve is explained on the basis of three contributions: x The electrophoretic effect x The relaxation effect x Specific interactions between ions The electrophoretic effect is the consequence of the fact that when a hydrated ion moves in a viscous medium it tends to drag other ions of the solution in its vicinity. Conversely, dragged ions will affect the original ion, restraining its advance, an effect whose impact will increase with ionic concentration. In understanding the relaxation effect, it can be mentioned that the function that describes the distribution of charges in the neighbourhood of a given ion, which was calculated by Debye and Hückel (Debye and Hückel, 1923) for dilute solutions, is, in the absence of an electric field, spherically symmetric. Thus, when an electric field is applied, ions with opposite charge are attracted in opposite directions, and the consequence of this fact is that the ionic atmosphere loses its symmetry and exerts a force, against that induced by the field, which will depend on concentration. Finally, as the concentration of an electrolyte increases, the probability that an anion and a cation get near enough to generate an electrostatic attractive force that overcomes thermal agitation is enhanced and “ionic pairs” with net zero charge can be formed, which will not contribute to conductivity. Furthermore, specific chemical interactions might exist, forming molecules with net zero charge. Weak electrolytes are a typical case of chemical interactions that affect conductivity. Figure 2.2 illustrates how equivalent conductance of acetic acid solutions briskly falls from the infinite dilution value because of the strong chemical interaction between the acetate anion and the H+ cation that impedes the complete dissociation of the acid.

Chapter Two

18

For dilute electrolytic solutions, in which ionic pair formation can be dismissed, the equations describing the electrophoretic and the relaxation effects were obtained by Onsager (Onsager, 1927) employing the distribution function of Debye and Hückel and the equivalent conductance is expressed as: Ȧ ൌ Ȧ° െ ሾ‫ܣ‬. Ȧ° ൅ ‫ܤ‬ሿ. ࣣଵ/ଶ

(2.23)

where A and B are constants, and ࣣ is the solution ionic force defined by: ଵ

ࣣ ൌ σ௜ ܿ௜ ‫ݖ‬௜ଶ ଶ

(2.24)

where ܿ௜ is the molar concentration of species i, ‫ݖ‬௜ its charge, and the sum is carried over all ionic components. Obviously, Kohlrausch´s equation is a particular case of Eqn. (2.24) for 1 – 1 electrolytes. As said above, Onsager´s equation is valid only at highly dilute solutions which are not typical of those found in electrochemical processes of practical interest. Therefore, several modifications have been proposed to be used at higher concentrations which can be considered approximate forms of the polynomial: Ȧ ൌ Ȧ° െ ሾ‫ܤ‬ଵ . Ȧ° ൅ ‫ܤ‬ଶ ሿ. ࣣଵ/ଶ ൅ ‫ܤ‬ଷ ࣣ െ ‫ܤ‬ସ . ࣣଷ/ଶ ൅ ‫ڮ‬

(2.25)

In concentrated mixtures of ionic compounds, it is usual to work employing ad hoc correlations developed on the basis of (2.25).

2.6. Anomalous conductance in aqueous media. Conductance in non-aqueous media and molten salts If the values of equivalent ionic conductance at infinite dilution in water, shown in Table 2.5, are examined, it is clear that those corresponding to H+ and OH- are considerably larger than what can be expected from the arguments given in the previous section. This “anomalous” behaviour can be understood if the particular structure of liquid water is considered. In fact, and as demonstrated by spectroscopical measurements and computational models, water molecules interact through hydrogen bonds as shown in Figure 2.3

The Electrolyte Resistance

19

Figure 2.3: Hydrogen bond between two water molecules.

Thus, when an acid molecule dissociates in water, the H+ ion forms the species H3O+ that, by the conductive mechanism originally suggested by Grotthuss (Grotthuss, 1806), schematically shown in Figure 2.4, explains the observed value for the ionic conductance of the H+ ion.

Figure 2.4: Grotthuss mechanism for the H+ ion.

Analogously, OH- anomalous mobility is also explained in terms of the Grotthuss mechanism as shown in Figure 2.5

Figure 2.5: Grotthuss mechanism for the OH- ion.

Chapter Two

20

In modern chemical industry there is an ever-growing number of electrochemical reactions that are performed employing organic solvents or molten salts. It is then pertinent to briefly consider ionic conduction in these media and its difference with the case of aqueous solutions. In the case of organic solvents, it is convenient to consider them as classified in two groups: protic solvents, which are those whose molecules have a relatively labile hydrogen atom, such as primary alcohols or carboxylic acids, and aprotic solvents, like acetonitrile or dimethylsulphoxide. In protic solvents the Grotthuss mechanism might have some importance and one should expect that the H+ ion have also a particular behaviour respect to the case of aprotic solvents. In Table 2.6 experimental data for ionic conductance at infinite dilution of three ions: H+, K+ y Cl- are shown which confirm this argument. Solvent Water Methanol Ethanol Ethylene glycol Formic acid Formamide Dimethylformamide Acetonitrile

Ȝ°H+ ࣭.cm2.eq-1 349.8 146.2 62.8 27.7 79.6 10.8 34.7 99.0

Ȝ°K+ ࣭.cm2.eq-1 73.5 52.4 23.6 4.6 50.0 12.8 30.8 83.6

Ȝ°Cl࣭.cm2.eq-1 76.4 52.3 21.8 5.1 26.5 17.1 55.1 98.7

Table 2.6: Equivalent conductance at infinite dilution of H+, K+ and Clin several solvents. In the case of molten salts, the solvent is formed by ions and, so, it is convenient to compare specific conductivities. Table 2.7 shows some experimental values obtained for different molten salts compared to a KCl 4 M solution. Solution KCl 4 M LiF (liq) NaF (liq) KF (liq) LiCl (liq) NaCl (liq)

T/K 298 1,200 1,300 1,300 1,000 1,200

ț/( ࣭.cm-1) 0.374 8.923 5.019 4.077 6.308 3.954

The Electrolyte Resistance

KCl (liq) LiBr (liq) NaBr (liq) KBr (liq)

1,200 1,000 1,100 1,100

21

2.517 5.667 3.141 1.844

Table 2.7: Conductivity of several molten salts and of a KCl 4M solution.

References Debye, Peter and Hückel, Erich. 1923. "Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen” Physikalische Zeitschrif 185-203 Fuoss, Raymond and Onsager, Lars. 1957. “Conductance of Unassociated Electrolytes” J. Phys. Chem. 668-682 Fuoss, Raymond and Onsager, Lars. 1958. “The Kinetic Term in Electrolytic Conductance” J. Phys. Chem. 1339-1340 Grotthuss, Christian. 1806. “Sur la decomposition de l´eau et des corps qu´elle tient en dissolution à l´aide de l´électricité galvanique” Annales de Chimie. 54-73 Kohlrausch, Friedrich and Grotrian, Otto. 1874. “Das elektrische Leitungsvermögen der Cholr-Alkalien und alkalischen Erden, sowie der Salpetersäure in wässerigen Lösungen”. Göttingen Nachrichten. 405418 Onsager, Lars. 1927. “Zur Theorie der Elektrolyte. II” Physikalische Zeitschrift. 277-298

Bibliography MacInnes, Duncan. 1939. The Principles of Electrochemistry. New York. Reinhold Publishing Corporation. Robinson, R and Stokes, R.1970. Electrolyte Solutions. 2nd edition. London. Butterworths

CHAPTER THREE THERMODYNAMICS OF ELECTRODE REACTIONS

3.1. Gibbs´ fundamental equation For any reversible process the combination of the first and second principles of thermodynamics leads to the so called fundamental equation of thermodynamics or Gibb´s fundamental equation ܷ݀ ൌ ܶ. ݀ܵ ൅ σ௝ ‫ݓ‬௝ . ܹ݀௝ ൅ σ௜ ߤ௜ . ݀݊௜

(3.1)

Where U, T and S are, respectively, the internal energy, temperature and entropy of the system. In this equation the product ܶ. ݀ܵ is the differential of heat and the first sum describes the total amount of work performed by the system which, according to the particular case, might include mechanical work, electrical work, magnetic work, etc. By its side, the second sum describes the internal energy variation associated with the change in the number of moles for each chemical species in the system. In Gibbs´ fundamental equation every differential of work is described as the product of an intensive parameter by the differential of an extensive parameter. Thus, work associated with a differential change of the system volume is the product of pressure, P, and the differential of volume, dV; work associated with a differential change of area results as the product of surface tension, ı and ݀ࣛ; electrical work is described by the product of electric potential Ȍ and the differential of charge, dq; etc. These pairs of parameters are called conjugated parameters and in the first sum in Eqn. (3.1) are symbolised with wj for intensive and Wj for extensive parameters. In the second sum the number of moles of each component are the extensive parameters ni and their conjugated, intensive, parameters, ȝi, are the chemical potentials of the different species i. Obviously the differential of heat is also the product of the intensive parameter T with the extensive parameter, S.

Thermodynamics of Electrode Reactions

23

Eqn. (3.1) explicitly shows that all the independent variables that describe the change of internal energy of a system are extensive parameters and further analysis leads to Gibbs’ equilibrium conditions: ሺߜܷሻௌ,ௐೕ ,௡೔ ൒ 0

(3.2)

(ߜSሻ௎,ௐೕ ,௡೔ ൑ 0

(3.3)

The meaning of Eqn. (3.2) is that an adiabatic system (constant entropy) that, while maintaining its chemical composition, does not exchange matter with its surroundings and whose extensive work parameters are constant, is at equilibrium if, and only if, any modification of the intensive parameters leads to an increase of internal energy (irreversible change) or to a conservation of its value (reversible change). By the same token, Eqn. (3.3) indicates that a system that is at constant internal energy and constant extensive parameters is at equilibrium if and only if any change of the intensive parameters leads to a decrease of entropy (irreversible change) or to the conservation of its value (reversible change).

3.2. Thermodynamic potentials Gibbs’ general equation (3.1) gives the complete thermodynamic description of the reversible evolution of a system and, in principle, any change between two equilibrium conditions could be studied with it. However, the use of this equation in solving real problems is hindered by the fact that several extensive parameters cannot be easily measured, nor controlled. In fact, no instrument exists that can yield direct entropy measurements and, although the volume of a given system is easily measured, volume changes might become very difficult to quantify or control, particularly when working with condensed matter. However, the corresponding intensive parameters, T and P, are quite simply measured or controlled and, for this reason, it is convenient to employ a mathematical procedure, known as Legendre transform, through which the intensive parameter of a conjugated pair becomes an independent variable. This transformation, that is performed by adding or subtracting to internal energy the product ‫ݓ‬௜ ή ܹ௜ defines new state functions called thermodynamic potentials. Thus, if the system volume, V, is the first extensive work variable, the first thermodynamic potential, enthalpy, H, is defined as:

Chapter Three

24

‫ ܪ‬ሺܵ, ܲ, ܹ௝ஷଵ , ݊௜ ሻ ൌ ܷ ൅ ܲ ࣰ

(3.4)

Analogously, considering now the conjugated pair T, S the Legendre transform leads to Helmholtz free energy, F: ‫ ܨ‬ሺܶ, ࣰ, ܹ௝ஷଵ , ݊௜ ሻ ൌ ܷ െ ܶܵ

(3.5)

Finally, Gibbs´ free energy, G, is defined as: ‫ ܩ‬ሺܶ, ܲ, ܹ௝ஷଵ , ݊௜ ሻ ൌ ܷ ൅ ܲ ࣰ െ ܶܵሻ

(3.6)

If differentials of these new functions are calculated one obtains: ݀‫ ܪ‬ൌ ܶ. ݀ܵ ൅ ࣰ. ݀ܲ ൅ σௐೕಯభ. ܹ݀௝ஷଵ ൅ σ௜ ߤ௜ . ݀݊௜

(3.7)

݀‫ ܨ‬ൌ െ ܵ. ݀ܶ ൅ ܲ. ݀ ࣰ ൅ σௐೕಯభ. ܹ݀௝ஷଵ ൅ σ௜ ߤ௜ . ݀݊௜

(3.8)

݀‫ ܩ‬ൌ െ ܵ. ݀ܶ ൅ ࣰ. ݀ܲ ൅ σௐೕಯభ. ܹ݀௝ஷଵ ൅ σ௜ ߤ௜ . ݀݊௜

(3.9)

It must be observed that the chemical potential, which is the intensive parameter linked with the variation of mole numbers becomes: ߤ௞ ൌ ൬ ൬

డ௎ డ೙ೖ

డ௎ డ೙ೖ

൰ ௌ,ௐೕ ,௡೔ಯೖ

ൌ ൬

డு డ೙ೖ

൰ ௌ,௉,ௐೕಯభ ,௡೔ಯೖ

൰

=

൬

డி డ೙ೖ

൰

= ்,ௐೕ ,௡೔ಯೖ

(3.10) ்,௉,ௐೕಯభ ,௡೔ಯೖ

The equilibrium conditions are transformed as: ሺߜ‫ܪ‬ሻௌ,௉,ௐೕಯభ,௡೔ ൒ 0

(3.11)

ሺߜ‫ܨ‬ሻ ்,ࣰ,ௐೕಯభ,௡೔ ൒ 0

(3.12)

ሺߜ‫ܩ‬ሻ ்,௉,ௐೕಯభ,௡೔ ൒ 0

(3.13)

From these equations it comes that: a) Enthalpy defines equilibrium stability for closed, adiabatic systems at constant pressure when all extensive work parameters, except volume, are constant and no chemical reaction occurs.

Thermodynamics of Electrode Reactions

25

b) Helmholtz free energy defines equilibrium stability for closed, isothermal systems when all extensive work parameters are constant, and no chemical reaction occurs. c) Gibbs free energy defines equilibrium stability for closed, isothermal systems at constant pressure when all extensive work parameters, except volume, are constant and no chemical reaction occurs.

3.3. Electrochemical equilibrium conditions Let us now consider a closed system in which some of its components have an electrical charge zi and the only possible work contributions are electrical work or volume change. In this case, Eqn. (3.1) can be written as: ܷ݀ ൌ ܶ. ݀ܵ െ ܲ. ࣰ݀ ൅ ߖ. ݀ँ ൅ ߑߤ௜ . ݀݊௜

(3.14)

݀ँ ൌ ࣠. σ௜ ‫ݖ‬௜ ݀݊௜

(3.15)

with

Consequently, ܷ݀ ൌ ܶ. ݀ܵ െ ܲ. ࣰ݀ ൅ σ௜ሺߤ௜ ൅ ‫ݖ‬௜ ࣠ߖሻ݀݊௜

(3.16)

which allows to define the electrochemical potential of component i as: ߤ௜‫ כ‬ൌ ሺߤ௜ ൅ ‫ݖ‬௜ ࣠ߖሻ

(3.17)

Considering now a closed system with ȗ components and ȟ phases, within which all phases are open, i.e., they may exchange entropy, volume and mass,the general equilibrium conditions indicate that the intensive parameters must satisfy: ܶ ሺఈሻ ൌ ܶ ሺఉሻ ൌ ‫ ڮ‬ൌ ܶ ሺకሻ

(3.18)

ܲሺఈሻ ൌ ܲሺఉሻ ൌ ‫ ڮ‬ൌ ܲሺకሻ

(3.19)

‫כ‬ሺఉሻ

ߤ௝‫כ‬ሺఈሻ ൌ ߤ௝

‫כ‬ሺకሻ

ൌ ‫ ڮ‬ൌ ߤ௝

(3.20)

Considering now a particular interphase, say į/İ, it comes that: ‫כ‬ሺఋሻ

ߤ௜

‫כ‬ሺఌሻ

ൌ ߤ௜

(3.21)

Chapter Three

26

which, from Eqn. (3.17) implies: ሺఋሻ

ሺఌሻ

ߤ௝ െ ߤ௝ ൌ ‫ݖ‬௝ ࣠൫Ȳ ሺఌሻ െ Ȳ ሺఋሻ ൯

(3.22)

Thus, at equilibrium, the difference of chemical potential of a charged species at each interphase permeable to it is directly related to the electric potential difference between both phases. In the case that the interphase is not at equilibrium: ሺఋሻ

ሺఌሻ

ߤ௝ െ ߤ௝ ് ‫ݖ‬௝ ࣠൫Ȳ ሺఌሻ െ Ȳ ሺఋሻ ൯

(3.23)

and the system will transfer the necessary number of moles of species i from one phase to the other, until equilibrium is attained, and Eqn. (3.22) is satisfied. It is important to point that, while chemical potential values can be obtained through experimental concentration measurement of the involved species, direct measurement of the electric potential difference between phases is not possible since, as shown in Figure 3.1, new interphases are created when voltmeter probes get in touch with each phase.

Figure 3.1: Direct measurement of the potential difference at an interphase is affected by the potential difference of the interphase between probes and phases į and İ.

Thermodynamics of Electrode Reactions

27

3.4. Galvanic cells Eqns. (3.22) and (3.23) indicate that it is possible to establish controlled electric potential differences at an interphase by adjusting the chemical potential difference of charged species and, conversely, controlled modifications of charged species concentration can be produced by modifying the electric potential at each side of the interphase. Devices that can effect these controlled variations of electric potential difference and concentrations are called galvanic cells. The most general description of a galvanic cell is a system of ȟ phases that satisfies the following conditions: x x x x

All phases are electrical conductors. At least one phase is an electrolytic conductor. Phases 1 and ȟ are metallic conductors of the same composition. Phase 1 is only in contact with phase 2 and phase ȟ is only in contact with phase ȟ – 1. All other phases are in contact with two phases. x In each interphase exists, at least, a common charged particle (ions or electrons) that permeates through the interphase. x Circulation of electrical current through the system implies chemical reactions. In such a system the electric potential difference ߖ(1) - ߖ(ȟ) can be measured, since both phases are identical. This measurement must be carried out under thermodynamic equilibrium conditions, which means that electrical current should be zero. This can be accomplished by means of a potentiometric circuit as shown in Figure 3.2

Chapter Three

28

Figure 3.2: Potentiometric circuit for measuring the galvanic cell potential difference.

In this circuit a known electric potential difference V is applied to resistance R through which current I circulates. One of the terminal phases of the galvanic cell, whose potential difference is to be measured, is connected to A while the other terminal phase is connected to the resistance by means of a sliding contact which is moved until the point is reached at which no current is detected by the galvanometer, which means that the electric potential difference is directly given by IR´: ‫ܫ‬. ܴᇱ ൌ หȲ ሺଵሻ െ Ȳ ሺకሻ ห

(3.24)

which is the electromotive force (emf) of the galvanic cell. For a galvanic cell that operates adiabatically and at constant volume Eqn. (3.1) yields: ‫כ‬ሺఈሻ ሺఈሻ ܷ݀ ൌ σ௜,ఈ ߤ௜ . ݀݊௜

(3.25)

where the sum is performed over all phases and species. When a galvanic cell works adiabatically and at constant pressure Eqn. (3.7) gives:

Thermodynamics of Electrode Reactions ‫כ‬ሺఈሻ ሺఈሻ ݀‫ ܪ‬ൌ σ௜,ఈ ߤ௜ . ݀݊௜

29

(3.26)

If the galvanic cell operates at constant volume and temperature we have, from Eqn. (3.8): ‫כ‬ሺఈሻ ሺఈሻ ݀‫ ܨ‬ൌ σ௜,ఈ ߤ௜ . ݀݊௜

(3.27)

Finally, for a galvanic cell working at constant pressure and temperature Eqn. (3.9) yields: ‫כ‬ሺఈሻ ሺఈሻ ݀‫ ܩ‬ൌ σ௜,ఈ ߤ௜ . ݀݊௜

(3.28)

Hence, if the electrochemical reaction associated with the galvanic cell is: ߭ଵ ‫ ܣ‬൅ ߭ଶ ‫ ܤ‬൅ ‫ ڮ‬ൌ ‫߭ ڮ‬௡ିଵ ܳ ൅ ߭௡ ܴ

(3.29)

where the number of electrons involved in the process is ȣ* it is concluded that, for adiabatic and isochoric processes: σ௝,ఈ ߭௝ ߤ௝ሺఈሻ ൌ ȟܷ ൌ െ߭ ‫࣠ כ‬ȟȲ௧௛௘௥௠

(3.30)

while for adiabatic and isobaric processes: σ௝,ఈ ߭௝ ߤ௝ሺఈሻ ൌ ȟ‫ ܪ‬ൌ െ߭ ‫࣠ כ‬ȟȲ௧௛௘௥௠

(3.31)

and for isothermal and isochoric processes: σ௝,ఈ ߭௝ ߤ௝ሺఈሻ ൌ ȟ‫ ܨ‬ൌ െ߭ ‫࣠ כ‬ȟȲ௧௛௘௥௠

(3.32)

Finally, for isothermal and isobaric operation it comes that: σ௝,ఈ ߭௝ ߤ௝ሺఈሻ ൌ ȟ‫ ܩ‬ൌ െ߭ ‫࣠ כ‬ȟȲ௧௛௘௥௠

(3.33)

Eqn. (3.33) is usually employed since, normally, galvanic cells work at (approximately) constant pressure and temperature but, for galvanic cells at very high temperatures employing solid electrolytes (3.32) is a more suitable equation to be used for thermodynamic analysis of the system. These equations indicate that, when the electrochemical reaction of Eqn. (3.29) occurs reversibly in a galvanic cell, the total change in chemical potential of the intervening components equals the electrical work

Chapter Three

30

performed. In the case that the electrochemical reaction is spontaneous the electromotive force is positive, and the system is able to deliver work to be applied on an external circuit and the galvanic cell operates as a battery. If the reaction is not spontaneous electrical work must be applied and the galvanic cell operates as an electrolyser. Remembering that the chemical potential of species i can be written as: ߤ௜ ൌ ߤ௜௢ ሺܶ, ܲሻ ൅ ܴܶ ݈݊ሺܽ௜ ሻ

(3.34)

where ܽ௜ is the activity of species i and ߤ௜௢ ሺܶ, ܲሻ is its chemical potential at a given standard state. The equilibrium constant of reaction (3.29), K, verifies: ܴܶ ݈݊‫ ܭ‬ൌ െ ߂‫ ܩ‬௢ ൌ ߭ ‫ ߖ߂࣠ כ‬௢

(3.35)

so, it comes that: ȟȲ௧௛௘௥௠ ൌ ȟȲ ௢ െ

ோ் జ‫࣠ כ‬

ሺఈሻ జೕ

ln ቀςఈ,௝ൣܽ௝ ൧ ቁ

(3.36)

This equation, which gives the electromotive force of the galvanic cell as a function of reactants and products concentrations, was originally formulated by Nernst. The temperature dependence of the electromotive force is obtained from Eqns. (3.35) and (3.36), applying the Gibbs – Helmholtz equation: ߭ ‫ ࣠ כ‬ሾȟȲ௧௛௘௥௠ – ܶ ቀ

డ௱అ೟೓೐ೝ೘ డ்

ቁ ሿ ൌ െ ߂‫ܪ‬ ௉

(3.37)

Analogously, one can calculate the electromotive force variation with pressure: ቀ

డ୼ஏ೟೓೐ೝ೘ డ௉

ቁ ൌ െ ்

ଵ జ‫࣠ כ‬

డ௱ீ

ቀ డ௉ ቁ

்

ൌ െ

௱ࣰ೘ జ‫࣠ כ‬

(3.38)

Where ǻࣰ௠ is the molar volume change associated with reaction (3.29). Usually, and except in the case that the system works at very high pressures, ǻࣰ௠ is significant only if gaseous components are involved. Thus, in a reaction involving k gaseous components with partial pressure ܲ௞ it comes that:

Thermodynamics of Electrode Reactions

߂ࣰ௠ ൌ σ௞ ‫ݒ‬௞

ோ்

31

(3.39)

௉ೖ

where vk stands for the molar volume of species ݇ and, hence: ቀ

డ୼ஏ೟೓೐ೝ೘ డ௉

ቁ

்

ൌ െ σ௞ ‫ݒ‬௞

ோ்

(3.40)

జ ‫ ࣠ כ‬௉ೖ

Integration of this equation gives: ȟȲ௧௛௘௥௠ ൌ ߂ߖ ௢ – σ௞ ‫ݒ‬௞ ቀ

ோ் జ‫࣠ כ‬

௉

ቁ ln ቀ ௉°ೖቁ

(3.41)

If a reaction such as: ‫ܪ‬ଶ ሺ݃ሻ ൅ ‫݈ܥ‬ଶ ሺ݃ሻ ՞ 2‫ ܪ‬ା ሺܽ‫ݍ‬ሻ ൅ 2‫ ି ݈ܥ‬ሺܽ‫ݍ‬ሻ

(3.42)

is considered, we have that for unit activity of both ions at 298 K the electromotive force, in terms of decimal logarithms, is: ଵ

௉ಹమ ௉಴೗మ

ଶ

௉೚

ȟȲ ൌ ȟȲ ୭ ൅ 0.059ܸ log ቀ

మ

ቁ

(3.43)

with Pº the reference pressure. In this particular case, electromotive force at gas pressures of 1 bar is 1.36 V; 1.43 V for gas pressures of 10 bar and 1.19 V at 1 mbar.

3.5. Electrode potentials Every chemical reaction which occurs in a galvanic cell implies the loss of a number of electrons by at least one reactant and the gain of those electrons by other reactants. Therefore, the reaction might be separated into two half – cell reactions: one accounting for the loss of electrons (oxidation) that are discharged in the metallic phase 1 (anode) and one describing the gain of electrons (reduction) that are provided by the metallic phase ȟ (cathode). Considering reaction (3.29) we may then write: ߭ଵ ‫ ܣ‬൅ ߭ଷ ‫ ܥ‬൅ ‫ ڮ‬൅ ߭ ‫ ି ݁ כ‬՞ ‫ ڮ‬൅ ߭௡ିଵ ܳ

(3.44)

߭ଶ ‫ ܤ‬൅ ߭ସ ‫ ܦ‬൅ ‫ ڮ‬՞ ‫ ڮ‬൅ ߭௡ ܴ ൅ ߭ ‫ି ݁ כ‬

(3.45)

The electric potential difference in the cathode is, then:

Chapter Three

32

߂Ȳ௧௛௘௥௠,௖ ൌ ߂Ȳ௖௢ –

ோ் జ‫࣠ כ‬

ഔ

݈݊ሾ

௔ೂ೙షభ … ഔ

ഔ

௔ಲభ ௔಴మ …

ሿ

(3.46)

while for the anode the electric potential difference is: ߂Ȳ௧௛௘௥௠,௔ ൌ ߂Ȳ௔௢ –

ோ் జ‫࣠ כ‬

ഔ ௔ೃ೙ … ഔమ ഔర ௔ಳ ௔ವ …

݈݊ ൤

൨

(3.47)

and the electromotive force of the galvanic cell is: ǻȲ௧௛௘௥௠ ൌ ȟȲ௧௛௘௥௠,௖ െ ȟȲ௧௛௘௥௠,௔

(3.48)

ȟȲ ௢ ൌ ȟȲ௖௢ െ ȟȲ௔௢

(3.49)

with:

Given a typical electrode such as: ‫ܯ‬ሃ‫݈ܥܯ‬ሃܰܽ‫݈ܥ‬ሺܽ‫ݍ‬ሻ where two interphases exist, one between metal M and the sparingly soluble salt MCl and the other between the solid salt MCl and a NaCl solution, direct experimental measurement of the electrode potential, which is the electric potential difference ߖM – ߖS between M and the solution, would require that one of the probes of the measuring instrument gets in contact with metal M and the other with the solution. Now, as mentioned before, two new interphases would appear, each one with a particular electric potential difference, which impede direct experimental determination of ߖM – ߖS. However, our interest relies on the measurement of the electromotive force of a galvanic cell, which is the electric potential difference between the metal of the cathode, Mc, and the metal at the anode, Ma: ߂ߖ௧௛௘௥௠ ൌ ൫ߖெ೎ – Ȳௌ ൯ െ ሺߖெೌ – ߖௌ ሻ

(3.50)

and this is possible if the electric potential difference of a particular electrode is taken as an arbitrary zero. As will be discussed later, the generally accepted reference electrode, for which electrode potential difference is defined to be zero, independently of the operating temperature, is the so – called standard hydrogen electrode:

Thermodynamics of Electrode Reactions

33

ܲ‫ݐ‬ሺ‫ܪ‬ଶ , 1ܽ‫݉ݐ‬ሻሃ‫ܪ‬ଶ ܱܵସ ሺܽு శ ൌ 1ሻ in which metallic platinum, with a coating of platinum black, is in contact with an acid solution with unit activity of H+ ions and in which hydrogen gas is bubbled at 1 atm. The electrode reaction in the standard hydrogen electrode is: 2‫ ܪ‬ା ൅ 2݁ ି ՞ ‫ܪ‬ଶ

(3.51)

and it is possible to obtain the electrode potential of any half – cell reaction by measuring the electromotive force of a galvanic cell formed by a standard hydrogen electrode and the unknown electrode. Taking the case of Cu2+ and Zn2+ ions, standard electrode potentials, i.e., those corresponding to unit activity of the ions, are obtained measuring, by means of a potentiometric circuit, the electromotive force of the galvanic cells: ܲ‫ݐ‬ሃ‫ܪ‬ଶ ሃ‫ܪ‬ା , ܱܵସଶି Œ‫ݑܥ‬ଶା , ܱܵସଶି ሃ‫ݑܥ‬

(3.52)

ܲ‫ݐ‬ሃ‫ܪ‬ଶ ሃ‫ܪ‬ା , ܱܵସଶି Œܼ݊ଶା , ܱܵସଶି ሃܼ݊

(3.53)

In the case of (3.52) it is observed that the copper electrode is the cell cathode and the electromotive force at 298 K results 0.3402 V. Hence, if this galvanic cell is applied to an external circuit the electrode reactions that will spontaneously occur are: ‫ݑܥ‬ଶା ൅ 2݁ ି ՜ ‫ݑܥ‬

(3.54)

‫ܪ‬ଶ ՜ 2‫ ܪ‬ା ൅ 2݁ ି

(3.55)

and

and the standard change in Gibbs´ free energy results: ȟ‫ ܩ‬௢ ൌ െ ߭ ‫࣠ כ‬. ȟȲ ௢ ൌ െ2࣠. ሺ0.3402ܸሻ ൌ െ65.66 ݇‫ܬ‬/݉‫( ݈݋‬3.56) On the other hand, the electromotive force of (3.53) is 0.7268 V and the zinc electrode is the anode which means that spontaneous reactions are: 2‫ ܪ‬ା ൅ 2݁ ି ՜ ‫ܪ‬ଶ and

(3.57)

Chapter Three

34

ܼ݊ ՜ ܼ݊ଶା ൅ 2݁ ି

(3.58)

and the standard Gibbs´ free energy change is: ȟ‫ ܩ‬௢ ൌ െ ߭ ‫࣠ כ‬. ȟȲ ௢ ൌ െ2࣠. ሺ0.7628ܸሻ ൌ െ147.21 ݇‫ܬ‬/݉‫( ݈݋‬3.59) Standard electrode potentials referred to the standard hydrogen electrode at 298 K are organized in tables where, following the internationally accepted convention, electrode reactions are written as reduction reactions. Considering the half – cell reaction: ‫ܯ‬ଶା ൅ 2݁ ି ՜ ‫ܯ‬

(3.60)

௢ for the galvanic cell: whose standard electrode potential is ȟȲெ

ܲ‫ݐ‬ሃ‫ܪ‬ଶ ሃ‫ܪ‬ା , ܱܵସଶି ሃ‫ܯ‬ଶା , ܱܵସଶି ሃ‫ܯ‬

(3.61)

with all components in the standard state, it is possible to decide whether electrode M is the anode or the cathode of the cell. Thus, if ǻߖ°M > 0 V: ௢ ‫ܯ‬ଶା ൅ 2݁ ି ՜ ‫ܯ‬ȟȲெ ൐ 0ܸ

(3.62)

and ‫ܪ‬ଶ ՜ 2‫ ܪ‬ା ൅ 2݁ ି

ȟȲு௢ ൌ 0ܸ

(3.63)

the electromotive force of the galvanic cell results: ௢ ௢ െ ȟȲு௢ ሻ ൌ ሺȟȲெ െ 0ሻ ൐ 0 ȟȲ ௢ ൌ ሺȟȲெ

(3.64)

and therefrom: ȟ‫ ܩ‬௢ ൌ െ߭ ‫࣠ כ‬. ȟȲ ௢ ൏ 0

(3.65)

and the reaction, with all species at unit activity, is spontaneous. Conversely, if ߂ߖ°ெ < 0 V the change in Gibbs’ free energy will be positive and the spontaneous reaction would be the oxidation of M. There are half – cell reactions whose standard potential cannot be experimentally measured. This might happen for different reasons, as is the case of metal reduction of an alkali metal, that violently reacts with water.

Thermodynamics of Electrode Reactions

35

In such cases, the only way to obtain the standard electrode potentials is through calculation of the Gibbs’ free energy change from: ȟ‫ ܩ‬௢ ൌ ȟ‫ ܪ‬௢ െ ܶ. ȟܵ ௢

(3.66)

where ǻH° and ǻS° are computed from thermodynamical data. Table 3.1 shows the standard electrode potential values for a number of half – cell reactions. Half – cell reaction Li (aq) + e- ĺ Li(s) Rb+ (aq) + e- ĺ Rb(s) K+ (aq) + e- ĺ K(s) Cs+ (aq) + e- ĺ Cs(s) Ba+2 (aq) + 2e- ĺ Ba(s) Sr+2 (aq) + 2e- ĺ Sr(s) Ca+2 (aq) + 2e- ĺ Ca(s) Na+ (aq) + e- ĺ Na(s) Mg+2 (aq) + 2e- ĺ Mg(s) H2 + 2e- ĺ 2 H+2 Be (aq) + 2e- ĺ Be(s) Al+3 (aq) + 3e- ĺ Al(s) Ti+2 (aq) + e- ĺ Ti(s) TiO (s) + 2H+ (aq) + 2e- ĺ Ti(s) + H2O Ti2O3 (s) + 2H+ (aq) + 2e- ĺ 2TiO(s) + H2O Ti+3 (aq) + 3e- ĺ Ti(s) Mn+2 (aq) + 2e- ĺ Mn(s) V+2 (aq) + 2e- ĺ V(s) Sn (s) + 4 H+ + 4 e- ĺ SnH4 (g) SiO2 (s) + 4 H+ + 4 e- ĺ Si(s) + 2 H2O B(OH)3 + 3 H+ + 3e- ĺ B(s) + 3 H2O TiO+2 + 2 H+ + 4 e- ĺTi (s) + H2O 2 H2O (l) + 2e- ĺ H2 (g) + 2 OHZn+2 (aq) + 2e- ĺ Zn(s) Cr+3 (aq) + 3e- ĺ Cr(s) Au(CN)2 (aq) + e- ĺ Au (s) + 2 CN- (aq) 2 TiO2 (s) + 2 H+ + 2e- ĺ Ti2O3 (s) + H2O Ga+3 (aq) + 3e- ĺ Ga(s) H3PO2(aq) + H+ + e- ĺ P(s) + 2 H2O H3PO3 (aq) + 3 H+ +3 e- ĺ P(s) + 3 H2O +

߂ߖ° ሺVሻ -3.05 -2.98 -2.93 -2.92 -2.91 -2.89 -2.76 -2.71 -2.38 -2.25 -1.85 -1.68 -1.63 -1.31 -1.23 -1.21 -1.18 -1.13 -1-07 -0.91 -0.89 -0.86 -0.83 -0.76 -0.74 -0.60 -0.56 -0.53 -0.51 -0.50

36

Chapter Three

H3PO3 (aq) + 2 H+ +2 e- ĺ H3PO2(aq) + H2O Fe+2 (aq) + 2e- ĺ Fe(s) 2 CO2 (g) + 2 H+ + 2e- ĺ HOOCCOOH (aq) Cr+3 (aq) + e- ĺ Cr+2 (aq) Cd+2 (aq) + 2e- ĺ Cd(s) PbSO4 (s) + 2 e- ĺ Pb(s) + SO4-2 (aq) GeO2 (s) + 2 H+ + 2e- ĺ GeO (s) + H2O In+3 (aq) + 3e- ĺ In(s) Tl+ (aq) + e- ĺ Tl(s) Ge (s) + 4 H+ + 4 e- ĺ GeH4 (g) Co+2 (aq) + 2e- ĺ Co(s) H3PO4 (aq) + 2 H+ +2 e- ĺ H3PO3(aq) + H2O V+3 (aq) + e- ĺ V+2 (aq) Ni+2 (aq) + 2e- ĺ Ni(s) As (s) + 3 H+ + 3 e- ĺ AsH3 (g) MoO2 (s) + 4 H+ + 4e- ĺ Mo(s) + 2 H2O Si (s) + 4 H+ + 4 e- ĺ SiH4 (g) Sn+2 (aq) + 2e- ĺ Sn(s) O2 (g) + H+ + e- ĺ HO2* (aq) Pb+2 (aq) + 2e- ĺ Pb(s) WO2 (s) + 4 H+ + 4e- ĺ W(s) + 2 H2O CO2 (g) + 2 H+ + 2e- ĺ HCOOH (aq) Se (s) + 2 H+ + 2e- ĺ H2Se (g) CO2 (g) + 2 H+ + 2e- ĺ CO (g) + H2O SnO(s) + 2 H+ + 2e- ĺ Sn(s) + H2O SnO2 (s) + 2 H+ + 2e- ĺ SnO(s) + H2O WO3 (s) + 6 H+ + 6e- ĺ W(s) + 3 H2O P(s) + 3 H+ + 3 e- ĺ PH3 (g) HCOOH (aq) + 2 H+ + 2e- - ĺ HCHO (aq) + H2O 2 H+ + 2e- ĺ H2 (g) H2MoO4 (aq) + 6 H+ +6 e- ĺ Mo(s) + 4 H2O Ge+4 (aq) + 4e- ĺ Ge(s) C(s) + 4 H+ + 4e- ĺ CH4 (g) HCHO (aq) + 2 H+ + 2e- ĺ CH3OH S (s) + 2 H+ + 2e- ĺ H2S (g) Sn+4 (aq) + 2e- ĺ Sn+2 (aq) Cu+2 (aq) + e- ĺ Cu+ (aq) HSO4 (aq) + 3 H+ + 2e- ĺ SO2(aq) + 2 H2O SO4-2 (aq) + 4 H+ + 2e- ĺ SO2(aq) + 2 H2O SbO++ 2 H+ + 2e- ĺ Sb(s) + H2O

-0.50 -0.44 -0.43 -0.42 -0.40 -0.36 -0.36 -0.34 -0.34 -0.29 -0.28 -0.28 -0.26 -0.25 -0.23 -0.15 -0.14 -0.14 -0.13 -0.13 -0.12 -0.11 -0.11 -0.11 -0.10 -0.09 -0.09 -0.06 -0.03 0.00 +0.11 +0.12 +0.13 +0.13 +0.14 +0.15 +0.16 +0.16 +0.17 +0.20

Thermodynamics of Electrode Reactions

H3AsO3- (aq) + 3 H+ + 3e- ĺ As(s) + 3 H2O GeO(s) +2 H+ + 2e-ĺ Ge(s) + H2O Bi+3 (aq) + 3e- ĺ Bi(s) +2 VO (aq) + 2 H+ + e- ĺ V+3(aq) Cu+2 (aq) + e- ĺ Cu(s) [Fe(CN)6]-3(aq) + e- ĺ [Fe(CN)6]-4(aq) O2(g) + 2 H2O + 4e- ĺ 4 OH- (aq) H2MoO4 (aq) + 6 H+ + 3e- ĺ Mo+3(aq) CH3OH(aq) + 2 H+ + 2e- ĺ CH4 (g) + H2O SO2(aq) + 4 H+ + 4e- ĺ S(s) + 2 H2O Cu+(aq) + e- ĺ Cu(s) CO(g) + 2 H+ + 2e- ĺ C(s) + H2O I2(s) + 2e- ĺ 2I-(aq) I3- (aq) + 2e- ĺ 3I-(aq) [AuI4]- (aq)+ 3e- ĺAu(s) + 4I-(aq) H3AsO4(aq) + 2 H+ + 2e- ĺ H3AsO3(aq) + H2O [AuI2]- (aq)+ e- ĺAu(s) + 2I-(aq) MnO4 (aq) +2 H2O + 3e- ĺ MnO2(s) + 4 OH- (aq) S2O3-2 + 6 H+ + 4e- ĺ 2 S(s) + 3 H2O H2MoO4 (aq) + 2 H+ + 2e- ĺ MoO2(s) + 2 H2O O2(g) + 2 H+ + 2e- ĺ H2O2(aq) Tl+3 (aq) + 3e- ĺ Tl(s) H2SeO3 (aq) + 4 H+ + 4e- ĺ Se(s) + 3 H2O Fe+3 (aq) + e- ĺ Fe+2 (aq) Hg2+2 (aq) + 2e- ĺ 2Hg(l) Ag+(aq) + e- ĺ Ag(s) NO3 (aq) + 2 H+ + 2e- - ĺ NO2 (g) + H2O [AuBr4]- + 3 e- ĺ Au (s) + 4 BrHg+2 (aq) + 2e- ĺ 2Hg(l) MnO4- (aq) + H+ + e- ĺ HMnO42 Hg+2 (aq) + 2e- ĺ Hg2+2 (aq) [AuCl4]- + 3 e- ĺ Au (s) + 4 ClMnO2(s) + 4 H+ + e- ĺ Mn+3 (aq) + 2 H2O [AuBr2]- + e- ĺ Au (s) + 2 BrBr2(l) + 2e- ĺ 2Br-(aq) Br2(aq) + 2e- ĺ 2Br-(aq) IO3 (aq) + 5 H+ + 4e- ĺ HIO (aq) + 2 H2O [AuCl2]- + e- ĺ Au (s) + 2 ClHSeO4 (aq) + 3 H+ + 2e- ĺ H2SeO3 (aq) + H2O Ag2O (s) + 2 H+ + 2e- ĺ 2 Ag (s) + H2O

37

+0.24 +0.26 +0.32 +0.34 +0.34 +0.36 +0.40 +0.43 +0.50 +0.50 +0.52 +0.52 +0.54 +0.54 +0.56 +0.56 +0.58 +0.59 +0.60 +0.65 +0.70 +0.72 +0.74 +0.77 +0.80 +0.80 +0.80 +0.85 +0.85 +0.90 +0.91 +0.93 +0.95 +0.96 +1.07 +1.09 +1.13 +1.15 +1.15 +1.17

38

Chapter Three

ClO3- (aq) + 2 H+ + e- ĺ ClO2 (aq) + H2O ClO2 (g) + H+ + e- ĺ HClO2 (aq) 2 IO3- (aq) + 12 H+ + 10 e- ĺ I2 (s) + 6 H2O ClO4- (aq) + 4 H+ + 2 e- ĺ ClO3- (aq) + H2O O2(g) + 4 H+ + 4 e- ĺ 2 H2O MnO2(s) + 4 H+ + 2 e- ĺ Mn+2 (aq) + 2 H2O Tl+3 (aq) + 2 e- ĺ Tl+ (aq) Cl2 (g) + 2 e- ĺ 2 Cl- (aq) -2 Cr2O7 (aq) + 14 H+ + 6 e- ĺ 2 Cr=3(aq) + 7 H2O CoO2(s) + 4 H+ + e- ĺ Co+3 (aq) + 2 H2O 2 HIO (aq) + 2 H+ + 2 e- ĺ I2 (s) + 2 H2O BrO3- (aq) + 5 H+ + 4 e- ĺ HBrO (aq) + 2 H2O 2 BrO3- (aq) + 12 H+ + 10 e- ĺ Br2 (l) + 6 H2O 2 ClO3- (aq) + 12 H+ + 10 e- ĺ Cl2 (l) + 6 H2O MnO4- (aq) + 8 H+ + 5 e- ĺ Mn+2 (aq) + 4 H2O O2* + 2 H+ + 2 e- ĺ H2O2 (aq) Au+3 (aq) + 3 e- ĺ Au (s) NiO2(s) + 4 H+ + 2 e- ĺ Ni+2 (aq) + 2 H2O 2 HClO (aq) + 2 H+ + 2 e- ĺ Cl2 (g) + 2 H2O Ag2O3 (s) + 6 H+ + 4 e- ĺ 2 Ag+ (aq) + 3 H2O HClO2 (aq) + 2 H+ + 2 e- ĺ HClO (aq) + H2O Pb+4 (aq) + 2 e- ĺ Pb+2 (aq) MnO4 (aq) + 4 H+ + 3 e- ĺ MnO2(s) + 2 H2O H2O2 (aq) +2 H+ + 2 e- ĺ 2 H2O Au+ (aq) + e- ĺ Au (s) BrO4 (aq) + 2 H+ + 2 e- ĺ BrO3- (aq) + H2O Co+3 (aq) + e- ĺ Co+2 (aq) Ag+2 (aq) + e- ĺ Ag+ (aq) S2O8-2 (aq) + 2 e- ĺ 2 SO4-2 HMnO4 (aq) + 3 H+ + 2 e- ĺ MnO2(s) + 2 H2O F2 (g) + 2 e- ĺ 2 F- (aq) F2 (g) + 2 H+ + 2 e- ĺ 2 HF (aq)

+1.18 +1.19 +1.20 +1.20 +1,23 +1.23 +1.25 +1.36 +1.36 +1.42 +1.44 +1.45 +1.48 +1.49 +1.51 +1.51 +1.52 +1.59 +1.63 +1.67 +1.67 +1.69 +1.70 +1.76 +1.83 +1.85 +1.92 +1.98 +2.07 +2.09 +2.87 +3.05

Table 3.1: Standard electrode potential values for a number of half – cell reactions.

Thermodynamics of Electrode Reactions

39

3.6. Galvanic cells under non – equilibrium conditions: The case of the water decomposition reaction A good example of how a given galvanic cell might work as a battery or as an electrolyser is the water decomposition/production reactions: ଵ

‫ܪ‬ଶ ൅ ܱଶ ՞ ‫ܪ‬ଶ ܱ ଶ

(3.67)

In May 1800 Nicholson and Carlisle (Nicholson and Carlisle, 1800) showed that water could be decomposed into hydrogen and oxygen by applying an electromotive force obtained from a Volta battery on two electrodes immersed in a sulfuric acid solution. Some forty years later, Grove (Grove, 1842) reported that electricity was obtained when gaseous oxygen and hydrogen were fed to a couple of platinum electrodes coated with platinum black. Figure 3.3 shows schematically how both devices work.

Figure 3.3: a) By applying an electromotive force to a couple of platinum electrodes in sulfuric acid, hydrogen is evolved at the right – side electrode (cathode) and oxygen is produced at the left – side electrode (anode); b) The electromotive force is withdrawn and when the circuit is closed, oxygen is reduced at the left – side electrode (cathode) and hydrogen is oxidized at the right side electrode (anode).

The galvanic cell in which the global reaction (3.67) occurs proceeds through two half – cell reactions whose electrode potentials are:

Chapter Three

40

2‫ ܪ‬ା ൅ 2݁ ି ՞ ‫ܪ‬ଶ

ȟߖு௢ ൌ 0ܸ

(3.68)

ଵ

ȟȲை௢ ൌ 1.23ܸ

(3.69)

ଶ

ܱଶ ൅ 2‫ ܪ‬ା ൅ 2݁ ି ՞ ‫ܪ‬ଶ ܱ

Thus, when the reaction (3.67) proceeds towards the production of water the electromotive force at standard conditions is: ௢ ȟȲ௕௔௧௧ ൌ ȟȲை௢ െ ȟȲு௢ ൌ 1.23ܸ

(3.70)

where reaction (3.68) takes place in the anodic direction. These reactions are spontaneous when combined and correspond to the process depicted in Figure 3.3.b) in which electric current in the external circuit circulates from the cathode (positive electrode of the battery), where oxygen reduction takes place, to the anode (negative electrode of the battery) where hydrogen is oxidized. As oxygen and hydrogen are consumed their pressures are lower and after a certain time the Gibbs’ free energy change is: ߂‫ ܩ‬ൌ ܹ ൌ െ ߭ ‫߂ ࣠ כ‬Ȳ௧௛௘௥௠ ൌ െ ߭ ‫ ࣠ כ‬ሾ߂ߖ ௢ ൅

ோ் ଶ࣠

భ/మ మ ௉°య/మ

௉ಹమ .௉ೀ

݈݊ ሺ

ሻሿ

(3.71)

with pressure values below 1 atm for both gases. W is the maximum amount of work that can be delivered by the system to an external load. When equilibrium is attained, and the battery is exhausted ǻG = 0 and: ߂ߖ ௢ ൌ െ

ோ் ଶ࣠

భ/మ

௉ಹమ ሺ೐೜ሻ .௉ೀ ሺ೐೜ሻ మ ሻ ௉°య/మ

݈݊ ሺ

(3.72)

This means that, for an oxygen partial pressure of 0.2 atm, equilibrium is attained at a very low hydrogen pressure (ca. 10-42 atm). In order that water be decomposed in oxygen and hydrogen, both at 1 atm, reaction (3.67) must occur leftwards and, since the potential difference of this process is -1.23 V, ο‫ ܩ‬is positive. Consequently, in order to produce the desired reaction, a potential difference larger than 1.23 V must be applied in opposition to the cell, connecting the negative terminal of the external power source to the hydrogen electrode (that will be the electrolyser cathode) and the positive pole to the oxygen electrode (that will be the electrolyser anode). If gas pressure values are different from 1 atm the thermodynamic cell ௖௘௟௟ and zero net current condition can be attained applying potential is ȟȲ௧௛௘௥௠ ௖௘௟௟ a potential difference ȟȲ ‫ כ‬ൌ െ ȟȲ௧௛௘௥௠ in a potentiometric circuit such as

Thermodynamics of Electrode Reactions

41

that of Figure 3.2. In such a case, if the system operates reversibly, the amount of work needed for producing hydrogen and oxygen is given by: ȟ‫ ܩ‬ൌ ܹ ൌ ߭ ‫࣠ כ‬. ȟȲ ‫כ‬

(3.73)

When involved currents are different from zero the system operates irreversibly and, in the case of electricity generation, gas pressure values will have to shift from equilibrium conditions in order to overcome the activation energy of the reaction and mass transfer effects and electrode potentials are: οߖைమ ൌ οȲ௧௛௘௥௠ ைଶ ൅ οȲ௔௖௧ ைଶ ൅ οȲ௠௧ ைଶ

(3.74)

οߖுమ ൌ οȲ௧௛௘௥௠ ுଶ ൅ οȲ௔௖௧ ுଶ ൅ οȲ௠௧ ுଶ

(3.75)

In this context οȲ௔௖௧ ைଶ and οȲ௔௖௧ ுଶ are the required electric potential difference values at cathode and anode, respectively, needed to overcome the activation barrier when a certain current value is circulating. We will call this amount the activation overpotential of the electrode, ߟ௔௖௧ , whose value will depend on the current intensity. Thus, for the cathode: οȲ௔௖௧ ை ൌ Ʉ௔௖௧ ை మ

మ

(3.76)

and for the anodic reaction: οȲ௔௖௧ ு ൌ Ʉ௔௖௧ ு మ

మ

(3.77)

In a similar analysis, mass transfer limitations are described in terms of the mass transfer overpotential, ߟ௠௧ and Eqns. (3.75) and (3.76) are written as: οߖைమ ൌ οȲ௧௛௘௥௠ ைଶ ൅ Ʉ௔௖௧ ைଶ ൅ ߟ௠௧ ைଶ

(3.78)

οߖுమ ൌ οȲ௧௛௘௥௠ ுଶ ൅ Ʉ௔௖௧ ுଶ ൅ ߟ௠௧ ுଶ

(3.79)

where anodic overpotentials are positive and cathodic overpotentials have a negative sign. From these results it is clear that when the galvanic cell operates as an electrolyser the applied potential difference value is: :

42

Chapter Three

ȟȲ௔௣௣௟ ൌ οߖுమ െ οߖைమ ൐ οȲ௧௛௘௥௠

(3.80)

If the electrolyte resistance is R the electric power needed for the reaction occurring at an electric current value I is: ܹሶ ൌ ‫ܫ‬. ȟȲ௔௣௣௟ ൌ ‫ܫ‬൫οߖுమ െ οߖைమ െ ‫ܫ‬. ܴ൯

(3.81)

When the system operates as a battery its potential difference is: ȟȲ௕௔௧௧ ൌ οߖைమ െ οߖுమ ൏ οȲ௧௛௘௥௠

References Grove, W.R. 1842. “On a Gaseous Voltaic Battery” The London and Edinburgh Philosophical Magazine and Journal of Science. 417-420 Nicholson, W. and Carlisle, A. 1800. “New Electrical Apparatus” The Journal of Natural Philosophy, Chemistry & the Arts.179-190

Bibliography Guggenheim, E. A. Thermodynamics 7th. ed. 1985. New York. Elsevier Münster, A. Classical thermodynamics. 1970. New York. John Wiley

CHAPTER FOUR THE ELECTRODE – SOLUTION INTERPHASE

4.1.- Introduction When the electromotive force of a commercial battery is measured employing the potentiometric circuit of Figure 3-2, it is found that an electric potential difference ǻȲ exists between anode and cathode. This measurement is performed when the electrical current is zero and this potential difference is solely due to the thermodynamic electrode potential at both electrodes, which only depends on the concentrations of the intervening species. The electrochemical reactions occurring at the interphase of the electrodes define, then, the electromotive force of the battery. Consequently, understanding, at least in its main aspects, the properties of these interphases is of the utmost importance. With this goal in mind let us consider a monovalent solid metal M, whose structure may be described as an ordered net of M+ ions through which electrons, responsible of metal conductivity, can move under an applied electric field. We assume now that when metal M is in contact with liquid water the only possible reaction is: ି ‫ܯ‬ሺ݉݁‫ݐ‬ሻ ՞ ‫ܯ‬ା ሺ‫݈݋ݏ‬ሻ ൅ ݁ெ

(4.1)

Let us further consider a solution of the salt MX, which completely dissociates into M+ and X- ions, in which a solid bar of metal M is introduced. At instant t = 0 the net charge in any point of the system is zero and the electrical situation corresponds to the sketch of Figure 4-1:

 M qM=0 ʗM=ʗM,0 

MX(sol) qsol=0 ʗsoln=ʗsoln,0 

Figure 4-1: Electrical state of the metal – solution interphase at t = 0.

Chapter Four

44

In order that equilibrium is attained Eqn. (3.21) must be satisfied, which in our case implies that: ‫כ‬ሺ௠௘௧ሻ

ߤெ

‫כ‬ሺ௦௢௟ሻ ൌ ߤெ ൅ ߤ௘‫כ‬ሺ௠௘௧ሻ ష శ

(4.2)

At ‫ ݐ‬ൌ 0 charge is zero at both phases but the chemical potential of ‫ܯ‬ା is not the same in the solution and in the solid metal and the system is thus at a non-equilibrium condition: ‫כ‬ሺ௠௘௧ሻ

ߤெ

‫כ‬ሺ௦௢௟ሻ ് ߤெ ൅ ߤ௘‫כ‬ሺ௠௘௧ሻ ష శ

(4.3)

‫כ‬ሺ௦௢௟ሻ

‫כ‬ሺ௠௘௧ሻ ൐ ߤெ శ According to Eqn. (4.1) if ߤெ

൅ ߤ௘‫כ‬ሺ௠௘௧ሻ the system will evolve ష

‫כ‬ሺ௦௢௟ሻ

‫כ‬ሺ௠௘௧ሻ ൏ ߤெశ ൅ ߤ௘‫כ‬ሺ௠௘௧ሻ ions will be deposited dissolving ions while, if ߤெ ష on the solid. In both cases, charge will be accumulated at each side of the interphase. Figure 4-2 shows the system situation at t ് 0 when oxidation proceeds as in reaction (4.1).

Figure 4-2: Electrical state of the metal – solution interphase at t > 0.

The electric potential difference at the interphase will continue evolving until, at sufficiently large time values, equilibrium is attained which, according to Eqn. (3.22), implies: ሺ௠௘௧ሻ

ߤெ

ሺ௦௢௟ሻ

ሺ௠௘௧ሻ

ൌൌ ߤெశ ൅ ࣠. ߖሺ௦௢௟ሻ ൅ ߤ௘ ష

െ ࣠. ߖሺ௠௘௧ሻ

(4.4)

The electrical state of the interphase at equilibrium is depicted by Figure 4-3.

The Electrode – Solution Interphase

45

Figure 4-3: Electrical state of the metal – solution interphase at equilibrium.

Thus, in order to be at equilibrium, the system has created an electrical double layer by accumulating charges of different sign at each side of the interphase and the resulting electric potential difference will be labelled ǻȲM/S. The foregoing analysis, which was circumscribed to the metal – solution interphase, is also valid for any kind of interphase in which charged particles are involved: metal – metal, metal – semiconductor, semiconductor – semiconductor, semiconductor – solution, solution – solution.

4.2. The electrical double layer The simplest model describing the electrical double layer, due to Helmholtz (Helmholtz, 1853), is the compact double layer. In this approach, it is assumed that charge accumulates on two well defined planes: one on the metal surface and the other at the solution side, defined by solvated ions. The distance between both planes is about one ion diameter as shown in Figure 4 – 4:

46

Chapter Four

Figure 4-4: The Helmholtz compact double layer model.

According to the Helmholtz model the electrical double layer behaves as an ideal capacitor and, thus, the electric potential linearly depends on the distance to the electrode plane. This approach dismisses the movement of ions in solution, whose thermal effect will affect the compact layer stability. The thermal effect was considered by Gouy (Gouy, 1910) and Chapman (Chapman, 1913) who analysed the double layer problem considering only the thermal effect and, through a mathematical approach similar to that employed some years later by Debye and Hückel for electrolyte solutions, obtained a function describing the change of electric potential in terms of distance to the electrode surface, thus depicting a diffuse double layer. Stern (Stern, 1924) pointed to the convenience of combining both models by assuming a Helmholtz plane, on which only ions with electric charge of different sign to that of the metal are present, wherefrom the diffuse layer is built. Thus, the change of electric potential between the electrode and the electrolyte solution can be written as: ȟȲெ/ௌ ൌ ሺȲெ െ Ȳୌ୔ ሻ ൅ ሺȲୌ୔ െ Ȳௌ ሻ Figure 4-5 gives an image of this model:

(4.5)

The Electrode – Solution Interphase

47

Figure 4-5: The Stern model.

The Stern model does not consider the effect of the solvent dipoles nor the existence of chemical interactions which might lead to specific adsorption effects. In fact, and as experimentally proven, different anions chemically interact with metal atoms and therefore show specific adsorption. Then, a

48

Chapter Four

more realistic model for the electrical double layer is such as the one shown in Figure 4–6 in which specific adsorption due to chemical interaction is recognized and, consequently, two planes are defined: the so - called internal Helmholtz plane, due to chemical anion effects, and the outer Helmholtz plane. This model, schematically shown in Figure 4-6, obviously implies a more involved mathematical description (Bockris et al., 1963).

Figure 4-6: The electrical double layer model with specific adsorption.

Since current flows through electrodes, it is clear that the Helmholtz model is an oversimplification because, if the interphase behaviour is that of a capacitor, when a potential difference is applied, current flow should fall to zero once the capacitor is charged. Hence, the electrical image of the electrode interphase might be described, in a first approximation, by an “equivalent circuit” in which a capacitor is connected in parallel with a resistance, called the “equivalent resistance”. Figure 4-7 shows the

The Electrode – Solution Interphase

49

equivalent circuit of an electrode which is connected to the electrolyte resistance Rs.

Figure 4-7: Equivalent circuit of an electrode.

Let us now consider an electrode at which the following electrochemical reaction takes place: ‫ ܣ‬൅ ݁ ି ՜ ‫ିܣ‬

(4.6)

In this case the equivalent resistance value will depend on the reaction velocity: for very low reaction rates the equivalent resistance is large and, in order to have a detectable current flow a significant electrical potential difference must be applied. In this case it is said that the electrode is polarizable. An ideally polarizable electrode is one in which current flow is zero at any applied potential difference value. On the other hand, if the equivalent resistance is low, i.e., the electrode reaction rate is high, it is said that the electrode is non-polarizable: very low displacements of the electric potential difference at the interphase can produce large values of current flow. An ideally non polarizable electrode is that in which there is no charge accumulation at the double layer at finite current values.

4.3. Reference electrodes It was shown in Section 3.5 that the electric potential difference at an interphase is only measurable with respect to another electrode, which must be taken as reference. From what has been said above it is clear that a good reference electrode to be employed in the measurement circuit shown in

50

Chapter Four

Figure 3-2 must be of the non-polarizable type, since its potential difference is constant when current flows through it. The hydrogen electrode was chosen as a primary reference electrode because, when properly built, it shows a non-polarizable behaviour over a large range of current values; however, its use is rather difficult in most applications. This happens because a well defined concentration of H+ ions must be maintained, hydrogen gas pressure should be constant, no impurities should exist in the gas, especially oxygen, and the platinum electrode must be periodically cleaned. For these reasons, a number of nonpolarizable interphases have been developed as reference electrodes of simpler construction and maintenance characteristics. Two widely used reference electrodes are the Ag/AgCl and the calomel (Hg/Hg2Cl2) electrodes. In both cases the metal is in direct contact with a slightly soluble salt of the corresponding metal and with a solution in which chloride ions are present at a known concentration: ‫ܯ‬ሃ‫݈ܥܯ‬ሺ‫݈݋ݏ‬ሻሃ‫ ି݈ܥ‬ሺܽ‫ݍ‬ሻ The electrode process in this type of electrodes corresponds to the reactions: ‫ ݈ܥܯ‬՜ ‫ܯ‬ା ൅ ‫ି ݈ܥ‬ ‫ܯ‬ା ൅ ݁ ି ՜ ‫ܯ‬ In the case of the Ag/AgCl/KCl 0,1 M electrode its potential difference is ȟȲ = 0.2368 V while for the calomel electrode in saturated solution of KCl ȟȲ is 0.2682 V, both measured with respect to the standard hydrogen electrode.

4.4. Electrode potential control and polarization curves Studying the electrochemical behaviour of reaction (4.1) at an electrode implies measuring the electric current value that flows at a given electrode potential difference. In order to achieve this, an experimental arrangement is needed in which electric current and electrode potential difference with respect to a reference electrode can be simultaneously measured. In such a device, care must be taken to assure that the reference electrode potential difference is kept constant throughout the experiment. Figure 4-8 shows the circuit usually employed when this type of measurement is performed employing a potentiostat.

The Electrode – Solution Interphase

51

Figure 4-8: Schematic description of a potentiostatic circuit for current intensity measurement as a function of electrode potential.

In this circuit the electrode whose behaviour is to be studied is called the working electrode (W.E.) and the electrode that closes the circuit through which current flows is the counterelectrode (C.E.). Total potential difference between them, ȟȲ௧௢௧௔௟ , is measured by a voltmeter which is contained in the potentiostat, and current intensity is measured by the ammeter A. Now, ȟȲ௧௢௧௔௟ includes the electric potential difference at the working electrode, the electric potential difference at the counterelectrode and the electric potential difference due to the electrical resistance of the electrolyte solution. The reference electrode (R.E.), who is chosen according to its specific properties, is introduced as a third electrode which is connected to the potentiostat and, through the Luggin capillary (L.C.), to the working electrode. The electric potential difference at the working electrode with respect to the reference electrode is measured by the voltmeter V. The current flow through the reference electrode is minimized introducing a large resistance by means of a salt bridge which is filled with the working solution and ends, on the reference electrode side, in a diaphragm of very low porosity, such that effects of ion diffusion can be

Chapter Four

52

dismissed, and, on the working electrode side, by the L.C. whose distance to the W.E. must be kept as low as possible to avoid significant ohmic losses. When operating this circuit, a potential difference value between the working and reference electrodes is set in the potentiostat and, to attain it, ȟȲ௧௢௧௔௟ is modified until the measurement in the voltmeter V coincides with the set value. The potential difference read in the voltmeter corresponds to the current intensity measured in the ammeter. In this way the current value corresponding to a given electrode overpotential, Ƨ, defined as the difference between the measured electrode potential with respect to the open circuit: ߟ ൌ ȟȲ െ ȟȲ௢.௖.

(4.21)

is measured and the polarization curve is obtained for the electrode under study. When, on the working electrode a unique reaction occurs, the open circuit potential is the equilibrium potential at the particular concentration and temperature conditions.

References Bockris, John, Devanathan, Michael and Müller, Klaus. 1963. “On the structure of charged interphases”. Proceedings of the Royal Society. Series A. 55 – 79 Chapman, Douglas. 1913. “A contribution to the theory of electrocapillarity”. Philosophical Magazine 475 - 481 Gouy, Louis. 1910. “Sur la constitution de la charge électrique à la Surface d´un électrolyte”. Journal de Physique 457-468 Helmholtz, Hermann. 1853. “Über einige Gesetze der Vertheilung elektrischer Ströme in körperlichen Leitern mit Awendung auf die thierisch – elektrischen Versuchen”. Annalen der Physik und Chemie 211 – 233 Stern, Otto. 1924. “Zur Theorie der Elektrolytischen Doppelschicht”. Zeitschrift für Elektrochemie 508 – 516

Bibliography Bockris, John, Reddy, Amulya and Gamboa – Adelco, María Modern electrochemistry. Vol. 2A. New York. Kluwer Academic Publishers, 2002

CHAPTER FIVE ELECTRODE KINETICS. I. CHARGE TRANSFER

5.1. The Butler – Volmer equation When studying the rate of electrochemical reactions Julius Tafel (Tafel, 1905) found that, within a range of current values, a linear relationship exists between the applied electric potential difference and the logarithm of current intensity: ߂ߖ ൌ ܽ ൅ ܾ. ݈‫ܫ݃݋‬

(5.1)

This relationship, known as the Tafel equation, was, for almost twenty years, an empirical result which found theoretical explanation through the works published during the 1924 – 1935 period by Butler (Butler, 1924), Eyring (Eyring, 1935) and Volmer and Erdey – Gruz (Volmer, 1928; Erdey-Gruz and Volmer, 1930). In this section a summary of these works is given. Let us consider, again, a metal electrode M which can only undergo the reaction: ‫ܯ‬ା ൅ ݁ ି ՞ ‫ܯ‬

(5.2)

which is in contact with an aqueous solution of the salt ‫ܺܯ‬, of unit activity, that completely dissociates according to: ‫ܺܯ‬ሺ‫݈݋ݏ‬ሻ ՜ ܺ ି ൅ ‫ܯ‬ା

(5.3)

As it was discussed in the previous chapter, at the instant (t=0) when a bar of the metal M, with area ࣛ, is immersed in the XM solution the net electrical charge on each phase is zero, reaction (5.2) is not in equilibrium and reduction and oxidation rates will be different ‫ݎ‬Ԧ  ‫ݎ‬ര

(5.4)

Chapter Five

54

Reaction rates are measured in mol.s-1.cm-2 and, hence, can be associated to the electric current through Faraday´s law: ଓԦ= ࣠ ‫ݎ‬Ԧ  രଓ = ࣠ ‫ݎ‬ര

(5.5)

Where i = I.ࣛ -1 is the current density on the electrode which is measured in A.cm-2. According to the activated complex theory the system energy situation at t = 0 is depicted as shown in Figure 5-1

Figure 5-1: Energy graph for system (5.2) at t = 0.

At t = 0 the difference of Gibbs free energy between the activated complex # # ሬሬሬሬሬሬሬሬԦ രሬሬሬሬሬሬሬሬ and, analogously, ο‫ܩ‬ is the Gibbs free energy and reactants is ο‫ܩ‬ ௧ୀ଴

௧ୀ଴

difference between activated complex and products. Employing the activated complex theory for the case of reaction (5.2) the current density for cathodic and anodic reactions at t=0 can be written as:

Electrode Kinetics. I. Charge Transfer

55

ሬሬሬሬሬሬሬሬሬሬԦ #

ሬሬሬሬԦܿ ܽ ൅ ൌ ࣠ ௞் exp ቆെο‫ݐܩ‬ൌ0 ቇ ܽ ൅ ଓԦሺ‫ ݐ‬ൌ 0ሻ ൌ ࣠݇ ‫ܯ‬ ‫ܯ‬ ‫݈݋ݏ‬

ோ்

௛

‫݈݋ݏ‬

(5.6)

and രሬሬሬሬሬሬሬሬሬሬ #

രሬሬሬሬ௖ ሺ‫ ݐ‬ൌ 0ሻܽெ = ࣠ ቀ௞்ቁ expሺοீ೟సబሻܽெ രଓሺ‫ ݐ‬ൌ 0ሻ = ࣠ ݇ ೘೐೟ ೘೐೟ ௛

ோ்

(5.7)

രሬሬሬሬ௖ are where k and h are Boltzmann’s and Planck’s constants and ሬሬሬሬሬԦ ݇௖ and ݇ the rate constants of the cathodic and anodic reactions. Since reaction (5.2) is the only path through which the system might evolve, electron transfer will be produced from one phase to the other until, at sufficiently large time values, equilibrium is attained with the energy situation shown in Figure 5-2.

Figure 5-2: Energy graph for system (5.2) at equilibrium.

Chapter Five

56

Now, ሺ௠௘௧ሻ

ߤெ

ሺ௦௢௟ሻ

ሺ௠௘௧ሻ

ൌ ߤெ శ ൅ ߤ௘ ష

൅ ࣠. ߖ௦௢௟ െ ࣠. ߖ௠௘௧

(5.8)

and the electric potential difference between electrode and solution corresponds to the equilibrium potential difference: Ȳ௘௤,௠௘௧ െ Ȳ௘௤,௦௢௟ ൌ ȟȲ௘௤

(5.9)

At equilibrium, cathodic and anodic current densities are equal: ଓԦ = രଓ = ݅଴

(5.10)

where io is the exchange current of reaction (5.2) which, from Eqns. (5.6) and (5.7), can be written as: ݅଴ ൌ ࣠

௞் ௛

݁‫ ݌ݔ‬ቆ

ሬሬሬሬሬሬሬԦ # ିοீ ೐೜ ோ்

ቇ ܽெశ ൌ ࣠ ೞ೚೗

௞் ௛

exp ቆ

രሬሬሬሬሬሬሬ # ିοீ ೐೜ ோ்

ቇ ܽெ೘೐೟

(5.11)

Thus, charge transfer between electrode and solution builds, as discussed in Chapter 4, an electrical double layer in which charge accumulates until the electric potential difference is such that activation Gibbs free energy for reduction and oxidation reactions varies from initial to equilibrium values. Clearly, if the electric potential difference is modified, important changes of the activation energies will be produced, and this is the reason why reactions which cannot be performed through modifications in temperature or reactants concentration can occur at high rates by controlling the electrode potential. In effect, if one takes the electrode of reaction (5.2) in equilibrium and an electric potential difference value, ȟȲ ୑/ୗ , such that oxidation is favoured is applied, activation energies are modified as shown in Figure 5-3 and then: ଓԦ ‫ ا‬രଓ

(5.12)

Electrode Kinetics. I. Charge Transfer

57

Figure 5-3. Energy graph for system (5.2) when displaced from equilibrium by applying an electric potential difference.

Since the concentration of M+ in solution has not been changed, the electrochemical potential difference at the new electric potential difference can be written as: ‫כ‬ ‫כ‬ ‫כ‬ ߤெ െ ߤெ ൌ ࣠ ൫ȟȲ ୑/ୗ െ ȟȲ௘௤ ൯ శ െ ߤ௘ ష ೘೐೟ ೘೐೟ ೞ೚೗

(5.13)

where the difference in parenthesis is the already mentioned activation overpotential that is applied to the electrode: ߟ௔௖௧ ൌ ȟȲ ୑/ୗ െ ȟȲ௧௛௘௥௠

(5.14)

Thus, ߟ௔௖௧ indicates how much, and in what direction, the system is displaced from equilibrium.

Chapter Five

58

As shown in Figure 5-3 the activated complex theory states that application of the overpotential ߟ௔௖௧ affects the activation energy of both reactions: a fraction ߚ is responsible of the change in activation energy of the reduction reaction while the complementary fraction (1 - ߚሻ affects the activation energy of oxidation of M. For this reason, ߚ is called the symmetry factor of the electrochemical reaction, since it is a measure of the relative impact on the activation energy of both reactions. The current density flowing through the electrode/solution interphase when an overpotential Ș is applied is, then: ݅ ൌ ଓԦ െ ଓരሬ ൌ ࣠

௞் ௛

exp ൬

ሬሬሬሬሬሬሬሬԦ # ି௱ீ

௞்

ோ்

௛

൰ ܽெశ െ ࣠ ೞ೚೗

exp ൬

രሬሬሬሬሬሬሬሬ # ି௱ீ ோ்

൰ ܽெ೘೐೟ (5.15)

Assuming a linear relationship with overpotential: # ሬሬሬሬሬሬሬሬԦ ሬሬሬሬሬሬሬሬሬԦ ȟ‫ ܩ‬# ൌ ȟ‫ܩ‬ ௘௤ ൅ ߚ࣠ߟ

(5.16)

# രሬሬሬሬሬሬሬሬሬ രሬሬሬሬሬሬሬሬሬ ȟ‫ ܩ‬# ൌ ȟ‫ܩ‬ ௘௤ െ ሺ1 െ ߚሻ࣠ߟ

(5.17)

and substituting in Eqn. (5.11) we reach to: ݅ ൌ ଓԦ െ ശଓ ൌ ݅଴ ቄexp ቀ

ିఉ࣠ఎ ோ்

ቁ െ exp ቂ

ሺଵିఉሻ࣠ఎ ோ்

ቃቅ

(5.18)

which is known as the Butler – Volmer equation. Clearly, for negative overpotential values the first exponential prevails and reaction (5.2) occurs cathodically while the anodic reaction dominates in the opposing case. Experimental data taken from a large number of cases indicate that the value of ߚ is usually near 0.5. The experimental polarization curve of reaction (5.2), obtained by applying controlled overpotential values and measuring the resulting current, yields results such as those shown in Figure 5-4, which are consistent with the Butler – Volmer equation.

Electrode Kinetics. I. Charge Transfer

59

Figure 5-4: Polarization curve of reaction (5.2) a) Ș vs i; b) Ș vs log i.

5.2. Tafel equation and linear polarization The Butler – Volmer equation takes simpler forms within certain overpotential ranges. Thus, if |ߟ| ‫ب‬

ோ் ࣠

ൌ 25.7ܸ݉ ሺ25°‫ܥ‬ሻ

(5.19)

one of the exponentials in Eqn. (5.18) can be dismissed and for |ߟ| values of 100 mV or larger we have ݅ ൌ ݅௢ exp ቂ

– ஒ࣠஗ ோ்

ቃ

(5.20)

if (Ʉ ൏ 0) and ݅ ൌ െ݅௢ exp ቂ

ሺଵ–ஒሻ࣠஗ ோ்

ቃ

(5.21)

if (Ʉ ൐ 0). Taking decimal logarithms ߟൌቀ

ଶ,ଷ଴ଷோ் ఉ࣠

ቁ log ݅௢ െ ቀ

ଶ,ଷ଴ଷோ் ఉ࣠

ቁ log|݅|

(5.22)

for the cathodic case and ଶ,ଷ଴ଷோ்

ଶ,ଷ଴ଷோ்

ߟ ൌ െ ቂ ሺଵିఉሻ࣠ ቃ log ݅௢ ൅ ቂ ሺଵିఉሻ࣠ ቃ log|݅|

(5.23)

for the anodic reaction. Both equations respond to the empirical relationship found by Tafel:

Chapter Five

60

ߟ ൌ ܽ ൅ ܾ. log|݅|

(5.24)

which is, then, a particular case of the Butler – Volmer equation. From (5.18) it is clear that a depends on temperature and exchange current, while b, usually named the Tafel slope is a function of temperature. The Tafel slope value for a simple reaction of the type of (5.2) and a 0.5 symmetry factor is 118 mV at 25o C. On the other hand, at low overpotential values, say ሃߟሃ ൏ 10 mV, both exponentials may be expressed in terms of a Taylor series and, keeping the linear term: ݅ ൌ ݅௢ ቄ1 െ

ஒ࣠஗ ோ்

െ ቂ1 ൅

ሺଵ–ஒሻ࣠஗

࣠ఎ

ோ்

ோ்

ቃቅ ൌ െ݅௢

(5.25)

from which the polarization resistance is defined as: ఎ

ோ்

௜

௜೚ ࣠

ܴ௣ ൌ െ ൌ

(5.26)

Hence, within this overpotential range there is a linear dependence of current with overpotential, as shown in Figure 5-5

Figure 5-5: Linear polarization curve in the range – 10 mV < Ș < 10 mV.

Electrode Kinetics. I. Charge Transfer

61

5.3. The exchange current meaning From Eqn. (5.11) it comes that the exchange current io is a measure of the rate of the anodic and cathodic reactions at equilibrium and its value, for a given reaction, might considerably vary with the nature of the electrode on which the process occurs. In other words, comparison of the catalytic properties of different electrode materials for a given reaction, can be established by measuring the exchange current values of each case. This point is particularly important when the electrochemical reaction involves a sequence of two or more steps as, for example, is the case of the hydrogen evolution reaction: 2‫ ܪ‬ା ൅ 2݁ ି ՞ ‫ܪ‬ଶ

ሺ5.27ሻ

for which the exchange current values on different metals vary through several orders of magnitude, as shown in Figure 5-6 where ݅௢ is correlated with the adsorption energy of a hydrogen atom on the metal.

Figure 5-6: Exchange current for the hydrogen evolution reaction vs the H-M bonding energy for different metals.

62

Chapter Five

In Figure 5-7 polarization curves for an electrochemical reaction with different values of the exchange current are shown. As it can be seen, for high exchange current densities, such as io = 1 mA.cm-2, the polarization curve almost coincides with the y – axis which means that anodic and cathodic reactions are very fast and, consequently, the electrode is non – polarizable, as discussed in section 4.2. At lower values of the exchange current the overpotential needed for attaining a given current value increases; in particular, in the case of io = 10-6 mA.cm-2 a span of about 600 mV exists in which the current density is almost nil, and the behaviour is that of a highly polarizable electrode.

Figure 5-7: Polarization curves for a simple reaction with three exchange current values. a) 1 mA.cm-2; b) 10-3 mA.cm-2; 10-6 mA.cm-2.

5.4. Consecutive reactions: Reaction mechanisms Very few electrochemical reactions of practical interest occur through a single charge - transfer step as in Eqn. (5.2); in many cases several steps are needed in which electrochemical and chemical reactions take place and catalytic properties of the electrode material have an important impact. For instance, in the rather simple case of hydrogen evolution the reaction mechanism depends on the particular metal on which it occurs. For a platinum electrode the mechanism has two steps, the first one electrochemical and the second purely chemical:

Electrode Kinetics. I. Charge Transfer

63

ି ‫ ܪ‬ା ൅ ݁ெ ՜ ‫ܪ‬ሺܽ݀‫ݏ‬ሻ

(5.28)

2‫ܪ‬ሺܽ݀‫ݏ‬ሻ ՜ ‫ܪ‬ଶ

(5.29)

However, if the reaction proceeds on a copper electrode, both steps are electrochemical: ି ՜ ‫ܪ‬ሺܽ݀‫ݏ‬ሻ ‫ ܪ‬ା ൅ ݁ெ ି ՜ ‫ܪ‬ଶ ‫ܪ‬ሺܽ݀‫ݏ‬ሻ ൅ ‫ ܪ‬ା ൅ ݁ெ

(5.28) (5.29´)

In general, several reaction steps are needed, and the analysis of the complete process might be complex. However, in many cases it is possible to simplify the kinetic analysis of the process. This happens when one of the steps has a rate constant considerably lower than the rest, in which case this particular reaction step is called the rate-determining step of the whole process. An example of this is a global reaction that comprises n electrochemical reactions, each one being a single electron transfer: ‫ܣ‬ଵ ൅ ݁ ି ՞ ‫ܣ‬ଶ ‫ܣ‬ଶ ൅ ݁ ି ՞ ‫ܣ‬ଷ ………….. ‫ܣ‬௦ ൅ ݁ ି ՞ ‫ܣ‬௦ାଵ ‫ܣ‬௦ାଵ ൅ ݁ ି ՞ ‫ܣ‬௦ାଶ

(5.30)

…………………

‫ܣ‬௡ ൅ ݁ ି ՞ ܲ Let us consider in this example that the reduction of ‫ܣ‬௦ to ‫ܣ‬௦ାଵ is the ratedetermining step which means that: ݅௢,௦ ‫݅ ا‬௢,௞ஷ௦

(5.31)

Clearly, the rate of the global reaction will be determined by the rate of the step ‫ ݏ‬and, consequently: ݅ ൌ ݊݅௦

(5.32)

From (5.31) it is valid to assume that the remaining reaction steps can be considered to be in quasi – equilibrium. ሬሬሬሬሬሬሬԦ ଓ௞ஷ௦ ଓ௞ஷ௦ ൎ രሬሬሬሬሬሬሬ

(5.33)

Chapter Five

64

Now, the net current density of the ‫ – ݏ‬step is: ݅௦ ൌ ଓሬሬԦ௦ െ ଓരሬሬ௦

(5.34)

where: ሬሬሬԦ௦ ܿ஺ exp ቀିఉ ࣠୼అቁ ଓሬሬԦ௦ ൌ ࣠݇ ೞ

(5.35)

രሬሬሬ௦ ܿ஺ exp ቂሺଵିஒሻ ࣠୼ஏቃ ଓരሬሬ௦ ൌ ࣠݇ ೞశభ

(5.36)

ோ்

and ோ்

The ܿ஺ೞ value can be obtained from the quasi – equilibrium condition of the previous steps which allows to consider that the equilibrium condition established by Nernst equation can be applied. Then, for the first step we have: ǻȲ ൌ ȟȲଵ଴ െ

ோ் ࣠

ln ቀ

௖ಲమ ௖ಲభ

ቁ

(5.37)

wherefrom: ܿ஺మ ൌ ܿ஺భ exp ቂ

ି࣠ ሺ୼ஏି୼ஏబ భሻ ோ்

ቃ

(5.38)

In a similar way we obtain for step 2: ܿ஺య ൌ ܿ஺మ exp ቂ

ି࣠ ሺ୼ஏି୼ஏబ మሻ ோ்

ቃ ൌ exp ቂ

బ ࣠൫୼ஏబ భ ା୼ஏమ ൯

ோ்

ቃ ܿ஺భ exp ቀ

ିଶ࣠୼ஏ ோ்

ቁ

(5.39)

and, by the same token we arrive to: ܿ஺ೞ ൌ exp ቀ

బ ࣠ σೞషభ ೖసభ ୼ஏౡ

ோ்

ቁ ܿ஺భ exp ቂ

ିሺ௦ିଵሻ࣠୼ஏ ோ்

ቃ

(5.40)

If we define: ଴ ൌ ȟȲ୮୰ୣ

ଵ ௦ିଵ

଴ σ௦ିଵ ௞ୀଵ ȟȲ୩

(5.41)

and replace in (5.35) బ

ሬሬሬԦ௦ c୅ଵ exp ൤ିሺ௦ିଵሻ࣠൫୼ஏି୼ஏ೛ೝ೐൯൨ exp ቀିఉ ࣠୼అቁ ଓሬሬԦ௦ ൌ ࣠݇ ோ்

ோ்

(5.42)

Electrode Kinetics. I. Charge Transfer

65

Taking into account that when the global reaction is at equilibrium we have: ିሺ௦ିଵሻ࣠൫୼ஏ೟೓೐ೝ೘ ି୼ஏబ ೛ೝ೐ ൯

ሬሬሬԦ௦ c୅ଵ exp ൤ ݅Ԣ௢,௦ ൌ ࣠݇

ோ்

൨ exp ቂ

ିఉ ࣠୼అ೟೓೐ೝ೘ ோ்

ቃ

(5.43)

and then: ´ exp ቂ ଓሬሬԦ௦ ൌ ݅଴,௦

ିሺ௦ିଵାఉሻிఎ ோ்

ቃ

(5.44)

For രሬ ଓఫሬ, Eq. (5.36), can be expressed as a function of P by a similar procedure: ሺ௡ି௦ሻ࣠൫୼ஏି୼ஏబ ೛೚ೞ೟ ൯

ܿ஺ೞశభ ൌ ܿ௉ exp ൤

ோ்

൨

(5.45)

with ଴ ȟȲ୮୭ୱ୲ ൌ

ଵ ௡ି௦

σ௡௞ୀ௦ାଵ ȟȲ୩଴

(5.46)

which, with (5.36) gives ´ രሬሬ exp ቂ ଓ௦ ൌ ݅଴,௦

ିሺ௡ି௦ାଵିఉሻிఎ ோ்

ቃ

(5.47)

where: ሺ௡ି௦ሻ࣠൫୼ஏ೟೓೐ೝ೘ ି୼ஏబ ೛೚ೞ೟ ൯

ሺଵିఉሻ࣠୼ஏ೟೓೐ೝ೘

ோ்

ோ்

´ രሬሬሬ௦ ܿ௉ exp ൤ ൌ ࣠݇ ݅଴,௦

൨ exp ቂ

ቃ

(5.48)

and the Butler – Volmer equation for this system results: ݅ ൌ ݅௢ ቄexp ቂ

ିሺ௦ିଵାఉሻிఎ ோ்

ቃ െ exp ቂ

ିሺ௡ି௦ାଵିఉሻிఎ ோ்

ቃቅ

(5.49)

Since ݊ electrons are transferred each time that the rate-determining step occurs we have that the global exchange current is: ݅௢ ൌ ݊ ݅Ԣ௢,௦

(5.50)

and defining: ߙԦ ൌ ‫ ݏ‬െ 1 ൅ ߚ

(5.51)

ߙര ൌ ݊ െ ‫ ݏ‬൅ 1 െ ߚ

(5.52)

Chapter Five

66

we have: ݅ ൌ ݅௢ ቂexp ቀ

ሬሬԦ ࣠ ఎ ିఈ ோ்

ቁ െ exp ቀ

രሬሬ ࣠ ఎ ఈ ோ்

ቁቃ

(5.53)

where ߙԦ and ߙര are, respectively, the transfer coefficients of the cathodic and anodic reactions. This is the same as the Butler – Volmer equation, Ec. (5.18), with transfer coefficients replacing symmetry coefficients. It must be noted that transfer coefficients do not have the physical meaning of symmetry coefficients and, in fact, might take values greater than 1. Also, from Eqs. (5.51) and (5.52) it is seen that they sum up to ݊. Analogously if a cathodic overpotential |ߟ| ‫ܴܶ ب‬/࣠ is applied we have, as before: ݅ ൌ ݅௢ exp ቀ

ሬሬԦ ࣠ఎ ିఈ ோ்

ቁ

(5.54)

and ߟ ൌ

ିଶ,ଷ଴ଷோ் ሬఈ ሬԦ࣠

log ݅௢ െ

ଶ,ଷ଴ଷோ் ሬఈ ሬԦ࣠

log|݅|

(5.55)

and, again, Tafel behaviour results. Thus, observing a linear dependence of current logarithm with overpotential does not imply that the reaction occurs in a single step. At very low overpotential values, expanding the exponentials in series and taking the linear term we have: ࣠ఎ

݅ ൌ െ݅௢ ቀ ቁ ோ்

ఎ՜଴

(5.56)

Polarization curves can be experimentally obtained with different reactant concentrations and at various temperature values which allow, for a given global reaction, to define the following kinetic parameters at constant electrode potential or at constant overpotential. Cathodic reaction order: ሬሬሬԦప ൌ ቀ ‫݌‬

ப୪୭୥పԦ ப௖೔

ቁ ௝ஷ௜,௉,்,୼అ

Anodic reaction order

(5.57)

Electrode Kinetics. I. Charge Transfer

‫݌‬ప ൌ ቀ ശሬሬሬ

ப ୪୭୥ ശప ப௖೔

ቁ

67

(5.58)

௝ஷ௜,௉,்,୼అ

Apparent cathodic order ‫ݍ‬ ሬሬሬԦప ൌ ቀ

ப୪୭୥పԦ ப௖೔

ቁ

(5.59)

௝ஷ௜,௉,்,ఎ

Apparent anodic order ശሬሬሬ ‫ݍ‬ప ൌ ቀ

ப୪୭୥పശ ப௖೔

ቁ

(5.60)

௝ஷ௜,௉,்,ఎ

Apparent activation energy ‫ܧ‬௔# ൌ ܴ ቀ

ப୪୬୧

ቁ

ப் షభ ఎ,௖ ೔

(5.61)

It must be noted that absolute values for activation energy cannot be obtained, because the electrode potential of the reference hydrogen electrode is arbitrarily defined as zero at any temperature. In Appendix 1 an example on how the mechanism of a complex reaction can be determined from experimental data is developed.

5.5. Competitive reactions: Faradaic efficiency In industrial applications, either in the production of substances or in energy generation, it is very unusual that the involved process imply the existence of a single electrode reaction. Normally, several reactions take place simultaneously on the electrode and the first consequence is that current efficiency is affected, since the total current flow will be the sum of the partial currents which correspond to each of the occurring reactions. In the case of k electrode reactions taking place on a given electrode, each one involving nk electrons, the total current density is: ݅ ൌ σ௡௞ୀଵ ݅௞

(5.62)

݅௞ ൌ ߭௥‫ݎ࣠ כ‬௞

(5.63)

with

Chapter Five

68

where ‫ݎ‬௞ is the rate of the slow step of the kth reaction. Hence, the faradaic efficiency ߦଵி for reaction 1, which is that of practical interest, is below 100%: ߦଵி ൌ

௜భ

(5.64)

௜

Let us consider the simple case of two parallel electrode reactions: ‫ ܣ‬൅ ݊ଵ ݁ ି ՞ ‫ܤ‬

(5.65)

ܲ ൅ ݊ଶ ݁ ି ՞ ܳ

(5.66)

that simultaneously occur on a given inert electrode in an electrolyte with particular concentration values for the four species A, B, P and Q, at overpotential values which correspond to the Tafel zone for both reactions. If the standard electrode potential values for both reactions, ǻȌo1 and ǻȌo2, are known it is possible to calculate, by means of Nernst equation, the thermodynamic electrode potential for each reaction, ߂Ȳ௧௛௘௥௠ଵ and ߂Ȳ௧௛௘௥௠ଶ . Then, if the electrode/solution electric potential difference, ǻȌel, is controlled, the overpotential value for each reaction results: ߟଵ ൌ ȟȲ௘௟ െ ȟȲ௧௛௘௥௠ଵ

(5.67)

ߟଶ ൌ ȟȲ௘௟ െ ȟȲ௧௛௘௥௠ଶ

(5.68)

And the total current density is: ݅ ൌ ݅ଵ ሺߟଵ ሻ ൅ ݅ଶ ሺߟଶ ሻ

(5.69)

where ݅ଵ and ݅ଶ are obtained from Eqn. (5.55)

References Butler, John. 1924. The kinetic interpretation of the Nernst theory of electromotive force. Trans. Faraday Soc. 729-733 Erdey-Gruz, Tibor and Volmer, Max 1930. Zur Theorie der Wasserstoff Überspannung. Zetschrift Physik. Chem. 203-213 Eyring, Henry. 1935. The Activated Complex and the Absolute Rate of Chemical Reactions. Chemical Reviews. 65-77 Tafel, Julius. 1905. Über die Polarization bei katodischer Wasserstoffentwicklung. Zeitschrift Physik. 641-712

Electrode Kinetics. I. Charge Transfer

69

Volmer, Max 1928. Zur Theorie der Vorgänge an unpolarisierbaren Elektroden. Zetschrift Physik. Chem. 597-604

Bibliography Bockris, John, Reddy, Amulya and Gamboa – Adelco, María Modern electrochemistry. Vol. 2A. 2002. New York. Kluwer Academic Publishers Bard, Allen and Faulkner, Larry. Electrochemical Methods. 2001. New York. John Wiley & Sons



CHAPTER SIX ELECTROCHEMICAL KINETICS. II. MASS TRANSFER

6.1. The mass transfer overpotential Let us consider the following electrochemical reaction, that takes place on a plane electrode: ܱሺܽ‫ݍ‬ሻ ൅ ߭ ‫ ି ݁ כ‬՞ ܴሺܽ‫ݍ‬ሻ

(6.1)

where O and R are dissolved species. In order that this reaction occurs the following sequence of events must happen: x The dissolved reactant species, which is in the bulk, must reach the reaction plane which is at a distance to the electrode that makes electron transfer possible. x Electron transfer occurs and a product species is obtained at the reaction plane. x The product species must leave the reaction plane and reach the bulk. At low current density values, reaction rate is low and thermal agitation is sufficient to permit that species concentration at the reaction plane be the same as in the bulk. This case has been discussed in the previous chapter and only activation overpotential is required. As reaction rate increases, thermal agitation is not sufficient to keep concentrations at the reaction plane equal to those in the solution bulk. Hence concentration gradients, whose characteristics depend on the system hydrodynamics, develop along the direction normal to the electrode plane, affecting the reaction rate. In order to overcome the concentration gradient, the electrode potential must be increased in order to satisfy the requirements of the mass transfer process: this increment is the mass transfer overpotential.

Electrochemical Kinetics. II. Mass Transfer

71

If the total applied potential is continuously increased, the rate at which the reacting species is consumed is so high that its concentration on the reaction plane is virtually zero, which is tantamount to say that every O species that reaches the reaction plane is instantaneously transformed into R. Under these conditions current density attains the maximum value that is compatible with the specific hydrodynamic conditions; this current value is called the limiting current density, ݅௅ . Figure 6-1 describes the reactant concentration gradient for increasing overpotential values and Figure 6-2 shows the polarization curve shape over a large range of current densities.

Figure 6-1: Reactant concentration gradient for increasing overpotential values.

Chapter Six

72

Figure 6-2: Polarization curve shape including the activation and the mass-transfer controlled regions

Consequently, the total electrode overpotential at a given current density value will be given by the contribution of the activation overpotential, discussed in the previous chapter, and the mass transfer overpotential. It must be remembered that, as shown in Chapter 2, the electrolyte resistance generates an ohmic overpotential that affects the total power to be applied on, or obtained from, a cell. All these contributions must be considered and the approach usually employed, when carrying out the preliminary analysis of an electrochemical cell, is to describe the system in terms of purely electrical circuits as discussed in Appendix 2. Turning now to the specific consideration of the mass transfer overpotential we write the current density for a process with nil activation overpotential as: ݅ ൌ ࣠ σ௝ ‫ݖ‬௝ Ȱ௝

(6.2)

where Ȱ௝ is the molar flux of species j that, generally, can be written as the sum of three contributions: convection, which is the consequence of applied mechanical forces on the fluid; diffusion, which is driven by concentration gradients, and migration, due to the electric field:

Electrochemical Kinetics. II. Mass Transfer

Ȱ௝ ൌ ߮௝ௗ ൅ ߮௝௠ ൅ ܿ௝ ‫ݒ‬Ԧ

73

(6.3)

In this equation ߮௝ௗ is the molecular flow due to diffusion, ߮௝௠ is the molecular flow due to migration and cj‫ݒ‬Ԧ is the convective contribution being cj the concentration of species j and ‫ݒ‬Ԧ the fluid velocity. From the equations of irreversible thermodynamics, it is shown that both ߮௝ௗ and ߮௝௠ depend on the electrochemical potential gradient (Ibl, 1983) through: ߮௝ௗ ൅ ߮௝௠ ൌ

ି஽ೕ ௖ೕ ோ்

஽ೕ ௭ೕ ௖ೕ ࣠ ሬሬሬሬሬሬሬԦ ሬሬሬሬሬሬԦ ‫ߤ׏‬ఫ‫ כ‬ൌ െ‫ܦ‬௝ ሬሬሬሬሬԦ ‫׏‬cఫ െ ‫׏‬Ȳ ோ்

(6.4)

where the diffusional contribution is given by the term including the concentration gradient, and the migrational contribution is described by the term that contains the electric potential gradient. For the sake of clearness, it has been assumed that activity coefficients for all species are equal to 1. From (6.2) it is possible to write, then: ݅ ൌ ࣠ σ௝ ‫ݖ‬௝ Ȱ௝ ൌ െ࣠ σ௝ ‫ݖ‬௝ ‫ܦ‬௝ ሬሬሬሬሬԦ ‫׏‬cఫ െ Ɉ ሬሬሬሬሬሬԦ ‫׏‬Ȳ ൅ ‫ݒ‬Ԧ࣠ σ௝ ‫ݖ‬௝ ܿ௝

(6.5)

where: ߢ ൌ ࣠ଶ

σೕ ௭ೕమ ஽ೕ ௖ೕ ோ்

(6.6)

is the solution conductivity.

6.2. The purely diffusional case In order to study the purely diffusional contribution, it is necessary to define experimental conditions which permit the dismissal of migrational and convective effects and, in this respect, it must be said that the validity of such experiments must be carefully evaluated in each case. Regarding migration effects it is to be recalled that, as it has been mentioned in Chapter 2, the contribution of an ionic species j to electrolyte conductance depends on the ionic transport number, IJj, which, from Eqn. (2.21), can be diminished by increasing the number and concentration of other ionic species that do not intervene in the electrode reaction. These ionic compounds, called supporting electrolytes must contain anions and cations

Chapter Six

74

that are stable within the electric potential range in which the experiments are performed. Alkaline perchlorates are among the most frequently used and its concentration should be, at least, one order of magnitude above the concentration of the reacting species. By its side, convective effects might be reduced by establishing conditions that guarantee that there are no mechanical forces applied on the fluid and, thus, forced convection is eliminated. However, as the experiment proceeds and concentration gradients are established, density gradients will be generated which induce mechanical movements in the fluid. This contribution, called natural convection, interferes with the diffusional measurements and, consequently, purely diffusional studies can only be developed over short time periods. Assuming that migrational and convective contributions can be dismissed, let us consider reaction (6.1) in purely diffusional conditions and, particularly, we will be interested in the case in which concentration of O and R on the electrode, ܿை௘ and ܿோ௘ , differ from those in the solution bulk, ܿைஶ and ܿோஶ . At equilibrium, total current is zero and, as described in Chapter 5, anodic and cathodic currents are equal to the exchange current: ݅଴ ൌ ߭ ‫࣠ כ‬

௞் ௛

exp ቆ

ሬሬሬሬሬሬሬԦ # ିοீ ೐೜

௞்

ோ்

௛

ቇ ܿைஶ ൌ ߭ ‫࣠ כ‬

exp ቆ

രሬሬሬሬሬሬሬ # ିοீ ೐೜ ோ்

ቇ ܿோஶ

(6.7)

However, when an overpotential ߟ is applied species concentrations on the electrode are modified and cathodic and anodic currents are different leading to the net current value: ݅ ൌ ߭‫࣠כ‬

௞் ௛

exp ൬

ሬሬሬሬሬሬሬሬሬԦ # ି୼ீ

௞்

ோ்

௛

൰ ܿை௘ െ ߭ ‫࣠ כ‬

exp ൬

രሬሬሬሬሬሬሬሬሬ # ି୼ீ ோ்

൰ ܿோ௘

(6.8)

From these equations and considering equations (5.16) and (5.17) we have: i = ݅଴ ൜

೐ ௖ೀ

ಮ ௖ೀ

݁‫ ݌ݔ‬ቂെ

ఈజ‫࣠ כ‬ఎ ோ்

ቃെ

ಮ ௖ೃ ೐ ௖ೃ

݁‫ ݌ݔ‬ቂ

ሺଵିఈሻజ‫࣠ כ‬ఎ ோ்

ቃൠ

(6.9)

Under mass transfer control the applied overpotential is significant and one of the exponentials can be dismissed. Therefore, for the case of a cathodic reaction we have: ߟൌ

ோ் ఈజ‫࣠ כ‬

݈݊݅௢ െ

ோ் ఈజ‫࣠ כ‬

݈݊݅ െ

ோ் ఈజ‫࣠ כ‬

݈݊

ಮ ௖ೀ ೐ ௖ೀ

(6.10)

Electrochemical Kinetics. II. Mass Transfer

75

where the first two terms correspond to the activation overpotential and the last one is the mass transfer overpotential for the purely diffusional case: ߟ஽ ൌ െ

ோ் ఈజ‫࣠ כ‬

ln ൬

ಮ ௖ೀ ೐ ௖ೀ

൰

(6.11)

Assuming a linear concentration gradient in the space between the electrode and the nearest plane at which the concentrations of ܱ and ܴ are the same than in the bulk we have, taking ݀ to be the distance between those planes: ߮ைௗ ൌ

೐ ಮ ஽ೀ ൫௖ೀ ି௖ೀ ൯

(6.12)

ௗ

߮ோௗ ൌ െ

೐ ಮ ஽ೃ ൫௖ೃ ି௖ೃ ൯

(6.13)

ௗ

The maximum value for the concentration gradient corresponds to the case ܿை௘ ൌ 0 in which case, and assuming constant values for ݀ and ‫ܦ‬ை , the limiting current in the cathodic direction can be written as: ݅௅ ൌ ߭ ‫࣠ כ‬

ಮ ஽ೀ ௖ೀ

(6.14)

ௗ

Analogously, for the anodic reaction: ݅௅ᇱ ൌ െ߭ ‫࣠ כ‬

ಮ ஽ೃ ௖ೃ

(6.15)

ௗ

Under these assumptions, and remembering that the current sign is positive for the cathodic reaction and negative for the anodic case, we have: ೐ ௖ೀ

ಮ ௖ೀ

ൌ1െ

௜

(6.16)

௜ಽ

and ೐ ௖ೃ

ಮ ௖ೃ

ൌ1െ

௜

(6.17)

௜ಽᇲ

Wherefrom mass transfer overpotential can be written as: ߟ஽ ൌ

ோ் ఈజ‫࣠ כ‬

ln ሺ1 െ

௜ ௜ಽ

ሻ

when reaction (6.1) proceeds cathodically, and as:

(6.18)

Chapter Six

76

ߟ஽ ൌ െ

ோ் ఈజ‫࣠ כ‬

ln ሺ1 െ

௜ ௜ಽᇲ

ሻ

(6.19)

in the anodic case. In the general case in which stoichiometric coefficients in Eqn. (6.1) are not 1 the mass transfer overpotential under purely diffusional conditions is: ߟ஽ ൌ െ

ோ் జ‫࣠ כ‬

݅ ܱ߭

ln ቈቀ1 – ቁ ݅ ‫ܮ‬

݅

ܴ߭

ቆ1 െ Ԣ ቇ ቉ ݅‫ܮ‬

(6.20)

Figure 6-3 shows the polarization curves obtained when the mass transfer overpotential is included as compared when it is dismissed.

Figure 6-3: Polarization curves for reaction (6.1) with ȟȲ௧௛௘௥௠ ൌ 0,25ܸ. The dashed line corresponds to the Butler – Volmer equation. The solid line includes the activation and mass transfer overpotentials.

As seen from this Figure, when ȟȲ ՜ െλ (ߟ ՜ െλ), the (cathodic) limiting current is ݅ ՜ ݅௅ , while for ȟȲ ՜ ൅λ (ߟ ՜ ൅λ), the (anodic) limiting current is ݅ ՜ ݅௅ᇱ .

Electrochemical Kinetics. II. Mass Transfer

77

In the cathodic case the boundary conditions are: ՜ ܿை௘ ൌ ܿைஶ ՜

݅ ൌ 0 ݅ ൌ ݅௅ ՜

ܿை௘ ൌ 0

՜

ߟ஽ ൌ 0 ߟ஽ ൌ െ λ

Nernst proposed to replace d by the parameter įN, which considers the thickness of the diffusion layer as determined by the intersection of the tangent of the concentration curve, under limiting current conditions, at the electrode with the bulk concentration line, as shown in Figure 6-4.

Figure 6-4: Nernst´s diffusion layer.

Nernst´s diffusion layer width, įN, approximately indicates the space in which the concentration profiles are developed, and its value can be estimated from the limiting current density: ߜே ൌ െ

జ‫כ‬ జೀ

࣠‫ܦ‬ை ቀ

ಮ ௖ೀ

௜ಽ

ቁ

(6.21)

78

Chapter Six

The purely diffusional model can be illustrated by considering reaction (6.1) on a plane electrode through which a fixed current value, ݅௚ , circulates. At time t = 0 we have ܿை௘ = ܿைஶ and ܿோஶ ൌ 0 and these concentration values will evolve as is qualitatively shown in Figure 6-5

Figure 6-5: Time evolution of concentration profiles at increasing reaction time for a) Reactant O; b) Product R with (1) t = 0; (2) t = t1; (3) t = t2 > t1; (4) t = t3 > t2.

Electrochemical Kinetics. II. Mass Transfer

79

In this one - dimensional diffusional case the flux of species j, whose stoichiometric coefficient is ߭௝ , can be written, according to Fick’s first law as: ߮௝ௗ ൌ െ‫ܦ‬௝

డ௖ೕ

(6.22)

డ௫

and, since the mass – balance must be satisfied, Fick´s second law is valid, and we have: డ௖ೕ డ௧

ൌ ‫ܦ‬௝

డమ ௖ೕ

(6.23)

డ௫ మ

with boundary conditions: െ‫ܦ‬௝

డ௖ೕ

ቚ

డ௫ ௫ୀ଴

௜

ൌ ߭௝

at ‫ ݔ‬ൌ 0 for ‫ ݐ‬൐ 0

జ‫࣠ כ‬

(6.24)

ܿ௝ = ܿ௝ஶ at ‫ ݔ‬՜ λ

(6.25)

ܿ௝ = ܿ௝ஶ for all ‫ ݔ‬at ‫ ݐ‬ൌ 0

(6.26)

These equations can be solved yielding: ܿ௝ ሺ‫ݔ‬, ‫ݐ‬ሻ ൌ ܿ௝ஶ ൅ ߭௝

ସ஽ೕ ௧

௜ జ‫࣠ כ‬஽ೕ

൥ට

గ

݁

ೣమ రವೕ ೟

ି

െ ‫ ݔ‬erfc ൬

௫ ඥସ஽ೕ ௧

൰൩

(6.27)

where: ௫

ଶ

݁‫݂ܿݎ‬ሺ‫ݔ‬ሻ ൌ 1 െ ‫׬‬଴ exp ሺെ‫ ݖ‬ଶ ሻ݀‫ݖ‬ గ

(6.28)

With the stoichiometry of Eq. (6.1), the concentration of O at the electrode surface is: ܿை ሺ0, ‫ݐ‬ሻ ൌ ܿைஶ െ

௜ జ‫࣠ כ‬

ସ௧

ටగ஽

ೀ

(6.29)

which is Sand´s equation. It can be seen that species O would be exhausted at time t = ߠ ߠൌ

ಮ గ஽ೀ జ‫࣠ כ‬௖ೀ

ସ

ቀ

௜

ଶ

ቁ

(6.30)

Chapter Six

80

Thus, for time values below ߠ ܿை at x = 0 is: ௧

ܿை ሺ0, ‫ݐ‬ሻ ൌ ܿைஶ ቆ1 െ ට ቇ ఏ

(6.31)

Substituting this in (6.18) the mass transfer overpotential is obtained as: ߟ஽ ൌ

ோ் జ‫࣠ כ‬

௧

ln ቆ1 െ ට ቇ ఏ

(6.32)

whose dependence on time is shown in Figure 6-6.

Figure 6-6: Mass transfer overpotential variation with time for the purely diffusional case.

At large time values, however, this result is not valid because, as mentioned, density gradients will be generated and natural convection will begin to operate. Since natural convection is detected over time intervals in the order of some seconds, the obtained equations are strictly valid at very short times.

6.3. Migration effects If it is not possible to disregard migration effects, we must consider its contribution to ionic conduction. In a first approximation, we will suppose

Electrochemical Kinetics. II. Mass Transfer

81

that the electric potential gradient in the solution bulk can be approximated by means of Ohm’s law (i.e., dismissing the second term at the right-hand side of Eq. (6.5) and recalling that the effect of convection in this equation is nil). Further, let us assume, for the sake of simplicity, that ‫ݖ‬ை ൌ 1, in which case: ݅ ൌ െ ࣠ ‫ܱܦ‬

߲ܱܿ ቚ ൅ ߲‫ݔ ݔ‬ൌ0

ܱ߬ ݅

(6.33)

where ߬ை is the transport number of species O. Eqn. (6.33) can be reordered: ‫ܦ‬

߲ܱܿ ቚ ܱ ሻ ߲‫ݔ ݔ‬ൌ0

െ ࣠ ሺ1െܱ߬

(6.34)

that, for limiting current density conditions becomes: ݅௅ ൌ ࣠

஽ೀ ሺଵିఛೀ ሻ

ܿλ ܱ ߜܰ

(6.35)

6.4. Convective effects As it has been pointed above, considering mass transport in an electrochemical reaction under a purely diffusional approach is not realistic and in all electrode processes of practical interest mass transfer must be analysed as the consequence of diffusional and convective characteristics of the system while, in some cases, migration effects must also be taken into consideration. Convective effects on electrolyte flow are not amenable to a unique description and its exact mathematical representation is, in many cases, impossible because of their dependence on mechanical forces applied on the fluid and on concentration and temperature gradients which affect local density. Forced convection is defined by the hydrodynamic characteristics of the specific system under study and in its theoretical treatment the hydrodynamic layer, within which velocity profiles are established, plays a central role. The Nernst diffusion layer, in turn, is central in defining the diffusional contribution to ionic flows. In order to show the difficulties found in obtaining an exact mathematical treatment of the diffusion – convection problem, two simple cases will be analysed for reaction (6.1): the plane plate semi-infinite electrode and the rotating disk electrode.

Chapter Six

82

6.4.1. The semi – infinite plate plane electrode Let us consider an electrode which is a plate plane in the y – z plane of dimensions y = ࣦ; z = ’ on which reaction (6.1) occurs with the electrolyte flowing in the y direction at velocity v ሬԦ, as shown in Figure 6-7

Figure 6-7: Hydrodynamic condition on a semi – infinite plate plane electrode.

In order to find an exact solution of the problem, the following boundary conditions are assumed: 1) 2) 3) 4) 5) 6) 7)

Steady state. Laminar fluid flow. Incompressible fluid, i.e., ߘሬԦ. ‫ݒ‬Ԧ = 0 Infinite width on z implies vz = 0 Isothermal and isobaric conditions. Only the component y intervenes in the momentum balance. Dependence of the y component of ‫ݒ‬Ԧ on ‫ ݕ‬is not relevant, which means that

డ௩೤ డ௫

൐൐

డ௩೤ డ௬

8) Convection on the electrode occurs only on the y direction and sliding is zero, i.e., |‫ݒ‬Ԧ| = 0 at x = 0 9) Diffusion is perpendicular to the electrode. 10) The reaction proceeds at limiting current density. With these conditions the diffusion – convection equation for O is: ‫ݒ‬௫ ቀ

డ௖ೀ

డ௖

డమ ௖

ቁ ൅ ‫ݒ‬௬ ቀ డ௬ೀ ቁ ൌ ‫ܦ‬ை ሺ డ௫మೀ ሻ డ௫

(6.36)

The functions that describe ‫ݒ‬௫ and ‫ݒ‬௫ in the hydrodynamic boundary layer were obtained by Blasius (Blasius, 1908) :

Electrochemical Kinetics. II. Mass Transfer

‫ݒ‬௬ ൌ ‫ݒ‬௫ ൌ

83

ಮ௫ ௩೤

(6.37)

ఋ೓ ఔ య ఋ೓

‫ݔ‬ଶ

(6.38)

In these equations ‫ݒ‬௬ஶ is the y component outside the hydrodynamic boundary layer, ߥ is the kinematic viscosity and ߜ௛ is the hydrodynamic boundary layer width given by: ߜ௛ ൌ 5,2 ሺ

ఔ

ಮ ௩೤

ሻଶ ඥ‫ݕ‬

(6.39)

With these parameters, and with the boundary conditions: at x = 0ܿை ൌ ܿைஶ

ܿை ൌ 0

for x ՜ λ (6.40)

ܿை ൌ ܿைஶ

at ‫ ݕ‬ൌ 0

Eqn. (6.35) has the exact solution (Fahidy, 1985): భ

௑

ఔ య

ఔ

ܿை ൌ 0.678 ቀ ቁ ܿைஶ ‫׬‬଴ expሺെ0.22 ߞ ଷ ሻ ݀ߞ ஽ ஽

(6.41)

where: ܺ ൌ ½ሺ

୴ಮ ౯ ଵ/ଶ ௫ ሻ ஝ ξ௬

(6.42)

The dependence of ܿை with x is: భ

భ

డ௖ೀ డ௫

ఔ య ୴ಮ ఔ ౮ మ ቁ exp ቂെ0.22 ቀ஽ቁ ܺ ଶ ቃ ܿைஶ ‫ି ݕ‬ଵ/ଶ ௩

ൌ 0.339 ቀ ቁ ቀ ஽

(6.43)

and, therefrom, the limiting current density results: భ

݅௅ ሺ‫ݕ‬ሻ ൌ ߭ ‫ܦ࣠ כ‬ை ቀ

ఔ య ୴ಮ ౯

డ௖ೀ

ቁ డ௫

௫ୀ଴

ଵ/ଶ

ൌ 0.339߭ ‫ܦ࣠ כ‬ை ቀ ቁ ቀ ቁ ஽ ఔ

and the Nernst layer width is:

ܿைஶ ‫ି ݕ‬ଵ/ଶ

(6.44)

Chapter Six

84 ஽ ଵ/ଷ

ߜே ሺ‫ݕ‬ሻ ൌ 2.95 ቀ ቁ ఔ

൬

ఔ ୴ಮ ౯

ଵ/ଶ

൰

‫ି ݕ‬ଵ/ଶ

(6.45)

The mean values for both quantities are: ଵ/ଶ ఔ ଵ/ଷ ௩ ಮ ೣ

ࣦ

ଓഥ௅ ൌ ࣦ ିଵ ‫׬‬଴ ݅௅ ሺ‫ݕ‬ሻ݀‫ ݕ‬ൌ 0.678߭ ‫ܦ࣠ כ‬ை ቀ ቁ ஽

ଵ/ଷ ௩ ಮ ଵ/ଶ ೣ

ࣦ ஽ തതത ߜே ൌ ࣦ ିଵ ‫׬‬଴ ߜே ሺ‫ݕ‬ሻ݀‫ ݕ‬ൌ 2 ቀ ቁ ఔ

ቀ ௩ࣦ ቁ

ቀ

௩

ቁ

ܿைஶ

(6.46) (6.47)

and, from (6.39): തതതത ఋಿ തതതത ఋ ೓

஽ ଵ/ଷ

ൎ 0.6 ቀ ቁ ఔ

(6.48)

Thus, for this particular electrode the relationship between the hydrodynamic boundary layer and Nernst´s layer is independent of fluid velocity. Figure 6-8 shows the dependence of both parameters on y.

Electrochemical Kinetics. II. Mass Transfer

85

Figure 6-8: Hydrodynamic boundary layer and Nernst´s layer width for the semiinfinite plate plane electrode as a function of 100(y/ ࣦሻ.

6.4.2. The rotating – disk electrode The rotating-disk electrode is built by inserting a cylindrical piece of the chosen metal, with radius ro, in a larger cylinder of an isolating material, with radius R >> ro, that rotates at a controlled speed. Figure 6-9 illustrates schematically this device.

Chapter Six

86

Figure 6-9: The rotating-disk electrode.

The diffusion – convection equation for the ion flow in this system: డ௖ೀ

ሬሬሬሬሬሬሬԦ ൌ ‫ܦ‬ை ߘ ଶ ܿை െ v ሬԦ ή ߘܿ ை

డ௧

(6.49)

was solved by Levich (Levich,1962) employing cylindrical coordinates: డ௖ೀ డ௧

ൌ ‫ܦ‬ை ቂ

డమ ௖ೀ డ௥ మ

൅

ଵ డ௖ೀ ௥ డ௥

൅

ଵ డమ ௖ೀ ௥ మ డఏమ

൅

డమ ௖ೀ డ௭ మ

ቃെቀ

୴౨ డ௖ೀ డ௥

൅

୴ಐ డ௖ೀ ୰ డఏ

൅ v୸

డ௖ೀ డ௭

ቁ (6.50)

In solving this equation, it must be considered that, because of the symmetry of the problem, ion concentration will not depend on the rotating angle and, since R >> ro, it might be assumed that concentration is independent of r. Thus, when the system works at steady-state conditions, concentration is independent of time and Eqn. (6.50) yields: ‫ܦ‬ை

ௗ మ ௖ೀ ௗ௭ మ

ൌ ‫ݒ‬௭

ௗ௖ೀ

(6.51)

ௗ௭

From the theory of hydrodynamics, the z component of velocity is: ‫ݒ‬௭ ൌ െሺ0.51

ఠయ ఔభ/మ

െ

௭మ ଶ

ሻ

(6.52)

where Ȧ is the rotational speed and ߥ is the kinematic viscosity. Hence:

Electrochemical Kinetics. II. Mass Transfer ௗ మ ௖ೀ ௗ௭ మ

൅ ቀ0.51

ఠయ ఔభ/మ

௭మ

െ

ଶ

ቁ ‫ܦ‬ைିଵ

ௗ௖ೀ ௗ௭

87

ൌ0

(6.53)

The boundary conditions in this system, if reduction occurs with rate constant ݇ ൌ ݇ሺߟሻ and oxidation is negligible, are: ݇ܿை ൌ ‫ܦ‬ை

ௗ௖ೀ

at ‫ ݖ‬ൌ 0

ௗ௭

ܿை ൌ ܿைஶ

(6.54)

at ‫ ݖ‬՜ λ

(6.55)

Employing Eqns. (6.51) – (6.54) Eqn. (6.50) can be solved, and the result is: ௭

ܿை ሺ‫ݖ‬ሻ െ ܿை ሺ‫ ݖ‬ൌ 0ሻ ൌ ‫׬ ܭ‬଴ exp ቀെ

௔௭ య ଷ

ቁ ݀‫ݖ‬

(6.56)

where ܽ ൌ 0.51‫ܦ‬ைିଵ ߱ ଷ/ଶ ߥ ିଵ/ଶ and ‫ ܭ‬is: ିଵ/ଷ

‫ ܭ‬ൌ 0.620 ‫ܦ‬ை

߱ଵ/ଶ ߥ ିଵ/଺ ሺܿைஶ െ ܿை௭ୀ଴ ሻ

(6.57)

and the concentration dependence on ‫ ݖ‬is, then: ௗ௖ೀ ௗ௭

௭య

ൌ ‫ ܭ‬exp ቀെܽ ቁ ଷ

(6.58)

Consequently, concentration of species ܱ at the electrode (‫ ݖ‬ൌ 0) is: మ/య

ܿை ሺ‫ ݖ‬ൌ 0ሻ ൌ

ಮ ଴,଺ଶ଴ ஽ೀ ఠభ/మ ఔషభ/ల ௖ೀ

(6.59)

మ/య

௞ା ଴,଺ଶ଴ ஽ೀ ఠభ/మ ఔషభ/ల

and the current density at steady-state condition can be written as: మ/య

݅ ൌ ࣠. ݇ܿை ሺ‫ ݖ‬ൌ 0ሻ ൌ ࣠. ݇

ಮ ଴.଺ଶ଴ ஽ೀ ఠభ/మ ఔషభ/ల ௖ೀ మ/య

௞ା ଴.଺ଶ଴஽ೀ ఠభ/మ ఔ షభ/ల

(6.60)

In the case of a very fast reaction ݇ ՜ λ, i.e., ܿை ሺ‫ ݖ‬ൌ 0ሻ ՜ 0 limiting current density is: ଶ/ଷ

݅௅ ൌ ࣠൫0.620 ‫ܦ‬ை ߱ଵ/ଶ ߥ ିଵ/଺ ܿைஶ ൯ and the current density is given by:

(6.61)

Chapter Six

88

݅ ൌ ݇. ࣠

ಮ ௖ೀ ௜ಽ

(6.62)

ಮା ௜ ௞.࣠௖ೀ ಽ

Defining ݅ ஶ ൌ ࣠݇ܿைஶ we finally get the Koutecky – Levich equation: ଵ ௜

ൌ

ଵ ௜ಮ

൅

ଵ ௜ಽ

ൌ

ଵ ಮ ࣠௞௖ೀ

൅

ଵ మ భ భ య ఠ మ ఔ షల ௖ ಮ ࣠଴.଺ଶ଴ ஽ೀ ೀ

(6.63)

The first term in Eqn. (6.63) only depends on the rate constant and is dominant when the reaction rate is controlled by the reaction activation energy. The second term becomes important when mass transfer is relevant. Thus, if current density is measured at a given overpotential with different rotational speed values, the reaction rate constant is calculated by plotting ݅ ିଵ as a function of ߱ ିଵ/ଶ . Figure 6-10 (a) displays a set of polarization curves obtained at different rotational speeds for a reaction of the type of (6.1) with an electrode equilibrium potential of – 0.250 V and Figure 6-10 (b) shows the Koutecky – Levich lines for three electrode potential values.

Electrochemical Kinetics. II. Mass Transfer

89

Figure 6-10: a) Polarization curves at different rotational speed values; b) Koutecky – Levich plots for electrode potential values 0 mV, 50 mV and 350 mV.

Figure 6.10 a) shows that at an electrode potential ǻȌ = 350 mV (Ș = 600 mV) reaction rate is controlled by mass transfer, while at ǻȌ = 0 mV (Ș = 250 mV) and at ǻȌ = 30 mV (Ș = 300 mV) both activation and mass transport control are significant. Figure 6.10 b) permits the calculation of the rate constant at a given overpotential, from the intercept of the corresponding line. Finally, the slope of the iL vs ߱ ିଵ/ଶ line allows to obtain the diffusion coefficient of the reacting species.

6.5. The mass transfer coefficient As can be concluded from the discussed cases, exact solution of the diffusion – convection equation is only possible in some cases but, normally, it is necessary to resort to approaches that imply correlation of variables which are significant in the specific case under study. A strategy that has proved adequate in searching for correlations of practical use is to take the purely diffusional case as standpoint. Thus, for an electrode reaction involving the transfer of ȣ* electrons and stoichiometric coefficients different from 1, Eqn. (6.12) becomes:

Chapter Six

90

݅ ൌ െ

జ‫כ‬ జೀ

࣠‫ܦ‬ை

డ௖ೀ

ቚ

డ௫ ௫ୀ଴

ൌെ

జ‫כ‬ జೀ

࣠‫ܦ‬ை

୼௖ೀ

ቚ

୼௫ ௫ୀ଴

ൌെ

జ‫כ‬ జೀ

࣠‫ܦ‬ை

೐ ಮ ି ௖ೀ ௖ೀ

ఋಿ

(6.64)

Defining the mass transfer coefficient ݇௠ as: ݇௠ ൌ

஽ೀ

(6.65)

ఋಿ

it is possible to describe current density as: ݅ ൌ െ݇௠

జ‫כ‬ జೀ

࣠ሺܿைஶ െ ܿை௘ ሻ

(6.66)

Consequently, limiting current density results: ݅௅ ൌ െ݇௠

జ‫כ‬ జೀ

࣠ܿைஶ

(6.67)

These expressions give a convenient formalization for calculating current density values, provided that km data can be obtained for diffusion – convection conditions. By its hand, from a large number of experimental measurements, it has been proposed that the mass transfer overpotential can be estimated, in initial stages of reactor design as:

ߟ௠௧ ൌ െ

௜ జೀ

ோ் ఈ೘೟ జ‫࣠ כ‬

݈݊ ൤ቀ1 – ቁ ௜ಽ

௜ జೃ

ቀ1 െ ᇲ ቁ ൨ ௜ಽ

(6.68)

where ߙ௠௧ is a correction factor with respect to the purely diffusional case and whose value is usually around 0.5. From what has been discussed it is clear that ݇௠ will be strongly dependent on the electrode geometry and the electrolyte flow conditions. In general, current and ݇௠ values will depend on the considered point of the electrode surface and, for the case of the plate plane electrode: ݅ሺ‫ݕ‬ሻ ൌ െ݇௠ ሺ‫ݕ‬ሻ

జ‫כ‬ జೀ

࣠ሾܿைஶ െ ܿை௘ ሺ‫ݕ‬ሻሿ

(6.69)

This is a further difficulty that, in preliminary design considerations, is circumvented by employing mean km values which for the case of the plate plane electrode of Figure 6.5 is: ଵ ࣦ തതതതത ݇௠ = ‫׬‬଴ ݇௠ ሺ‫ݕ‬ሻ݀‫ݕ‬ ࣦ

(6.70)

Electrochemical Kinetics. II. Mass Transfer

91

Values of the mass transfer coefficient have been obtained through the work of many research groups that, over several decades, have developed experimental work in which the obtained തതതതത ݇௠ values were correlated with critical dimensions of the system, as well as with its hydrodynamic characteristics. These correlations are experimentally obtained for a specific electrode geometry and well-defined hydrodynamics by measuring the limiting current of a very fast reaction such as: ସି ି ‫݁ܨ‬ሺ‫ܰܥ‬ሻଷି ଺ ൅ ݁ ՞ ‫݁ܨ‬ሺ‫ܰܥ‬ሻ଺

(6.71)

and systematically modifying different parameters, such as electrode dimensions, flow velocity, electrolyte viscosity, temperature, etc. and the obtained results are correlated employing dimensionless numbers. Thus, removing from now on the average bar, the Sherwood number is defined as: ௜ಽ ࣦ ഔ‫כ‬ ࣠஽௖ ಮ หഔೀ ห

݄ܵ ൌ

ൌ ݇௠

ࣦ ஽

(6.72)

where ࣦ is a characteristic length of the system and the Sherwood number is correlated with other dimensionless numbers: Reynolds ܴ݁ ൌ

|௩|ࣦ ఔ

(6.73)

Schmidt: ܵܿ ൌ

ఔ

(6.74)

஽

Péclet: ܲ݁ ൌ ܴ݁. ܵܿ

(6.75)

Grashof: ‫ ݎܩ‬ൌ

ࢍఈࣦ య ఔమ

(6.76)

Chapter Six

92

where ݃ is the acceleration of gravity, and ߙ is the densification coefficient defined as ߙൌ

ఘಮ ିఘ೐

(6.77)

ఘಮ

where ߷ is the fluid density, and Rayleigh: ܴܽ ൌ ‫ݎܩ‬. ܵܿ

(6.78)

In Tables 6-1 and 6-2 some of the correlations obtained for a number of electrodes of different geometry and hydrodynamic conditions are shown. Geometry Plate of length L Parallel plate of length L

Sherwood

ख

0.677ܴ݁ ଵ/ଶ ܵܿ ଵ/ଷ

L

Flow regime or range Laminar

݀௘

Laminar

݀௘

Laminar

ௗ

1.849(Pe ௅೐ ሻଵ/ଷ 1.467(

భ

ଶ

ଵାఊ

2.54 ሺܲ݁ 0.027ܴ݁

Inside curved area of cylindrical tube Rotating disk

ሻయ ሺܲ݁

ௗ೐ ଵ/ଷ ሻ ௅

ௗ೐ ଵ/ଷ ሻ ௅

଴.଼଻ହ

ܵܿ

݀௘ ଴.ଶଵ

ோ

2.035 ሺܲ݁ ௅ ሻଵ/ଷ 0.023ܴ݁

଴.଼

ܵܿ

ଵ/ଷ

0.6205 ܴ݁ ଵ/ଶ ܵܿ ଵ/ଷ 0.0117 ܴ݁ ଴.଼ଽ଺ ܵܿ ଴.ଶସଽ

10ହ ൑ ܲ݁

Wide electrodes Finite width electrodes ݀௘ ‫ܮ‬

ܴ

൑ 10଻ 10ସ ൑ ܴ݁ ൑ 10ହ and L/݀௘ ൐ 10 Laminar

ܴ

Turbulent

݀௘

Comment

High Pe

ܴௗ ܴௗ

Laminar 34൏ ܵܿ ൏ 8.9ή 10ହ ൏ 1400 ܴ݁ ൏ 1.18 ή 10଻ ݀௘ : equivalent diameter = 4 x cross-sectional area perpendicular to flow/wetted circumference R: radius Rd: disk radius ߛ: ratio of distance between electrodes to electrode width

Table 6-1: Correlations for calculation of km under forced convection.

Electrochemical Kinetics. II. Mass Transfer

Geometry Vertical plate

Sherwood 0.671ܴܽଵ/ସ 0.677

ௌ௖ భ/మ ீ௥ భ/ర

ख h h

ሺ଴.ଽହାௌ௖ሻభ/ర

Horizontal disk

Horizontal plate Horizontal cylinder Sphere

0.59ܴܽ଴.ଶ଺ 0.64 ܴܽଵ/ସ

h ݀஽

0.16 ܴܽଵ/ଷ

݀஽

0.24 ܴܽ଴.ଷଵଷ

κ

0.53 ܴܽଵ/ସ

2ܴ஼

2 ൅ 0.59 ܴܽଵ/ସ

2ܴௌ

2.3 + 0.58 ܴܽଵ/ସ

2ܴௌ

0.15 ܴܽ଴.ଷଷ

2ܴௌ

93

Validity range ܴܽ ൏ 4 ήή 10ଽ 2 ή 10ଵଵ ൏ ܴܽ 3ή 10ସ ൏ ܴܽ ൏ 2.5 ή 10଻ 2.5ή 10଻ ൏ ܴܽ ൏ 10ଵଶ 10଼ ൏ ܴܽ ൏ 10ଵ଴ ܴܽ ൏ 10ଽ

Coments High Sc Medium to high Sc

2.10଼ ൏ ܴܽ ൏ 2 ή 10ଵ଴ 3.10଺ ൏ ܴܽ ൏ 3.5 ή 10ଽ 2.85ή 10ଵ଴ ൏ ܴܽ ൏ 2.15 ή 10ଵଵ

݀஽ : disk diameter κ: anode – cathode distance ܴ஼ : Cylinder radius

Table 6-2: Correlations for calculation of km under free convection.

6.6. The current – potential relationship in the general case In the previous sections the analysis of reaction (6.1) has been considered under the assumption that the reaction activation energy is low or, equivalently, that the activation overpotential is negligible. However, in most of the processes of industrial interest this is not the case and the activation overpotential must be explicitly included in the calculations. In the general case, at low current density values the Butler – Volmer equation is valid and, at overpotentials of some decades of mV, is well approached by the Tafel equation. As the overpotential increases a

94

Chapter Six

concentration gradient is generated which determines the existence of a mass transfer overpotential that is additional to the activation overpotential and whose value will depend on hydrodynamic condition s. Thus, three regions can be identified: the first one is linear polarization, at very low overpotentials; at higher overpotential, but with constant species concentrations in the Nernst layer, Tafel description prevails while, at higher current densities, a concentration gradient is established and the mass transfer overpotential must be considered. Thus, in the case of overpotentials of more than 50 – 60 mV and assuming a large limiting current value for the anodic branch the current – potential relationship for reaction (6.1) will be: ߂ߖ ൌ ߂ߖ௧௛௘௥௠ ൅ ܽ ൅ ܾ ݈‫݃݋‬ሺ݅ሻ ൅

ோ் ఈ೘೟ జ‫࣠ כ‬

݅ ݅‫ܮ‬

ln ሺ1 െ ሻ

(6.79)

where a and b are the Tafel parameters. In Figure 6-11 the general polarization curve is displayed, and activation and mass transfer contributions to the total overpotential are plotted.

Figure 6-11: General polarization curve indicating the overpotential contributions of activation and mass transfer.

References Blasius, Hans. 1908. Grenzschichtenin Flüssigkeiten mit Kleiner Reibung. Z. Angew. Math. Phys. 1-37 Fahidy, Thomas. Principles of electrochemical reactor analysis. Elsevier, Amsterdam 1985 Chapter 1, p. 14

Electrochemical Kinetics. II. Mass Transfer

95

Norbert. 1983. Fundamentals of transport phenomena in electrochemical systems in Comprehensive Treatise of Electrochemistry (E. Yeager, J.O´M. Bockris, B.E. Conway y S. Sarangapani Eds.) Vol. 6, Chapt. 1. New York. Plenum Press Levich, Veniamin. Physicochemical hydrodynamics. 1962. New Jersey. Prentice Hall. Pp. 286-293

Ibl,

Bibliography Manzanares, José and Kontturi, Kyösti. 2007. Transport phenomena” in Electrochemistry Encyclopedia Vol. 2. New York. John Wiley & Sons Newman, John and Thomas – Alyea, Karen. Electrochemical systems 3rd edition. 2004. New York. John Wiley & Sons

CHAPTER SEVEN THE PERFECTLY STIRRED ELECTROCHEMICAL TANK REACTOR

7.1. Characteristics and analysis of the perfectly stirred electrochemical tank reactor The perfectly stirred electrochemical tank reactor (PSETR) is a tank in which electrodes are installed and where the electrolytic solution is stirred in such a way that electrolyte composition is homogeneous over the whole volume. Let us consider a process at a PSETR in which the occurring reaction at the cathode is

߭஽ ‫ ܦ‬൅ ߭௖‫ ି ݁ כ‬՞ ߭௉ ܲ

(7.1)

while the anodic process, written as the standard half-reaction, is:

߭ோ ܴ ൅ ߭௔‫ ି ݁ כ‬՞ ߭ௌ ܵ

(7.2)

which correspond to the overall reaction: ߭஽ ߭௔‫ ܦ כ‬൅ ߭ௌ ߭௖‫ ܵ כ‬՞ ߭௉ ߭௔‫ ܲ כ‬൅ ߭ோ ߭௖‫ܴ כ‬

(7.3)

In Figure 7-1 a PSETR of volume ࣰோ is shown in which an electrolyte, containing species D and S, with input concentrations ܿ஽భ and ܿௌభ, , enters at a flow rate Q. An electric current I circulates and, consequently, both species are consumed and output concentrations become ܿ஽మ and ܿௌమ which, because of the perfectly stirred condition, are the same as those inside the reactor. Accordingly, our analysis will be restricted to the direct branch of the cathodic reaction and the inverse branch of the anodic reaction. Thus, the mass balance for D at the cathode and S at the anode are:

The Perfectly Stirred Electrochemical Tank Reactor

97

 Figure 7-1: Sketch of the PSETR

ܳሺܿ஽భ െ ܿ஽మ ሻ ൌ െ ܳሺܿௌభ െ ܿௌమ ሻ ൌ െ

జವ ௜೎ జ೎‫࣠ כ‬

జೄ ௜ೌ ‫࣠כ‬ జೌ

ࣛ௖

(7.4)

ࣛ௔

(7.5)

where ic and ia are the cathodic and anodic current densities and ࣛ௖ and ࣛ௔ are, respectively, the cathode and anode areas. Since we have a single reaction at each electrode it is possible to write, recalling that the anodic current has been defined as negative: ‫ ܫ‬ൌ ݅௖ ࣛ௖ ൌ െ݅௔ ࣛ௔

(7.6)

Assuming that the electrode area is the same for cathode and anode, current density can be written: ݅ ൌ ݅௖ ൌ െ݅௔

(7.7)

Let us now consider the cathodic reaction case. As it has been discussed in Chapters 5 and 6, mass transfer and electron exchange at the electrode are

Chapter Seven

98

occurring at the same current density and, hence, we have from the mass transfer viewpoint that: ݅ൌെ

జ೎‫כ‬ జವ

࣠݇௠ ൫ܿ஽మ െ ܿ஽௘ ൯

(7.8)

with ܿ஽௘ the reactant concentration on the electrode while, at the same time, the Butler – Volmer reaction must be satisfied which can be generically written as: ௘ ௘ ିఉ௙ఎ೎,ೌ೎೟ ݁ െ ܿ௥௘ௗ ݁ ሺଵିఉሻ௙ఎ೎,ೌ೎೟ ൯ ݅ ൌ ߭௖‫݅ כ‬଴ ൫ܿ௢௫

(7.9)

where ܿ௢௫ and ܿ௥௘ௗ stand f or the oxidised and reduced species of the rate determining step of the cathodic half-reaction and f =࣠/ܴܶ. Similar equations can be written for the anodic process and, for the PSETR as a whole, we have: ߂ߖ ൌ ߂ߖ௧௛௘௥௠ ൅ ߟ௖,௔௖௧ ൅ ߟ௖,௠௧ െ ߟ௔,௔௖௧ െ ߟ௔,௠௧ െ ܴ‫ܫ‬

(7.10)

where the voltage drop due to electrolyte resistance has been included. It is worthwhile to stress that, according to the electrode potential convention ߂ߖ௧௛௘௥௠ is the difference ȟȲ௧௛௘௥௠,௖ െ ȟȲ௧௛௘௥௠,௔ and ߂ߖ is the electric potential difference between cathode and anode at operating conditions. Also, it must be reminded that electrode potentials are written as reduction reactions and, correspondingly, cathodic overpotentials have a negative sign and the opposite occurs for the anodic case. In the simple case in which the electrolyte resistance is low, the activation overpotentials are negligible and the only mass transfer overpotential is that due to the cathodic reaction, we have: ߂ߖ ൌ ߂ߖ௧௛௘௥௠ ൅ where ݅௅௖ = െ

జ‫כ‬ జವ

జವ ோ் ఈ೘೟ జ೎‫࣠ כ‬

݈݊ ቀ1 –

௜ ௜ಽ೎

ቁ

(7.11)

࣠݇௠஽ ܿ஽మ is the limiting current density of the cathodic

reaction. From Eqn. (7.10) it comes that, for very large total applied potential values ( ȟȲ ՜ െλ), the maximum attainable current density is: ݅ ൌ ݅௅௖

(7.12)

Under these conditions the electrode area, ࣛ min, is the required minimum to attain a given conversion degree and, employing (7.4), we have:

The Perfectly Stirred Electrochemical Tank Reactor

ܿ஽మ ൌ

99

௖ವభ

(7.13)

ଵାక

where ߦൌ

௞೘ ࣛ೘೔೙

(7.14)

ொ

Thus, the conversion at limiting current is: ߝ௉ௌா்ோ ൌ

௖ವభ ି௖ವమ ௖ವభ

ൌ

ஞ

(7.15)

ଵାஞ

If the PSETR operates at lower current density values calculation of i requires knowledge of all the involved overpotentials. In the general case, if activation overpotentials follow the Tafel equation the function expressing the total electric potential difference as a function of current density can be approximated as: ȟȲ ൌ ȟȲ௧௛௘௥௠ ൅ ܽ௖ ൅ ܾ௖ logሺ݅ሻ െ ܽ௔ െ ܾ௔ logሺ݅ሻ ൅ జೄ ோ் ‫࣠כ‬ ఈ೘೟ జೌ

జವ ோ் ఈ೘೟ జ೎‫࣠ כ‬ ௜

ln ൬1 െ

ᇲ ௜ಽೌ

ln ቀ1 – ௛

൰െ ݅ ఑

௜ ௜ಽ೎

ቁെ

(7.16)

where, for the sake of clearness, it has been assumed that the cross-sectional ௛ area of the reactor is the same as the electrode area and then ‫ ܴܫ‬ൌ ݅, where ఑ h is the distance between electrodes and the limiting current for the cathodic and anodic reactions is: ݅௅௖ ൌ െ ᇱ ൌെ ݅௅௔

జ೎‫כ‬ జವ ‫כ‬ జೌ

జೄ

࣠݇௠஽ ܿ஽

(7.17)

࣠݇௠ௌ ܿௌ

(7.18)

Eqn. (7.16) allows the calculation of current density at a given electric potential difference ߂ߖ whether the reactor acts as a battery or if it proceeds as an electrolyser. In the first case the reactor operates delivering electric energy to an external load and then, ߂ߖ ൌ ‫ܫ‬. ܼ

(7.19)

where I is the total current and Z the load impedance. In the second case, the potential difference is applied from an external source in order that

Chapter Seven

100

reactions proceed in opposition to the spontaneous evolution of the galvanic cell. When the reactor operates as a battery, the electric potential difference is a function of the concentrations of the different species and of the current density while, when the reactor is an electrolyser, the potential difference is defined by the external source applied. In any case, Eqn. (7.16) describes the relationship of the electric potential difference with current density and the internal parameters in the reactor. Since Eqn. (7.16) is not a differential equation it is convenient to apply the implicit function theorem and transform it as: ݀ȟȲ ൌ σ௝ ൅

డ డ௜

డ డ௖ೕ

௛

ቀ߂ߖ௧௛௘௥௠ ൅ ߟ௖,௔௖௧ ൅ ߟ௖,௧௠ െ ߟ௔,௔௖௧ െ ߟ௔,௧௠ െ ఑ ݅ቁ ݀ܿ௝ ൅ ௛

ቀ߂ߖ௧௛௘௥௠ ൅ ߟ௖,௔௖௧ ൅ ߟ௖,௧௠ െ ߟ௔,௔௖௧ െ ߟ௔,௧௠ െ ఑ ݅ቁ ݀݅

(7.20)

By its side, if an external source is employed for establishing the potential difference value we have: ݀ȟȲ ൌ

డ୼ஏ డ௧

݀‫ ݐ‬൅

డ௱ஏ డ௜

݀݅

ሺ7.21ሻ

Thus, when the reactor is an electrolyser, we can equate these two equations and it comes: ௗ௜ ௗ௧

σೕ

ൌ

ങ ቀ௱అ೟೓೐ೝ೘ ାఎ೎,ೌ೎೟ ା ఎ೎,೟೘ ିఎೌ,ೌ೎೟ ങ೎ೕ

೓ ഉ

೏೎ೕ ങ౴ಇ ି ങ೟ ೏೟

ି ఎೌ,೟೘ ି ௜ቁ

ങ౴ಇ ങ ି ቀ௱అ೟೓೐ೝ೘ ାఎ೎,ೌ೎೟ ା ఎ೎,೟೘ ିఎೌ,ೌ೎೟ ങ೔ ങ೔

೓ ഉ

ି ఎೌ,೟೘ ି ௜ቁ

(7.22)

7.2. Cascade of PSETR A possible way to improve the efficiency is to employ a cascade of reactors. Figure 7-2 shows a series of N identical PSETR in which the output flow of a given reactor is the input flow of the following.

The Perfectly Stirred Electrochemical Tank Reactor

101

Figure 7-2: N identical PSETR operating in cascade.

Taking the simplest case of operation at limiting current density for the jth reactor we have, from Eqn. (7.12), that: ܿ஽ೕశభ ൌ

௖ವೕ

(7.23)

ଵାక

where ܿ஽௝ and ܿ஽ೕశభ are, respectively, the input and output concentrations of D in this reactor. Thus, the final output concentration is: ܿ஽ಿశభ ൌ

௖ವಿ ଵାஞ

ൌ

௖ವಿషభ ሺଵାஞሻమ

ൌ

௖ವಿషమ ሺଵାஞሻయ

ൌ‫ڮ‬ൌ

௖ವభ ሺଵାஞሻಿ

(7.24)

and, for the total conversion degree we have: ߝேାଵ =

௖ವభ ష ௖ವಿశభ ௖ವభ

=1െ

ଵ

(7.25)

ሺଵାஞሻಿ

This equation indicates that, as new reactors are added to the cascade conversion tends asymptotically to 1. However, the efficiency of this procedure strongly depends on the value of Ɍ because of the added cost for every new reactor included. Table 7-1 shows the variation of the conversion with the number of reactors for different Ɍ values. N 1 2 3 4 5 6 7 8

ࢿ૚ (Ɍ ൌ 0,01ሻ 0.0099 0.0197 0.0294 0.0390 0.0485 0.0580 0.0673 0.0765

ࢿ૛ (Ɍ ൌ 0,05ሻ 0.0476 0.0930 0.1362 0.1773 0.2165 0.2538 0.2893 0.3232

ࢿ૜ (Ɍ ൌ 0,1ሻ

ࢿ૝ (Ɍ ൌ 0,5ሻ

0.0909 0.1736 0.2487 0.3170 0.3791 0.4355 0.4868 0.5335

0.3333 0.5556 0.7037 0.8025 0.8683 0.9122 0.9415 0.9610

Chapter Seven

102

9 …… 100

0.0857 ……. 0.6303

0.3554 …….. 0.9924

0.5759 …….. 0.9999

0.9740 …… 1.0000

Table 7-1: Conversion degree change with reactor numbers in a cascade for different ૆ values

7.3. The PSETR under batch operation Considering now the PSETR of Figure 7-1 in the case that there is no electrolyte flow, assuming that the solution volume remains constant during the process and that both electrodes have the same area, the mass balance for species D and S yields: ࣰோ ࣰோ

ௗ௖ವ ௗ௧ ௗ௖ೄ ௗ௧

ൌ

జವ ௜ జ೎‫࣠ כ‬

ൌെ

ࣛ

జೄ ௜ ‫࣠כ‬ జೌ

(7.26)

ࣛ

(7.27)

If, as is the practice in many cases, the reactor operates at constant current conditions, when reactant’s concentrations diminish it is necessary to increase the applied potential difference and, from Eqn. (7.22), its value is given by: ௗ୼ஏ

ൌ σ௝

ௗ௧ ௗ௖ೕ ௛ ఑

݅ቁ

డ డ௖ೕ

ቀ߂ߖ௧௛௘௥௠ ൅ ߟ௖,௔௖௧ ൅ ߟ௖,௧௠ െ ߟ௔,௔௖௧ െ ߟ௔,௧௠ െ (7.28)

ௗ௧

The necessary time to reach a given concentration, ܿ஽௙ from an initial value ܿ஽௜ is obtained from the corresponding mass balance, Eqn. (7.26): ‫ݐ‬௙ ൌ

ࣰೃ జ ‫࣠ כ‬ జವ ࣛ

௖

ଵ

‫׬‬௖ ವ೑ ௜ ݀ܿ஽ ವ೔

(7.29)

If the reactor operates galvanostatically, i is constant and the required time is: ‫ݐ‬௙ ൌ

ࣰೃ జ ‫࣠ כ‬ జವ ࣛ௜

൫ܿ஽௙ െ ܿ஽௜ ൯

(7.30)

The Perfectly Stirred Electrochemical Tank Reactor

103

The minimum time to attain a given conversion implies that the reactor ᇱ | it comes works at limiting current conditions and, assuming that ݅௅௖ ൏ |݅௅௔ that: ‫ݐ‬௠௜௡ ൌ െ

ࣰೃ ௞೘ವ ࣛ

ln ቀ

௖ವ೑ ௖ವ೔

ቁ

(7.31)

7.4. The PSETR in semi-batch operation This case corresponds to an PSETR that is divided by a diaphragm in two compartments, with the first one operating in batch while the other works with a continuous electrolyte flow. Figure 7-3 shows a sketch of this device for the case in which (7.1) is the cathode reaction, while in the anodic compartment the electrolyte is a solution of the acid HA that reacts evolving oxygen: ‫ܪ‬ଶ ܱ ՜ ܱଶ ൅ 4‫ ܪ‬ା ൅ 4݁ ି and H+ ions migrate towards the cathode through a membrane which is not permeable to other ions. In the cathode compartment the input solution contains species D and P at a definite pH value.

Figure 7-3: Sketch of a PSETR in semi-batch operation

The mass balances for each species are, then, the following: ࣰ௖

ௗ௖ವ೑ ௗ௧

ൌ ܳ൫ܿ஽௜ െ ܿ஽௙ ൯ ൅ ߭஽ ݅

ࣛ ఔ೎‫࣠ כ‬

(7.32)

Chapter Seven

104

ࣰ௖ ࣰ௖ ࣰ௔

ௗ௖ು೑

ൌ ܳ൫ܿ௉௜ െ ܿ௉௙ ൯ ൅ ߭௉ ݅

ௗ௧ ௗ௖ಹ೑ ௗ௧ ௗ௖ಹ

ൌ ܳ൫ܿு௜ െ ܿு௙ ൯ ൅ ߬ு ݅

ൌ െ߭ு ݅

ௗ௧

ࣛ

ࣛ ఔೌ‫࣠ כ‬

െ ߬ு ݅

ࣛ ௭ಹ ࣠

(7.33)

ఔ೎‫࣠ כ‬ ࣛ

(7.34)

௭ಹ ࣠

ൌ ሺ1 – ߬ு ሻ݅

ࣛ

(7.35)

࣠

where ‫ݖ‬ு = 1 and ߬ு is the transport number of the H+ ion. Reordering Eqn. (7.32): ߠ௖

ௗ௖ವ೑

ൌ ܿ஽௜ െ ܿ஽௙ ൅

ௗ௧

జವ ௜ࣛ

(7.36)

జ೎‫୕࣠ כ‬

where: ߠ௖ ൌ

ࣰ೎

(7.37)

ொ

is the residence time of the catholyte. Eqns. (7.33) to (7.35) can be reordered analogously. If the reactor operates near the limiting current, from Eqns. (7.12), (7.13) and (7.30) we have: ߠ௖

ௗ௖ವ ௗ௧

ൌ ܿ஽௜ െ ሺ1 ൅ ߦሻܿ஽

(7.38)

and, if the input concentration of D is constant, we obtain: ܿ஽ ሺ‫ݐ‬ሻ ൌ

௖ವ೔ ଵାక

ቂ1 െ exp ቀെ

ሺଵାకሻ௧ ఏ೎

ቁቃ ൅ ܿ஽଴ exp ቀെ

ሺଵାకሻ௧ ఏ೎

ቁ

(7.39)

For sufficiently large operating times the steady-state output concentration results: ܿ஽ஶ೑ ൌ

௖ವ೔ ଵାక

(7.40)

7.5. The PSETR with simultaneous chemical reaction In many cases reactants or products of an electrochemical reaction simultaneously participate in purely chemical processes and, if such is the case, it is necessary to know the characteristics of the involved reactions.

The Perfectly Stirred Electrochemical Tank Reactor

105

Let us consider a PSETR of volume ࣰோ which operates in batch, where the electrochemical reaction ‫ ܦ‬൅ ߭௖‫ ି ݁ כ‬՜ ܶ

(7.41)

occurs while, simultaneously, D is transformed into R according to the sequence: ‫ܦ‬՜ܲ՜ܴ

(7.42)

Assuming that both chemical reactions are of the first order, with rate constants ݇௉ and ݇ோ , the mass balance for species D is: ࣰோ

ௗ௖ವ ௗ௧

ൌെ

௜ జ೎‫࣠ כ‬

ࣛ െ ݇௉ ܿ஽ ࣰோ

(7.43)

while for species P and R the mass balance gives: ௗ௖ು ௗ௧ ௗ௖ೃ ௗ௧

ൌ ݇௉ ܿ஽ െ ݇ோ ܿ௉

(7.44)

ൌ ݇ோ ܿ௉

(7.45)

It is clear that, in such a case, as time evolves, cP passes through a maximum whose value will depend on the rate constant of the involved chemical reactions and on the current density at which the electrochemical reaction occurs. Solution of Eqn. (7.44) yields the time at which concentration of D is ܿ஽௙ : ௖

‫ ݐ‬ൌ െࣰோ ‫׬‬௖ బವ೑ ವ

ௗ௖ವ ࢏ ࣛା௞ು ௖ವ ࣰೃ ‫כ‬ ࣏ࢉ ࣠

(7.46)

while, from Eqn. (7.5), it comes that, assuming ܿ௉ ሺ0ሻ = 0 ௧

ܿ௉ ሺ‫ݐ‬ሻ ൌ ݇௉ exp ሺെ݇ோ ‫ݐ‬ሻ ‫׬‬଴ ܿ஽ ሺ‫ݑ‬ሻ expሺ݇ோ ‫ݑ‬ሻ ݀‫ݑ‬

(7.47)

These equations do not have a general analytic solution but, if the electrochemical reaction operates at limiting current density, we have: ࣰோ

ௗ௖ವ ௗ௧

wherefrom:

ൌ െሺ݇௠஽ ࣛ ൅ ݇௉ ࣰோ ሻܿ஽

(7.48)

Chapter Seven

106

ܿ஽ ൌ ܿ஽଴ exp ቀെ

௞೘ವ ࣛା௞ು ࣰೃ ࣰೃ

‫ݐ‬ቁ

(7.49)

And, from Eqn. (7.42) ܿ௉ ሺ‫ݐ‬ሻ is explicitly obtained: ܿ௉ ሺ‫ݐ‬ሻ ൌ ܿ௉଴ expሺെ݇ோ ‫ݐ‬ሻ െ

బ ௞ು ௖ವ ࣰೃ

௞೘ವ ࣛା௞ು ࣰೃ ି௞ೃ ࣰೃ

ቂexp ቀെ

௞೘ವ ࣛା௞ು ࣰೃ ࣰೃ

‫ݐ‬ቁ െ expሺെ݇ோ ‫ݐ‬ሻቃ

(7.50)

The maximum concentration of species P is obtained equating (7.44) to zero which gives, for the time value at which the maximum is achieved: ‫ ݐ‬௠௔௫ ൌ

ࣰೃ ௞೘ವ ࣛା௞ು ࣰೃ ି௞ೃ ࣰೃ

ln ቀ1 ൅

௞೘ವ ࣛା௞ು ࣰೃ ି௞ೃ ࣰೃ ௞ೃ ࣰ ೃ

ቁ

(7.51)

that corresponds to a concentration value of P: ܿ௉௠௔௫ ൌ ܿ஽଴

௞ು ௞ೃ

భశ೘ ೘

ሺ1 ൅ ݉ሻି

(7.52)

with ݉ൌ

௞೘ವ ࣛା௞ು ࣰೃ ି௞ೃ ࣰೃ

(7.53)

௞ೃ ࣰ ೃ

Finally, the mass balance for product T is: ࣰோ

ௗ௖೅ ௗ௧

ൌ

௜ జ೎‫࣠ כ‬

ࣛ ൌ ݇௠஽ ܿ஽ ࣛ

(7.54)

and, after integration ்ܿ ൌ ்ܿ଴ ൅ ܿ஽଴

௞೘ವ ࣛ ௞೘ವ ࣛା௞ು ࣰೃ

ቂ1 െ exp ቀെ

௞೘ವ ࣛା௞ು ࣰೃ ࣰೃ

‫ݐ‬ቁቃ

(7.55)

7.6. Parallel reactions on an electrode In the foregoing we have considered several types of PSETR under the assumption, in all cases, that at each electrode a unique reaction takes place. This, however, is a quite unusual situation for industrial processes and, normally, several electrochemical reactions proceed simultaneously. Consequently, the observed current density corresponds to the sum of all the electrons that are transferred at the electrode – solution interphase in the unit time, through all the occurring reactions. The industrial process of nickel electrodeposition is an example of such a case.

The Perfectly Stirred Electrochemical Tank Reactor

107

The usual way to produce nickel deposits on metallic surfaces is the electrochemical reduction of the Ni+2 ion from the so-called Watts bath which essentially contains nickel sulphate, nickel chloride and boric acid at pH 5. In this process the anode is a 98% purity nickel bar, which normally contains several impurities among which iron is the most important. Ideally the process should imply the cathodic deposition of nickel: ܰ݅ ଶା ൅ 2݁ ି ՞ ܰ݅

(7.56)

and nickel dissolution at the anode: ܰ݅ ՞ ܰ݅ ଶା ൅ 2݁ ି

(7.57)

However, because the electrolyte is acid at the cathode reduction of H+ ions will happen: 2‫ ܪ‬ା ൅ 2݁ ି ՞ ‫ܪ‬ଶ

(7.58)

and at the anode iron will also dissolve: ‫ ݁ܨ‬՞ ‫ ݁ܨ‬ଶା ൅ 2݁ ି

(7.59)

So, when a potential difference ȟȲ is applied, the measured electric current ‫ ܫ‬is the sum of the contributions of the two reactions occurring at each electrode. At the cathode: ‫ ܫ‬ൌ ൫݅ே௜ ൅ ݅ுమ ൯ࣛ௖

(7.60)

and, at the anode: ‫ ܫ‬ൌ ሺ݅ே௜ మశ ൅ ݅ி௘ శమ ሻࣛ௔

(7.61)

Total current density at the cathode is: ݅௖ ൌ ݅ே௜ ൅ ݅ுమ

(7.62)

and at the anode: ݅௔ ൌ ݅ே௜ మశ ൅ ݅ி௘ మశ

(7.63)

Chapter Seven

108

Now, since potential difference between the electrode and the solution is the same for each of the parallel reactions, it is clear that at the cathode: ȟȲ௖ ൌ ȟȲ௧௛௘௥௠,ே௜ ൅ ߟ௔௖௧,ே௜ ൅ ߟ௧௠,ே௜ ൌ ȟȲ௧௛௘௥௠,ுమ ൅ ߟ௔௖௧,ுమ ൅ ߟ௧௠,ுమ (7.64) and at the anode: ȟȲ௔ ൌ ȟȲ௧௛௘௥௠,ே௜ ൅ ߟ௔௖௧,ே௜ మశ ൅ ߟ௧௠,ே௜ మశ ൌ ȟȲ௧௛௘௥௠,ி௘ ൅ ߟ௔௖௧,ி௘ మశ ൅ ߟ௧௠,ி௘ మశ (7.65) The total potential difference applied on the reactor in order to produce nickel deposits at a certain rate will be defined by solving the equation: ȟȲ ൌ ȟȲ௖ െ ȟȲ௔ െ ‫ܫ‬. ܴ

(7.66)

In this case it is important to note that the faradaic efficiency of the nickel reactions at each electrode will not be the same, which means that, at large operation times, it might be necessary to take measures in order to adjust the electrolyte nickel concentration.

Bibliography Fahidy, Thomas. Principles of electrochemical reactor analysis”. 1985. Amsterdam. Elsevier Pickett, David. Electrochemical reactor design. 1979. Amsterdam. Elsevier Wendt, Hartmut and Kreysa, Gerhard. Electrochemical engineering. 1999. New York. Springer

CHAPTER EIGHT THE PLUG-FLOW ELECTROCHEMICAL REACTOR

8.1. Mass balance in the plug-flow electrochemical reactor The plug-flow electrochemical reactor (PFER) is a tube of constant section that contains electrodes of length ࣦ and width a, separated by a distance h, through which the electrolyte solution circulates parallel to the electrodes at flow rate Q, while an electric current I circulates, by which the following reactions occur: ߭஽ ൅ ߭௖‫ ି ݁ כ‬՞ ߭௉ ܲ

(8.1)

at the cathode and ߭ோ ܴ ൅ ߭௔‫ ି ݁ כ‬՞ ߭ௌ ܵ

(8.2)

at the anode. Generally, in a system like this, temperature, pressure, and concentration values will depend on the particular point of the reactor in a way that might be quite complex, according to the specific hydrodynamical characteristics of each case. In our analysis we will assume that variations of temperature, pressure or species concentration are only produced along the electrolyte flow direction, as shown in Figure 8-1. Hence, values of temperature, pressure and concentration depend on y, but not on x or z. These considerations are, of course, not valid at a very short distance from the electrodes where species concentration varies along the Nernst´s layer.

110

Chapter Eight

Figure 8-1: Sketch of a PFER

Let us now consider a differential volume in the PFER, as shown in Figure 8-2. This differential volume is defined between y and y+dy with crosssectional area S and its electrode area is ǻࣛ = ࣵ.ǻy.

Figure 8-2: Differential volume in a PFER

The Plug-Flow Electrochemical Reactor

111

If at each electrode a single reaction occurs and if the species that participate in the anodic reaction are not involved in the cathodic processes, and vice versa, the mass balance for species j at steady state, when current density is i yields: జೕ೎

జೕೌ

೎

‫כ‬ జೌ

ቀ߶௝೤ െ ߶௝೤శ೩೤ ቁ ࡿ ൌ െ ቀ జ‫ כ‬െ

௜

ቁ ࣠ ȟࣛ

(8.3)

In these equations ߶௝೤ is the molar flow of species ݆ at y and the ߭௝௞ are the stoichiometric coefficients of species ݆ in the cathodic and anodic reactions. Dividing by ȟ‫ݕ‬, with ȟ‫ ݕ‬՜ 0, leads to ௗథೕ ௗ௬

ൌቀ

జೕ೎ జ೎‫כ‬

జೕೌ

െ

‫כ‬ జೌ

௜ ௔

ቁ࣠ ࡿ

(8.4)

Since in the ideal PFER liquid flow is purely convective, ߶௝ ൌ ‫ܿݒ‬௝ and since fluid velocity is constant, we finally get the differential equation of mass balance that can be written as: ௗ௖ೕ ௗ௬

ൌቀ

జೕ೎ జ೎‫כ‬

െ

జೕೌ ‫כ‬ జೌ

௜ ௔

ቁ࣠ொ

(8.5)

where ܳ ൌ ࡿ‫ݒ‬. Integration of Eqn. (8.5) gives the total mass balance for species ݆: ௖

జೕ೎

జೕೌ

೎

‫כ‬ జೌ

೑ ‫׬‬௖ ೕ ݀ܿ௝ ൌ ቀ జ‫ כ‬െ ೕ೔

ଵ௔

ࣦ

ቁ ࣠ ொ ‫׬‬଴ ݅ሺ‫ݕ‬ሻ݀‫ݕ‬

(8.6)

Total current can, then, be written as: ࣦ

‫ ܫ‬ൌ ܽ ‫׬‬଴ ݅ሺ‫ݕ‬ሻ݀‫ݕ‬

(8.7)

wherefrom: ܿ௝ ೑ ൌ ܿ௝ ೔ ൅ ቀ

జೕ೎ జ೎‫כ‬

െ

జೕೌ ‫כ‬ జೌ

ூ

ቁ ࣠ொ

(8.8)

where ܳ ൌ ܵ‫ݒ‬. Integration of Eqn. (8.8) gives the total mass balance for species ݆ at the cathode: ௖

೑ ‫׬‬௖ ೕ ݀ܿ௝ ൌ ೕ೔

జೕ೎ ࣵ ࣦ ‫݅ ׬‬ሺ‫ݕ‬ሻ݀‫ݕ‬ ொ జౙ‫ ࣠ כ‬଴

(8.9)

Chapter Eight

112

and a similar formula is obtained for the anode. Then, for each electrode: ௙

ܿ௝ ൌ ܿ௝௜ ൅ ௙

ܿ௝ ൌ ܿ௝௜ െ

జೕ೎ ூ

(8.10)

ொ జౙ‫࣠ כ‬ జೕೌ ூ

(8.11)

ொ జ౗‫࣠ כ‬

These equations give the output concentration of species j as a function of the total current and the flow rate. The preceding equations indicate that, if the function i(y) is known, it is possible to calculate the electrode area required to achieve a given conversion. Thus, employing Eqn. (7.16) the total potential difference of the reactor can be written as: ȟȲ ൌ ȟȲ௧௛௘௥௠ ൅ ܽ௖ ൅ ܾ௖ logሺ݅ሻ െ ܽ௔ െ ܾ௔ logሺ݅ሻ ൅ జೄ ோ் ‫࣠כ‬ ఈ೘೟ జೌ

జವ ோ்

ln ቀ1 –

ఈ೘೟ జ೎‫࣠ כ‬ ௜ ௛

ln ൬1 െ

ᇲ ௜ಽೌ

൰െ ݅ ఑

௜ ௜ಽ೎

ቁെ

(8.12)

Taking the case of the EPFR operating as an electrolyser and assuming that the applied potential is constant, by a similar procedure than that performed in the previous chapter we get: ௗ௜ ௗ௬

ൌെ

σೕ

ങ ೓ ೏೎ೕ ቀ୼ஏ೟೓೐ೝ೘ ାఎ೎,ೌ೎೟ ା ఎ೎,೘೟ ିఎೌ,ೌ೎೟ ି ఎೌ,೘೟ ିഉ௜ቁ ೏೤ ങ೎ೕ ങ ೓ ቀ୼ஏ೟೓೐ೝ೘ ାఎ೎,ೌ೎೟ ା ఎ೎,೘೟ ିఎೌ,ೌ೎೟ ି ఎೌ,೘೟ ିഉ௜ቁ ങ೔

(8.13)

The minimum reactor length needed for a given conversion corresponds, of course, to operation under limiting current conditions. In this case if, for simplicity, it is assumed that the mass transfer overpotential for the anodic reaction is negligible and defining the mean value of the mass transfer coefficient over the electrode length as: ଵ ࣦ തതതത ݇ ௠ ൌ ‫׬‬଴ ݇௠ ሺ‫ݕ‬ሻ݀‫ݕ‬ ࣦ

(8.14)

it is possible to obtain the current density dependence on position as: ݅ሺ‫ݕ‬ሻ ൌ െ

జ೎‫כ‬ జವ

തതതതതത ࣠݇ ௠,஽ ܿ஽ ሺ‫ݕ‬ሻ

(8.15)

The Plug-Flow Electrochemical Reactor

113

where it was assumed, as in Section 7.4, that the limiting current for the cathodic reaction was lower than that of the anodic reaction. Introducing ݅ሺ‫ݕ‬ሻ in Eqn. (8.7) we obtain: ொ

௖೔

௞೘,ವ ௔

௖ವ

ࣦ௠௜௡ ൌ തതതതതതത ln ൬ ವ೑ ൰

(8.16)

8.2. The PFER with recycle The direct use of a PFER in a single-pass operation, as discussed in the preceding section, may require, in some cases, the use of large electrode areas in order to achieve an acceptable conversion. It is of interest, then, to analyse the behaviour of this type of electrode if the electrolyte is recycled, as shown in Figure 8-3.

Figure 8-3: Sketch of a PFER with recycle.

Introducing a recycle in a PFER generates two opposing effects: on one side, reactants concentrations at the input diminish, affecting the efficiency but, conversely, the increase in the flow rate improves mass transfer and thus promotes higher conversion values. It is worthwhile to consider this case in some detail. If the REFP of Figure 8-3 operates under steady-state conditions the mass balance at the input node is: ௙

ܳܿ஽௜ ൅ ܳோ ܿ஽ ൌ ሺܳ ൅ ܳோ ሻܿ஽ோ

(8.17)

Defining the recycle factor ݂ோ as: ݂ோ ൌ

ொೃ ொ

(8.18)

Chapter Eight

114

we have: ܿ஽ோ ൌ

೑

೑

೔ ொ௖ವ ାொೃ ௖ವ

ൌ

ொାொೃ

೔ ௖ವ ା௙ೃ ௖ವ

(8.19)

ଵା௙ೃ

Operating at limiting current density conditions for the cathodic reaction we have, from Eqn. (8.16), that the minimum electrode length to attain a given conversion degree is: ொሺଵା௙ೃ ሻ ௖ೃ ln ൬ ವ೑ ൰ തതതതതതത ೃ ௞೘,ವ ௔ ௖ವ

ோ ࣦ௠௜௡ ൌ

(8.20)

ோ is the mass coefficient in the PFER with recycle. Consequently: where തതതതതത ݇௠,஽ ೃ ࣦ೘೔೙

ࣦ೘೔೙

೑

തതതതതതത ௞ ଵା௙ೃ

೘,ವ ൌ തതതതതതത ೃ

೔ ௞೘,ವ ୪୬൭೎ವ ൱ ೑ ೎ವ

ln ൬

೔ ௖ವ ା௙ೃ ௖ವ ೑ ೑ ௖ವ ା௙ೃ ௖ವ

൰

(8.21)

Defining conversion as: ೑

ߝ ൌ1െ

௖ವ

(8.22)

೔ ௖ವ

Eqn. (8.21) can be written as: ೃ ࣦ೘೔೙

ࣦ೘೔೙

തതതതതതത ௞

ଵା௙ೃ

೘,ವ ൌ തതതതതതത ೃ

షభ ௞೘,ವ ୪୬ ሺଵିఌሻ

ln ቂ

ሺଵିఌሻషభ ା௙ೃ ଵା௙ೃ

ቃ

(8.23)

where, for a given conversion, the right-hand side depends only on the recycle factor. Taking into account that the mass transfer coefficient is proportional to ܴ݁ ఊ and, consequently, to ܳ ఊ , where usually Ȗ < 1, it comes that: തതതതതതത ೃ ௞೘,ವ തതതതതതത ௞೘,ವ

ൌ

ሺொାொೃ ሻം ொം

ൌ ሺ1 ൅ ݂ோ ሻఊ

(8.24)

and (8.23) can be written as: ݃ሺ݂ோ ሻ ൌ

ೃ ࣦ೘೔೙

ࣦ೘೔೙

ൌ

ሺଵା௙ೃ ሻభషം ୪୬ ሺଵିఌሻషభ

ln ቂ

ሺଵିఌሻషభ ା௙ೃ ଵା௙ೃ

ቃ

(8.25)

At this point, it is cogent to ask what is the convenient recycle factor to employ. Assuming that rate flow and conversion are constant values, and

The Plug-Flow Electrochemical Reactor

115

the main interest is to reduce the electrode length, calculations performed for ߛ ൌ 0.5 yield the curves shown in Figure 8-4

Figure 8-4: Recycle factor impact on the PFER efficiency for different conversion values(İ = 0.1; 0.2; 0.4; 0.8; 0.9)

Introducing a recycle in a PFER will be convenient if g Rp. When the activation overpotential is included, it is possible to assume, from a strictly electrical consideration, that an additional resistance, Ra, exists which is defined as: ܴ௔ ൌ

ௗఎೌ

(10.21)

ௗ௜

This resistance, which is not of ohmic nature but describes the opposition to charge transfer, can be written as: ܴ௔ ൎ

୼ఎ

(10.22)

୼௜

Since catalytic differences between peaks and recesses can be assumed to be dismissible, it is possible to consider that ܴ௔ is the same in both cases and then, Eqn. (10.20) takes the form: ௜೛ ௜ೝ

ൌ

ோೌ ାோ౨ ோೌ ାோ೛

൏

ோೝ ோ೛

(10.23)

thus providing further arguments to the homogenizing effect of the activation overpotential on current and electric potential distribution. Accepting, then, that the activation overpotential impact on the uniformity of current density is related to the relationship between activation pseudo resistance and electrolyte resistance the Wagner number is defined as: ܹܽ ൌ

ோೌ ோಈ

(10.24)

In this dimensionless number ܴஐ is the electrolyte resistance per unit area that can be calculated as: ܴஐ ൌ ߢ/ࣦ

(10.25)

where ߢ is the electrolyte conductivity and ࣦ is a characteristic distance that, in the case of Figure 10-3 is the distance between anode and cathode.

Current and Potential Distribution

143

In electroplating the Wagner number is related to deposit uniformity: high Wagner numbers indicate that surface conditions have little influence on the current distribution and the deposit will be homogeneous. A low Wagner number indicates that current lines will approach the primary distribution. In the case of the reactor of Figure 10-3 the secondary current distribution obtained for different values of the Wagner number varies as shown in Figure 10-10 (Wagner, 1951)

Figure 10-10: Secondary current distribution in the cell of Figure 10-3 For different Wagner number values. I, Wa=0.8; II, Wa=0.4; III, Wa=0.2; IV, Wa= 0.1; V, Wa=0.

10.5. Tertiary distribution When calculating the tertiary electric potential and current distribution, mass transfer overpotential is included in the problem and Eqn. (10.4) must be solved including the concentration dependent term since, although in the solution bulk composition is homogeneous, concentration gradients will be generated in Nernst´s layer.

144

Chapter Ten

The effect of mass transfer overpotential is qualitatively different from that of activation overpotential and its impact differs according to the thickness of the diffusional layer, įN. In fact, according to surface roughness characteristics and the hydrodynamics of the system, different outcomes might result. In order to illustrate this point, let us consider an electrode whose roughness can be approximated by a sawtooth function of height a for which, as shown in Figure 10-11, two limiting possibilities can be defined: a > įN (macroprofile).

Figure 10-11: Surface roughness and size relation with diffusion layer thickness: a) Microprofile; b) Macroprofile

Considering the case of a microprofile, when a metal is deposited on the electrode, growth on peaks overcomes a lesser resistance than deposition on recesses and, as a consequence, mass transfer influence on current distribution will be similar to that of primary distribution. This means that non uniform deposits are favoured in opposition to the activation overpotential effect. In the case of a macroprofile the Nernst’s layer thickness is small and, then, the diffusion layer copies the surface roughness and mass transfer overpotential is approximately the same over the whole surface promoting a more uniform deposition.

Current and Potential Distribution

145

When treating tertiary distribution, it must also be recognized that solution conductivity will vary over the zone in which non-zero concentration gradients occur and, hence, the theoretical treatment of the problem indicates that Eqn. (10.4) must be replaced by Poisson’s equation: ଵ

ሬሬሬሬሬሬԦ൯ ‫׏‬ଶ Ȳ ൌ െ ൫࣠ σ௝ ‫ݖ‬௝ ‫ܦ‬௝ ‫׏‬ଶ ܿ௝ ൅ ሬ‫׏‬Ԧߢ ή ‫׏‬Ȳ ఑

(10.26)

which must be solved together with the corresponding mass balances in order to obtain the electric potential and current distribution. The mathematical challenge of Eqn. (10.26) is quite complex, particularly when convection effects are included, but in some cases it has been possible to obtain analytical solutions. One of these cases is the rotating disk electrode when the cation of a binary electrolyte (CuSO4) is discharged (Newman, 1967) and it is illustrative to consider the results obtained for this particular example. Calculations were performed assuming that the symmetry coefficient in the Butler – Volmer equation is 0.5 and that exchange current is proportional to the square root of concentration. The curves obtained with these considerations correspond to different values of the dimensionless number N, which is defined as: జ‫ כ‬௓࣠ మ ஽಴ೠమశ ௖ ಮ మశ ௥೚మ ఠ ଵ/ଶ ଴.ହଵ଴ଶଷఔ ଵ/ଷ ಴ೠ ሻ ሺ ሻ ൤ ൨ జ಴ೠమశ ோ்ሺଵିఛ಴ೠమశ ሻ఑ಮ ఔ ଷ஽಴ೠమశ

ܰ ൌ െሺ

(10.27)

where ro is the electrode radius, Ȧ the rotation speed, ߥ the kinematic viscosity, ‫ܦ‬஼௨మశ the diffusion coefficient of the discharging ion, ߭ ‫ כ‬the ஶ number of electrons in the reaction, c஼௨ మశ the ion concentration in the solution bulk, ɓେ୳మశ the stoichiometric number of the ion in the cathodic reaction, ߬஼௨మశ the transport number, assumed to be 0.5, Ɉஶ the bulk solution conductivity and Z is defined as: ܼൌെ

௭శ ௭ష ௭శ ା|௭ష |

(10.28)

where subindex (+) corresponds to Cu2+ and (-) to sulphate. It must be noted that, except ߱, all other terms in Eqn. (10.27) are constant parameters. When N is zero, which means that there is no rotation of the electrode, the problem is that of a plane electrode. Solving Eqn. (10.26) for different

146

Chapter Ten

rotation velocities the concentration distribution of the cation is described in Figure 10-12

Figure 10-12: Cu2+ concentration distribution for a rotating disk electrode as a function of distance to its centre

As seen from this graph, ion concentration diminishes as the points are farther from the centre and may attain zero values at high rotation velocities. Once concentration distribution is known, current distribution can be calculated.

References Klingert, John, Lynn, Scott and Tobias, Charles. 1964. Evaluation of current distribution in electrode systems by high-speed digital computers Electrochim. Acta 9 297-311

Current and Potential Distribution

147

Newman, John. 1966. Resistance for flow of current to a disk J. Electrochem. Soc. 501-502 Newman, John. 1967. Current distribution on a rotating disk below the limiting current. J. Electrochem. Soc. 1235-1240 Wagner, Carl. 1951. Theoretical analysis of the current density distribution in electrolytic cells 116-128

Bibliography Ibl, Norbert. 1983. Current distribution in Comprehensive Treatise of Electrochemistry (E. Yeager, J.O´M. Bockris, B.E. Conway y S. Sarangapani Eds.) Vol. 6, Chapter 6. New York. Plenum Press Newman, J. and Thomas – Alyea, K.E. 2004. Electrochemical systems. 3rd edition John Wiley

SECOND PART AREAS OF PRACTICAL INTEREST

CHAPTER ELEVEN CORROSION

11.1. The corrosion phenomenon and its importance Corrosion can be defined as the destructive attack of a metal because of oxidation reactions that are produced as a consequence of interaction with its environment. The wide spread of corrosion produces significant consequences on three levels: on the economy, on personal safety and on the ecological impact of human activities. Corrosion of a metal is generally associated with mass loss and change of the properties (mechanical, electrical, etc.) of the affected part. Many modern catastrophes, which have had a heavy toll on human lives and produced enormous environmental and economic damages, have been caused by the corrosion-induced collapse of critical elements. Several examples which illustrate this point can be mentioned. The oil spill of the Erika tanker that contaminated 400 km of the Brittany coast in 1999 and the 40,000 tons of oil that the Prestige poured out in Galicia were just two of the large number oil spills caused by corrosion of tanks. Corrosion of structural parts that undergo mechanical efforts have been the cause of many catastrophic events among which the flight 562 of American S.E. Airlines, that crashed at Georgia in 1995 because of the fracture of bolts affected by stress corrosion cracking, and the Silver Bridge collapse in West Virginia, in which 46 people died in 1967, as a consequence of corrosion problems in structural pieces, can be mentioned. However, the most terrible accident associated with corrosion is that of the Union Carbide plant in Bhopal (India) where, on December 2, 1984, corrosion of safety valves led to the emission of 40 tons of methyl isocyanate and phosgene to the environment: 30,000 people died and more than 500,000 were seriously affected. The engineer´s role in this field is to prescribe specific actions that should be performed to impede corrosion or, more frequently, achieve its control in a way that permits to intervene before serious consequences are produced.

Corrosion

151

Of course, corrosion prevention management implies direct and indirect costs and, in defining the specific activities to be included in a corrosion control program, they must be carefully considered. Direct costs include the need to use more expensive materials and larger structures, pipes and parts, as well as establishing maintenance programs in which preventive replacements are defined and protection systems installed and maintained. These costs, however, are well below those that would occur if there were not an adequate corrosion prevention program: unscheduled plant closures, product spills, contamination, lower efficiency, etc., besides the possibility of personal accidents. In this respect it is to be pointed that a quite detailed study carried out by the American Society of Metals in 2005 estimated that corrosion related costs in the U.S. amount almost to 3.1% of the NGP. As we will see in this chapter, corrosion can be produced by widely diverse causes and, thus, it is essential that those responsible for establishing and managing a corrosion control program have a good knowledge and understanding of the causes and factors involved in this phenomenon.

2. Corrosion thermodynamics 2.1. General equations The simplest form through which a metallic part undergoes a corrosive attack is the metal atom oxidation to form a dissolved ion: ‫ ܯ‬՞ ‫ܯ‬௠ା ൅ ߭ெ ݁ ି

(11.1)

whose electrode potential employing standard electrode potentials, which as mentioned in Chapter 3, correspond to the reduction process, is given by Nernst’s equation: ଴ οȲ௧௛௘௥௠,ெ ൌ οȲெ െ

ோ் జಾ ࣠

ln ൬

ଵ ௔ಾ೘శ

൰

(11.2)

Clearly, in order that oxidation of M might occur, a reduction reaction, able to take the electrons lost by ‫ ܯ‬must happen at a site which is not far away from the place where oxidation of M occurs: ܴ ൅ ߭ோ ݁ ି ՞ ܴ௥ି whose equilibrium potential is:

(11.3)

Chapter Eleven

152

οȲ௧௛௘௥௠,ோ ൌ οȲோ଴ െ

ோ் జೃ ࣠

lnሺܽோೝష ሻ

(11.4)

Electroneutrality implies that, at every point: ߭ெ ݀ߦெ ൅ ߭ோ ݀ߦோ ൌ 0

(11.5)

where ߦ௞ is the advancement degree of reaction k. Thus, the corrosion process proceeds through the reaction: ߭ெ ܴ ൅ ߭ோ ‫ ܯ‬՞ ߭ோ ‫ܯ‬௠ା ൅ ߭ெ ܴ௥ି

(11.6)

and its spontaneity will be defined by the Gibbs´ free energy change: ଴

ο‫ ܩ‬ൌ ο‫ ܩ‬൅ ܴܶ ln ቆ

ഔ ഔ ௔ ೃ೘శ ௔ ಾ ೝష ೃ ಾ ഔ ഔ ௔ೃಾ ௔ಾೃ

ቇ

(11.7)

If ο‫ ܩ‬is negative reaction (11.6) will be spontaneous and metal M will corrode. In aqueous media hydrogen ion and dissolved oxygen are usually the electron receptors in corrosion processes, through the reduction reactions: 2‫ ܪ‬ା ൅ 2݁ ି ՞ ‫ܪ‬ଶ

(11.8)

½ܱଶ ൅ ‫ܪ‬ଶ ܱ ൅ 2݁ ି ՞ 2ܱ‫ି ܪ‬

(11.9)

Thus, in acid media corrosion of a divalent ion is described as: ‫ ܯ‬൅ 2 ‫ ܪ‬ା ՞ ‫ܯ‬ଶା ൅ ‫ܪ‬ଶ

(11.10)

and, in neutral or alkaline solutions: ‫ ܯ‬൅ ½ܱଶ ൅ ‫ܪ‬ଶ ܱ ՞ ‫ܯ‬ଶା ൅ 2ܱ‫ି ܪ‬

(11.11)

In aqueous media, then, thermodynamic stability of a metal will depend not only on the redox potential of the metal, but also on pH.

2.2. Pourbaix diagrams From the above arguments, and considering that most corrosion processes occur in aqueous media, the Belgian chemist Marcel Pourbaix suggested that thermodynamic data for a given metal could be conveniently compiled

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153

by means of graphs in which electrode potential is plotted as a function of pH, in such a way that it might be correlated with metal stability. These graphs, known as Pourbaix diagrams, show, at defined temperature, pressure and composition conditions, the electrode potential – pH regions in which the metal is immune, i.e., no corrosion occurs, and those in which the metal reacts forming soluble ions or specific oxides. Obviously, if other chemical species are present, the plot should take into consideration the chemical equilibria in which such species participate. It is important, and it must be kept in mind, that Pourbaix diagrams represent thermodynamic equilibria and, consequently, indicate what particular species are stable at a given electrode potential and pH values, but do not provide information on the rate at which the stable condition is attained. In fact, in some cases corrosion rate can be so low that a metal may be considered as stable for practical purposes although, according to thermodynamics, oxidation is spontaneous. In building the Pourbaix diagram for a given metal it is necessary to know what chemical species of the metal can exist and in what chemical or electrochemical equilibria they might participate. As will be discussed below, the decision to include, or not, a particular equilibrium depends on the significance that it may have according to its reaction rate. An important point, that deserves some consideration, is that of the concentration at which metal ions are present. In this respect two typical cases can be mentioned. The first one corresponds to the situation in which the metal part is in contact with a flowing solution, like in a valve. In this case, dissolved ions are washed down and ion concentration at the edge of Nernst´s layer will tend to a constant value which depends on the reaction kinetics and the hydrodynamic characteristics of the system. The second case is that of a tank which must store a corrosive solution; in this case metal dissolution will proceed at a certain rate, but ions will not accumulate near the walls of the tank and will diffuse toward the bulk and, again, concentration at the edge of Nernst´s layer may be taken as approximately constant during relatively large time periods. For this reason, published Pourbaix diagrams employ low constant concentration values for the dissolved ion, usually 10-6 M. With these considerations, if the only reaction in which metal M participates is:

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154

‫ܯ‬ା ൅ ݁ ି ՞ ‫ܯ‬

(11.12)

The equilibrium potential in Pourbaix diagram is: ଴ οȲ௧௛௘௥௠,ெ ൌ οȲெ ൅

ோ் ࣠

lnሺ10ି଺ ሻ

(11.13)

Hence corrosion behaviour will depend on the electrode potential value of the metal: x οȲ ൐ οȲ௧௛௘௥௠,ெ (ߟெ ൐ 0): the metal part corrodes. x οȲ ൌ οȲ௧௛௘௥௠,ெ (ߟெ ൌ 0): the solid metal is in equilibrium with its ions, no net reaction occurs. x οȲ ൏ οȲ௧௛௘௥௠,ெ (ߟெ ൏ 0): metal ions are reduced and metal atoms deposit on the metal part. Thus, by modifying the electrode potential, it is possible to take the solid metal from an active zone, where there is dissolution, to an immune zone, where no corrosion occurs. Most metals can react with water and other species, that differ from the simple dissolved ion, might exist and the stability zone for each of them will be defined by the characteristics of the corresponding equilibria. In order to explicitly address to this point, let us discuss the Pourbaix diagram for Ni, assuming that activity coefficients are unity for all species. Although nickel reactions with water are rather complex, in order to make the example as simple as possible we will assume the existence of only three species containing nickel atoms: Ni. Ni2+ and NiO, that will be considered as interacting with the water derived species: H+, OH-, H2O, H2 and O2 We have then eight species (N = 8) that are made up from three independent elements Ni, H, O (Ne = 3) and, therefore, in order to define the state of the system at a point in the Pourbaix diagram we will need a number, NR, of independent reactions linking the existing species: ܰோ ൌ ܰ െ ܰ௘ ൌ 8 െ 3 ൌ 5

(11.14)

Among the number of chemical reactions in which these species participate, it is possible to select the following set of five reactions whose thermodynamic data are known: 1) ܱܰ݅ ൅ 2‫ ܪ‬ା ൅ 2݁ ି ՞ ܰ݅ ൅ ‫ܪ‬ଶ ܱ οߖଵ଴ ൌ ൅0,11ܸ

(11.15)

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155

2) ܰ݅ ଶା ൅ 2݁ ି ՞ ܰ݅

οߖଶ଴ ൌ െ0,24ܸ

(11.16)

3) 2‫ ܪ‬ା ൅ 2݁ ି ՞ ‫ܪ‬ଶ

οߖଷ଴ ൌ 0ܸ

(11.17)

4) ܱଶ ൅ 4‫ ܪ‬ା ൅ 4݁ ି ՞ 2‫ܪ‬ଶ ܱ

οߖସ଴ ൌ ൅1,23ܸ

(11.18)

5) ‫ܪ‬ଶ ܱ ՞ ‫ ܪ‬ା ൅ ܱ‫ି ܪ‬

‫ܭ‬ௐ ൌ 10ିଵସ

(11.19)

Taking reaction 1) we have: οȲ௧௛௘௥௠,ଵ ൌ οȲଵ଴ െ

ோ் ଶ࣠ ୪୭୥ሺ௘ሻ

log ൬

ଵ ௖ಹశ

൰

(11.20)

that can be written: οȲ௧௛௘௥௠,ଵ ൌ οȲଵ଴ െ

଴,଴ହଽ௏ ଶ

‫ܪ݌‬

(11.21)

and corresponds to the line shown in Figure 11-1 of the electrode potential - pH diagram.

Figure 11-1: Equilibrium line for reaction 1)

The physical meaning of this line is that at a given pH value points that are above the line correspond to electrode potential values with ߟଵ ൐ 0, which means that reaction 1) proceeds to oxidation and, hence, the stable species is NiO. Below the line, of course, the stable species is Ni.

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156

The equilibrium potential for reaction 2) is: οȲ௧௛௘௥௠,ଶ ൌ οȲଶ଴ െ

ோ் ଶி

ln ൬

ଵ ௖ಿ೔మశ

൰

(11.22)

Taking, as mentioned above, the nickel ion concentration to be 10-6 this equilibrium is described by a horizontal line for which the stable species are Ni2+ in the upper region and Ni in the region below the line. In Figure 11-2 this line is plotted together with that corresponding to reaction 1)

Figure 11-2: Equilibrium lines for reactions 1) and 2)

Let us now consider the validity range over which the obtained lines have physical meaning. In doing this it is to be noted that, at pH values between 0 and the pH corresponding to the crossing point of both lines, metallic Ni cannot exist above the horizontal line; hence, in this pH range the equilibrium between Ni and NiO cannot take place and the oblique line is not valid. At pH values larger than that corresponding to the intersection point, metallic Ni cannot be present above the oblique line and, consequently, the equilibrium describing Ni/Ni2+ does not occur and the horizontal line has no physical meaning. By its side the stability zone for the Ni2+ - NiO pair must be duly considered, but in the selected set of reactions such equilibrium is not present. However, since the chosen set of five reactions contains the complete thermodynamic information of the system, a linear combination of these equations should yield the needed relationship which, in fact, is obtained by subtracting reaction 2) from reaction 1) which yields reaction 6):

Corrosion

6) ܱܰ݅ ൅ 2‫ ܪ‬ା ՞ ܰ݅ ଶା ൅ ‫ܪ‬ଶ ܱ

157

(11.23)

This is a chemical reaction and, as such, its equilibrium does not depend on the electrode potential and will be, hence, described as a vertical line in the Pourbaix diagram. By its side, since reaction 6) is a linear combination of reactions 1) and 2) the intersection of the lines corresponding to these reactions must be a point of the line depicting reaction 6) as shown in Figure 11-3. ǻ‫଺ܩ‬௢

Figure 11-3: Equilibrium lines for reactions 1), 2) and 6)

Since ο‫ ଺ܩ‬ൌ ο‫ܩ‬ଵ െ ο‫ܩ‬ଶ

(11.24)

and the equilibrium constant of this reaction is: K6 = ൬

௖ಿ೔మశ మ ௖ಹశ

൰

(11.25)

it is easily seen that: -RTlnK6 = ǻ‫଺ܩ‬௢ = ǻ‫ܩ‬ଵ௢ - ǻ‫ܩ‬ଶ௢

(11.26)

Remembering that: ǻ‫ܩ‬ଵ௢ =- 2 ࣠ȟȲଵ଴ and

ሺ11.27ሻ

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158

ǻ‫ܩ‬ଶ௢ =- 2 ࣠ȟȲଶ଴

ሺ11.28ሻ

we have: RTlnK6 = 2 ࣠ሺȟȲଵ଴ െ ȟȲଶ଴ ሻ

(11.29)

wherefrom at 298.15 K: ଴,଴ହଽ௏ ଶ

log ൬

௖ಿ೔మశ మ ௖ಹశ

൰ ൌ ሺȟȲଵ଴ െ ȟȲଶ଴ ሻ

(11.30)

and ‫ ܪ݌‬ൌ

೚ ୼ஏ೚ భ ି ୼ஏమ

଴,଴ହଽ

ଵ

െ ݈‫ܿ݃݋‬ே௜ మశ

(11.31)

ଶ

which indicates that the equilibrium pH value does not depend on ȟȲ. With ܿே௜ మశ ൌ 10ି଺ M equilibrium pH is 8.9 and the resulting vertical line with origin at the crossing point describes the equilibrium between Ni2+ and NiO at this particular ion concentration. Figure 11-3 indicates the stability zones for the three considered species which are defined by the lines with origin in the common point to the three equilibria (CP). Up to now, stability considerations have been limited to species that contain nickel but, as mentioned above, it is necessary to take into account the stability zones of the species linked to water, i.e., reactions 3) and 4). The corresponding equilibria are defined by: οȲ௧௛௘௥௠,ଷ ൌ ȟȲଷ଴ െ

ோ்

οȲ௧௛௘௥௠,ସ ൌ ȟȲସ଴ െ

ோ்

ி

ி

ln ൬ ln ൬

ଵ ௖ಹశ ଵ ௖ಹశ

൰ ൌ െ0,059ܸ ‫ܪ݌‬

(11.32)

൰ ൌ 1,23ܸ െ 0,059ܸ ‫ܪ݌‬

(11.33)

which yield two parallel lines that can be overlapped to Figure 11-3 as shown in Figure 11-4.

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159

Figure 11-4: Equilibrium lines for reactions 1), 2) 3), 4) and 6)

The included parallel lines describe electrode potential – pH equilibrium conditions for reactions 3) (lower line) and 4) (upper line). Taking this latter case, it is easy to see that those points which are above the equilibrium line correspond to the zone in which water is unstable and oxygen, which is the stable species, is evolved. Below this line oxygen is unstable and undergoes reduction. Analogously, in the case of the lower parallel line, that corresponds to reaction 3), ‫ܪ‬ଶ is stable below the line while H+ is stable above it. Thus, at potential – pH values below this line H2 will be evolved from H+ ions provided by Reaction 5). The region between these lines is the stability zone of water. The analysis carried out for the case of nickel leads to the following conclusions regarding Pourbaix diagrams. x When building a Pourbaix diagram, it is necessary to resort to as many equilibrium reactions as needed according to the number of involved elements and existing species. x Chemical equilibria which do not involve electron transfer between reactants are described by vertical lines. x Electrochemical reactions equilibria in which pH is not involved are horizontal lines. x Electrochemical reactions equilibria in which pH is involved are oblique lines. The first of these points is important because chemical reactions of metals with water might generate, particularly in the case of transition metals,

160

Chapter Eleven

different oxide and hydroxide forms whose stability depends on pH and electrode potential. Consequently, the diagram might change significantly as seen when Figure 11-4 is compared to the diagram obtained when the existence of Ni(OH)3 and HNiO2- is included, which is shown in Figure 115.

Figure 11-5: Pourbaix diagram of nickel including the species Ni(OH)3 and HNiO2-

Thus, Pourbaix diagrams provide information about the electrode potential – pH range in which a given species is stable. In Figure 11-5 the region in which metallic nickel is stable is called the immunity zone and the region in which Ni2+ is stable is called the corrosion zone. The regions in which the stable form is an oxide, or a hydroxide, are usually called passive zones although, as it will be seen later, corrosion will not be impeded in these zones except for certain cases in which the oxide film is compact enough. Pourbaix diagrams for most metals have been compiled (Pourbaix, 1974)

11.3. Corrosion kinetics 11.3.1. Direct measurement of corrosion rate If a metal M is immersed in an aqueous media in which species R is present, we know, from Chapter 3, that M will be spontaneously oxidized, and R reduced, if the electrical potential difference of the galvanic cell described

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161

by the reaction of Eqn. (11.6) is positive. This result, based on purely thermodynamic considerations, gives no information about the rate at which the process occurs; however, when corrosion of a metal part is under analysis, oxidation rate is a key point. In effect, the question that an engineer must be able to answer is not only whether the metal part will corrode but also, in case that corrosion exists, if its rate will be slow enough as to accept the use of metal M for the expected lifetime of the metallic part considered. When corrosion is uniform, which means that oxidation rate does not depend on the specific point of the metal surface, corrosion rate can be obtained by measuring the amount of metal dissolved in the unit time per unit area: ‫ݒ‬௖௢௥௥,௠ ൌ

ο௠

(11.34)

ο௧.ࣛ

where ࣛ is the area on which measurement is made and ο݉ is the mass loss in time interval ο‫ݐ‬. The usual international units for mass corrosion rate are mg.year-1.cm-2 and g.day-1.m-2 . An alternative way for corrosion rate evaluation is to determine the loss of thickness, d, of a metal part, ‫ݒ‬௖௢௥௥,ௗ . Of course, there is a direct relation between both ways of measuring corrosion rate as seen from Eqn. (11.35): ‫ݒ‬௖௢௥௥,௠ ൌ

ο௠ ο௧.ࣛ

ൌ ߩࣛ

οௗ ο௧.ࣛ

ൌ ߩ‫ݒ‬௖௢௥௥,ௗ

(11.35)

where ȡ is the metal density. Usual units for ‫ݒ‬௖௢௥௥,ௗ are mm.year-1 and mils.year-1 (mpy). Direct measurement of mass or thickness loss has several drawbacks: large times are required to reach reliable mass loss values and, moreover, in situ measurements are not possible. These shortcomings can be overcome remembering that, according to Faraday´s law, the mass of dissolved metal is proportional to the number of electrons transferred by the metal. Hence, if it were possible to determine the number of electrons transferred per unit time and unit area, we would have the current density associated with metal dissolution, icorr, from which the mass corrosion rate is readily obtained: ‫ݒ‬௖௢௥௥,௠ ൌ

௜೎೚ೝೝ ௐಲ జ‫࣠ כ‬

(11.36)

Where ܹ஺ is the atomic mass of the metal and ߭ ‫ כ‬the number of electrons involved in the oxidation reaction. Thus, since very low currents can be

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162

precisely measured, determining ݅௖௢௥௥ makes possible fast determinations of corrosion rates.

11.3.2. Electrochemical measurement of corrosion rate As has been said, corrosion is the consequence of spontaneous reactions that take place on the metal surface; now, since the potential difference between metal and solution is unique both, the oxidation and the reduction reactions must occur at the same electrode potential and electrons that are lost by the metal will move through it until they find a site in which the electron accepting species captures those charges, and a microscale galvanic cell is formed. Assuming that the corrosion process takes place via a single anodic and a single cathodic reaction we have that: ݅௡௘௧ ൌ ݅௥௘ௗ ൅ ݅௢௫ ൌ 0

(11.37)

and the corrosion current density is: ݅௖௢௥௥ ൌ ݅௥௘ௗ ൌ െ݅௢௫

(11.38)

Now, the electric potential difference between metal and solution at which corrosion occurs, ȟȲ௖௢௥௥ , is the same for both reactions and the overpotentials for each electrode are: ௥௘ௗ ߟ௥௘ௗ ൌ ȟȲ௖௢௥௥ െ οȲ௧௛௘௥௠

(11.39)

௢௫ ߟ௢௫ ൌ ȟȲ௖௢௥௥ െ οȲ௧௛௘௥௠

(11.40)

Hence, since it has been assumed that only one oxidation and only one reduction reactions occur, it is possible to determine the corrosion rate from the knowledge of the corresponding polarization curves. Taking the case of an iron pipe through which an acid solution flows, we have that the possible reactions are: Fe2+ + 2 e- ՞ Fe

(11.41)

2 H+ + 2 e- ՞ H2

(11.42)

and

If we can assume that for both reactions mass transfer overpotential is negligible, the corresponding polarization curves are described by the Butler

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163

– Volmer equation discussed in Chapter 5 and can be represented as shown in Figure 11-6





Figure 11-6: Plot of the anodic and cathodic branches for iron corrosion in acid media.

Obviously, net external current density is zero and Eqn. (11.37) can be written as: ݅௡௘௧ = ൫ଓሬሬሬሬሬԦ ுమ ൅ ଓരሬሬሬሬሬ൯ ுమ ൅ ሺଓሬሬሬሬሬԦ ி௘ ൅ ଓരሬሬሬሬሬሻ ி௘ ൌ 0

(11.43)

From Figure 11-6 it is clear that this condition can be satisfied only when the electrode potential is between the equilibrium electrode potential of both reactions: ௥௘ௗ ௢௫ ൐ ȟȲ௖௢௥௥ ൐ οȲ௧௛௘௥௠ ȟȲ௧௛௘௥௠

(11.44)

It is also clear that, at the corrosion potential, ቚ ሬሬሬሬሬሬԦ ݅‫ ݁ܨ‬ቚ ‫ ا‬ቚരሬሬሬሬሬሬ ݅‫ ݁ܨ‬ቚ and หଓሬሬሬሬሬԦห ுమ ‫ب‬ หଓരሬሬሬሬሬห ுమ then it is possible to write:

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164

െଓരሬሬሬሬሬ ி௘ = ଓሬሬሬሬሬԦ ுమ = ݅௖௢௥௥

(11.45)

Thus, it is concluded that when a piece of iron is immersed in an acid solution its electrode potential will evolve towards the corrosion potential. This type of graph is known as an Evans’ diagram and is useful in considering different aspects of the corrosion phenomenon. For instance, if we look at the impact of an increase in the difference of thermodynamic potentials (if pH diminishes, for example) corrosion rate will also increase, as can be seen by comparing Figure 11-6 with the new situation, depicted in Figure 11-7.

 Figure 11-7: Evans diagram for the system of Figure 11-6 when the equilibrium potential of the cathodic reaction moves to more positive values.

Analogously, if the catalytic properties of iron are modified and the exchange current is increased, corrosion rate also increases, as shown in Figure 11-8

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165

Figure 11-8: Evans diagram for the system of Figure 11-6 if the exchange current density of the cathodic reaction is increased.

If the change of catalytic properties does not affect exchange current values, but modifies the Tafel slope, corrosion rate is also changed with respect to the original case, as shown in Figure 11-9

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Figure 11-9: Evans diagram for the system of Figure 11-6 when the Tafel slope of the cathodic reaction is modified.

In many cases of practical interest, the difference between thermodynamic potentials of oxidation and reduction reactions is greater than 0.5 V and, then, Tafel´s equation is valid, and the corrosion potential expressed in terms of the anodic and cathodic reactions is: ௥௘ௗ ൅ ܽ௥௘ௗ ൅ ܾ௥௘ௗ logሺ݅௖௢௥௥ ሻ ȟȲ௖௢௥௥ ൌ οȲ௧௛௘௥௠

(11.46)

௢௫ ȟȲ௖௢௥௥ ൌ οȲ௧௛௘௥௠ ൅ ܽ௢௫ ൅ ܾ௢௫ logሺ݅௖௢௥௥ ሻ

(11.47)

These equations show that, if polarization curves for oxidation and reduction reactions are known, corrosion potential and corrosion current density can be calculated. When one of the reactions is controlled by mass transfer, corrosion rate cannot be obtained employing only Tafel parameters. One very common

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167

case in which this situation happens is when a metal is in contact with solutions that are saturated with air and pH is neutral or alkaline. In such a situation the cathodic reaction is oxygen reduction and, as discussed in Chapter 6 the overpotential for this reaction will be given by: ߟ ൌ ܽ ൅ ܾ logሺ݅ሻ ൅

ோ் ସఈ೟೘ ࣠

௜

ln ቀ1 െ ቁ ௜ ಽ

(11.48)

where, from Eqn. (6.15): ݅௅ ൌ 4࣠݇௠ ܿைమ

(11.49)

Figure 11-10 shows cathodic and anodic polarization curves for this case at different hydrodynamic conditions.

Figure 11-10: Evans diagram for corrosion of metal M with oxygen reduction as cathodic process. (a) Corrosion potential variation for different fluid velocity values. (b) Corrosion rate change with fluid velocity.

Up to now we have considered corrosion as being due to unique reactions occurring one at the anode and the other at the cathode. If this is the case, it is clear that the reaction whose electrode potential is lower will proceed towards oxidation, while the other reaction will occur towards reduction. However, when three or more reactions might take place at the metal surface it is necessary to take a careful look at the characteristics of each case. Let us consider a system in which the following reactions take place: ଵ ଶ

ܱଶ ൅ ‫ܪ‬ଶ ܱ ൅ 2 ݁ ି ՞ 2 ܱ‫ି ܪ‬

‫ܯ‬ଶା ൅ 2 ݁ ି ՞ ‫ܯ‬

(11.50) (11.51)

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168

‫ ܬ‬൅ ‫ܪ‬ଶ ܱ ൅ ݁ ି ՞ ‫ ܪܬ‬൅ ܱ‫ି ܪ‬

(11.52)

for which ௃ ை ெ οȲ௧௛௘௥௠ ൏ οȲ௧௛௘௥௠ ൏ οȲ௧௛௘௥௠

(11.53)

Since the metal solution potential difference, ȟȲ௖௢௥௥ , is unique and the net current density is zero: |݅ெ | െ |݅ை | േ ห݅௃ ห ൌ 0

(11.54)

According to the electrode potential values, reaction (11.50) will proceed as a reduction reaction while (11.51) will occur as oxidation reaction. In order to determine what will be the behaviour of reaction (11.52) it is necessary to consider the set of equations: ெ ܽெ ൅ ܾெ ݈‫݅|݃݋‬ெ | ൌ ȟȲ௖௢௥௥ െ οȲ௧௛௘௥௠

(11.55)

ை െ ȟȲ௖௢௥௥ ܽை ൅ ܾை ݈‫݅|݃݋‬ை | ൌ οȲ௧௛௘௥௠

(11.56)

௃ ห ܽ௃ ൅ ܾ௃ ݈‫݃݋‬ห݅௃ ห ൌ േ หȟȲ௖௢௥௥ െ οȲ௧௛௘௥௠

(11.57)

that can be solved if Eqn. (11.54) is included. It is clear that Tafel parameters in Eqn. (11.55) are those of the anodic polarization curve while those of Eqn. (11.56) correspond to the cathodic branch of the Butler – Volmer equation. For Eqn. (11.57) the system will have a physically meaningful solution only for one branch of the Butler – Volmer equation. This kind of situation is observed, for example, in the case of iron corrosion in a slightly acid solution where the reduction equations that occur are, simultaneously, hydrogen evolution and dissolved oxygen reduction. This analysis can be extended to the case of more than three reactions and, in all cases, consistent solution will be obtained when, for all reactions with electrode potentials between the most positive and the most negative, the adequate branch of the polarization curve is considered.

11.4. Oxide formation and passivation Although Pourbaix diagrams are very important in understanding the corrosion phenomenon, it must not be forgotten that these graphs are obtained from thermodynamic calculations and correspond to systems in equilibrium, while in almost all cases of practical interest one must deal with

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169

systems that are not in equilibrium, but evolving, with different rates, towards it. Thus, when the Pourbaix diagram for nickel indicates that, at certain pH and electrode potential values, NiO is the stable form for the metal in contact with an aqueous solution, no information is provided with respect to the rate at which the oxidation of Ni to NiO will proceed. These considerations are particularly relevant when studying the behaviour of metal and alloys in actual systems. Taking the case of iron, we have that the Pourbaix diagram for this metal, that is shown in Figure 11-11

Figure 11-11: Potential – pH diagram for iron.

indicates that at pH values below 7, iron is stable in ionic form, and hence will dissolve, if the electrode potential is above the thermodynamic potential of the reaction: Fe2+ + 2 e- ՞ ‫݁ܨ‬

(11.58)

However, if the dependence of current density on electrode potential is experimentally studied for iron in a 0.5 M H2SO4 solution, it is found that the anodic branch of the polarization curve gives results of the type shown in Figure 11-12

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Chapter Eleven

Figure 11-12: Schematic representation of the anodic polarization curve of iron in 0.5 M H2SO4. The dashed curve indicates the shape of the expected polarization curve if there is no passivation.

Initially, as the electrode potential is increased from the thermodynamic value for reaction (11.58), it is seen that, as expected, current density increases and Fe2+ ions are dissolved. However, as electrode potential gets more positive, current density values raise at a lower rate until, at the critical value icrit, which corresponds to the critical potential P, further increments of electrode potential lead to a brisk current diminution until an almost constant value, ipassive, which is several orders of magnitude lesser than icrit, is attained. The electrode potential range at which ipassive is maintained is called the passive zone and chemical analysis shows that current density is due to a slight dissolution of Fe3+ ions. When electrode potential increases above 1.5 V current density increases again and the electrode is said to have entered the transpassive zone in which dissolution to Fe3+ ions is larger and, above 1.6 – 1.65 V oxygen evolution also occurs. This behaviour was explained by Morris Cohen (Cohen, 1959) who identified the growth of a thin film of Ȗ-Fe2O3 which develops from Fe3O4 formed by chemical reaction of Fe with water. The Ȗ-Fe2O3 film is quite

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compact and hinders ion diffusion towards solution, leading to very low passivation currents. The stability of this oxide film is strongly dependent on pH, as shown in Figure 11-13, where data from polarization curves obtained at different pH values are shown (Sato, 1978)

 Figure 11-13: Anodic polarization curves of iron in phosphate solutions at several pH values. (1) pH = 1.85; (2) pH = 3.02; (3) pH = 4.55; (4) pH = 7.45; (5) pH = 8.42; (6) pH = 9.37; (7) pH = 11.5.

Passive film formation permits that a metal is not corroded at significant rate although, according to thermodynamics, it should oxidize. Aluminium is a very clear example of this behaviour since, according to its standard electrode potential (-1.7 V), it should violently react with water; however, the compact oxide film that is almost instantaneously formed creates a passive oxide layer. It is interesting, then, to compare the thermodynamic scale of metal stability (immunity) with the scale displayed in Table 11-1, in which the effect of passivity is included.

172

Order 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

Chapter Eleven

Thermodynamic immunity Au Pt Rh Pd Hg Ag Os Cu Bi C Pb Ni Co Cd Fe Sn Mo W Zn Nb Ta Cr V Mn Zr Al Hf Ti Mg

Immunity passivity Rh Nb Ta Au Pt Ti Pd Os Hg Zr Ag Sn Cu Hf Al Cr Bi W Fe Ni Co C Pb Cd Zn Mo V Mg Mn

+

Table 11.1: Metal stability for corrosion in water containing dissolved oxygen at pH ~7. As seen from this Table, significant modifications are observed regarding corrosion resistance when passivity is included. For instance, titanium, which in contact with water develops a very compact TiO2 film, moves from the 28th place in the thermodynamic corrosion scale to the sixth place if

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passivity is included; by its side tantalum moves from the 21st place in thermodynamic stability to the third place, overcoming gold and platinum. The oxide film stability is an important feature with respect to passive behaviour, and it can be qualitatively estimated through an experimental technique first applied by Flade (Flade, 1911) that consists in polarizing the electrode into the transpassive zone and then, after disconnecting the power source, proceed to measure the electrode potential evolution with time. The result of such an experiment on iron in 0.5 M H2SO4 solution is shown in Figure 11-14

Figure 11-14: Electrode potential change of passivated iron when power source is disconnected after being polarized at 2.2 V.

The electrode potential falls, rapidly at first and then at a much slower rate, until it attains a critical value, ǻȌF, wherefrom a sharp fall takes place until the electrode potential is in the active zone. This critical potential is called the Flade potential and, as shown by a large number of experiments, varies linearly with pH. For iron it has been determined that:

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߂ߖி ሺ‫݁ܨ‬ሻ ൌ 0,63ܸ െ 0,059ܸ ή ‫ܪ݌‬

(11.59)

while, for chromium, the change in Flade potential with pH is: ߂ߖி ሺ‫ݎܥ‬ሻ ൌ െ0,20ܸ െ 0,059ܸ ή ‫ܪ݌‬

(11.60)

This linear relationship is observed for many metals and alloys although, particularly in the case of alloys, the slope not always is 0.059 V. It must be mentioned that the Flade potential value is near that at which passivation begins although it is not exactly coincident. The Flade potential is related to oxide film stability: more negative values of ߂ߖி indicate greater stability. This is illustrated in Figure 11-15 in which ߂ߖி has been measured for a series of Fe – Cr alloys (Franck, 1949; Rocha and Lennartz, 1955; King and Uhlig, 1959)

Figure 11-15: Flade potential of Fe-Cr alloys of varying composition.

11.5. Types of corrosion Corrosion of metal parts is produced under different forms and because of a variety of causes. In general terms, when corrosion takes place

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homogeneously on a surface it is relatively easy to implement maintenance and preventive measures. However, when corrosion attack concentrates in specific points of the surface its action might lead to severe failures from which a particular piece or even a complex structure might collapse. It is important, then, to identify the types of attack that may happen under different conditions.

11.5.1. Uniform corrosion Uniform corrosion is the kind of attack that is usually seen in metallic structures that are in direct contact with the atmosphere or immersed in rivers, lakes or the sea. This type of corrosion produces rust on iron or steel elements and is also seen in a vast number of cases such as the blackening of silver pieces. It is the consequence of a homogeneous attack on the metal surface in which an oxide layer is formed that is either not compact, and cannot impede that corrosion continues, or not adherent and is not attached to the surface. In both cases, mass and thickness of the metal part will diminish with time. As mentioned in Section 11.3.- uniform corrosion rate can be obtained from measurements of mass loss. However, this type of technique is usually limited to perform tests of different materials exposed to natural surroundings or to specific environments. To carry on these tests, samples of the material to be evaluated are prepared according to specific protocols, that are defined for each case, and exposed to the chosen environment. In doing this, a statistically significant number of samples are weighed before the test begins and, at programmed intervals, a predetermined number of samples are picked out, treated according to the protocol, and weighed again to determine the mass loss. Figure 11-16 shows a device employed for evaluating atmospheric corrosion.

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Figure 11-17: Device for atmospheric corrosion evaluation of several alloys. Photograph courtesy of Mahdi Shiri (Metallurgy Dept. – Niroo Research Institute).

When it is necessary to select the material with which a structure that is to be exposed to a specific environment should be built, it is necessary to guarantee that the test will be carried out under conditions that copy, as precisely as possible, the characteristics to which the structure will be exposed, which can be an aggressive atmosphere in a chemical plant, the seaside atmosphere in a tropical installation, etc. Such a simulation is usually performed employing the so called cloud chambers whose characteristics are sketched in Figure 11-17.

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Figure 11-17: Cloud chamber for atmospheric corrosion studies.

As mentioned before, mass loss determinations have practical limitations, the most important of them being the necessary time to obtain reliable data. Thus, if a corrosion rate of 1 mm.year-1, which is a quite critical value, is to be determined more than a month must pass to have a mass loss of 30 mg.cm-2. As a consequence, whenever it is technically possible, electrochemical techniques are preferred, especially when the environment is a liquid solution since, in this case the corresponding Evans diagram, and hence icorr, can be readily obtained from the anodic and cathodic polarization curves. This procedure, although relatively simple to be performed at the laboratory is not always adequate when corrosion rate of a metal part in a complex medium, such as an iron rod in concrete or a metal piece in a bone or a dental implant, is to be measured. In such cases, an option to determine icorr is the use of the linear polarization technique developed by Stern and Geary (Stern and Geary, 1957) whose theoretical foundations are based on the use of the Butler – Volmer equation at low overpotentials. As it has been discussed in Section 11.3, when a metal corrodes its electrode potential is ȟȲ௖௢௥௥ and net current density is zero. Then, if a low intensity current Iappl is externally imposed, electrode potential will be displaced with respect to corrosion potential in an amount ߳ ൌ ȟȲ െ ȟȲ௖௢௥௥ and it is possible to define the polarization resistance as:

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178

ܴ௣௢௟ ൌ െ

ఢ

(11.61)

ூೌ೛೗

which is easily obtained experimentally. Now, ߳ may be linked to icorr if it is observed that anodic and cathodic currents can be written as: ൫୼ஏି୼ஏ೟೓೐ೝ೘,ೝ೐೏ ൯

‫ܫ‬௥௘ௗ ൌ ࣛ݅௢,௥௘ௗ exp ൤ ‫ܫ‬௢௫ ൌ െࣛ݅௢,௢௫ exp ቂ

൨

ᇲ ௕ೝ೐೏

(11.62)

൫୼ஏି୼ஏ೟೓೐ೝ೘,೚ೣ ൯

ቃ

ᇲ ௕೚ೣ

(11.63)

ᇱ ᇱ (negative) and ܾ௢௫ (positive) are the where ࣛ is the electrode area and ܾ௥௘ௗ Tafel slopes (employing natural logarithms) for the cathodic and anodic reactions of the corrosion process. From Eqn. (11.38) it is possible to establish that: ൫୼ஏ೎೚ೝೝ ି୼ஏ೟೓೐ೝ೘,ೝ೐೏ ൯

‫ܫ‬௖௢௥௥ ൌ ࣛ݅௢,௥௘ௗ exp ൤ െࣛ݅௢,௢௫ exp ቂ

ᇲ ௕ೝ೐೏ ൫୼ஏ೎೚ೝೝ ି୼ஏ೟೓೐ೝ೘,೚ೣ ൯

ቃ

ᇲ ௕೚ೣ

൨ൌ (11.64)

From this equation, the definition of ߳, and Eqns. (11.62) and (11.63) the cathodic and anodic components of Iappl are: ᇲ

‫ܫ‬௥௘ௗ ൌ ‫ܫ‬௖௢௥௥ ݁ ఢ/௕ೝ೐೏

(11.65)

ᇲ

‫ܫ‬௢௫ ൌ െ‫ܫ‬௖௢௥௥ ݁ ఢ/௕೚ೣ

(11.66)

hence: ᇲ

ᇲ

‫ܫ‬௔௣௣௟ ൌ ‫ܫ‬௖௢௥௥ ቀ݁ ఢ/௕ೝ೐೏ െ ݁ ఢ/௕೚ೣ ቁ

(11.67)

Since ߳ is small it is possible to expand this equation in series and, taking the linear term: ‫ܫ‬௔௣௣௟ ൌ െ‫ܫ‬௖௢௥௥ ߳

ᇲ ᇲ ሻ ሺି௕ೝ೐೏ ା௕೚ೣ ᇲ ௕ᇲ ௕ೝ೐೏ ೚ೣ

(11.68)

Writing Tafel parameters in decimal logarithms we finally obtain the Stern – Geary equation: ݅௖௢௥௥ ൌ

௕ೝ೐೏ ௕೚ೣ

ଵ

ଶ,ଷሺି௕ೝ೐೏ ା௕೚ೣ ሻ ோ೛೚೗

(11.69)

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179

In the case that the cathodic reaction is controlled by mass transfer, as it happens for oxygen reduction, the cathodic slope is much larger than that of the anodic reaction and we have ݅௖௢௥௥ ൌ

௕೚ೣ ଶ,ଷோ೛೚೗

(11.70)

The linear polarization technique allows fast corrosion current measurements even in difficult environments, as is the case depicted in Figure 11-18 where the corrosion current of an iron rod in reinforced concrete is measured.

Figure 11-18: In situ corrosion current measurement in reinforced concrete by linear polarization.

Linear polarization techniques are very useful in practical cases, but should be applied with some care because, normally, commercial devices based on this technique employ predefined Tafel slope values. Also, electrode areas are not always well defined and resistance effects are dismissed. Anyhow, with due caution, linear polarization is a valuable technique.

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11.5.2. Localized corrosion When corrosion proceeds uniformly it is a relatively simple matter to plan protection actions, maintenance programs and parts or systems replacement; on the contrary, in those cases in which corrosion is located in specific areas of structures, parts or equipment an adequate prevention and maintenance program is mandatory for preventing failures that can seriously affect operations and, even, lead to catastrophic events. It is necessary then, for a process engineer, to have a good understanding of the causes underlying different forms of localized corrosion. Localized corrosion occurs in a variety of situations which obey to specific causes. For this reason, it is essential to understand the mechanisms which trigger corrosion in each case. In what follows a brief description of the most frequent types of localized corrosion is given, which should be supplemented with further reading if design decisions are to be taken. 11.5.2.1. Differential aeration (or crevice) corrosion This is a very common case of localized corrosion and its occurrence is due to differences in the oxygen concentration at different sites of the metal. For example, if over a plane sheet of iron a water drop has formed at a given point, we have the situation that is depicted in Figure 11-19

Figure 11-19: Schematic description of the differential aeration corrosion mechanism.

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181

As it is seen, oxygen partial pressure at the drop border is higher than that at the inside and, consequently, the metal - solution potential difference at the border will differ from that in the drop centre, and a local galvanic cell will be created in which the oxygen rich zone will act as a cathode and, in the centre, metal will be dissolved. Another case of this type of corrosion occurs in metal pillars which support docks, as schematically shown in Figure 11-20

Figure 11-20: Differential aeration localized corrosion in a dock pillar.

As it is seen, the higher oxygen concentration near the water line makes this zone cathodic, and corrosion will occur in areas of the pillar on which oxygen concentration is lower. A particular and very frequent differential aeration corrosion case is the one produced in crevices, such as those formed when two pieces of the same metal are in contact with each other. As shown in Figure 11-21 the external zone of the crevice has free access to air and, consequently, oxygen concentration there will be much larger than that at the inner region, where oxygen access is more difficult. The external zone of the crevice will act as a cathode and corrosion will be produced in the internal region of the crevice. Figure 11-22 shows a case of this type of localized corrosion.

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Figure 11-21: Differential aeration corrosion in a crevice.

Figure 11-22: Corrosion at a crevice between two metal plates (Photograph courtesy of Kelley, Drye & Warren LLP).

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The usual way to avoid crevice corrosion is the use of non-conducting gaskets and O-rings. 11..5.2.2. Galvanic corrosion This type of localized corrosion is observed when metallic surfaces of different composition are in direct contact. A very common case is that of a bronze faucet that is screwed into a steel pipe for water service: in such a case a potential difference is established in which the metal with lower electrode potential (the less noble metal) oxidizes while the more noble (bronze in this case) acts as a cathode for oxygen reduction. Figure 11-23 shows a case of galvanic corrosion.

Figure 11-23: Galvanic corrosion at a bronze – steel contact.

Main points to be taken into account in galvanic corrosion are the following: x Standard electrode potential of the metals that are in direct contact. x Hydrogen evolution and oxygen reduction reaction kinetics. x Characteristics of the solution which is in contact with both metals: pH, conductivity, nature of existing ions, temperature, hydrodynamics. x System geometry: size of the parts, distance, shape. Consideration of these factors allows the evaluation of different actions that can be taken for preventing corrosion when, for particular reasons, it is necessary to connect different metals in a conducting medium. The most obvious is to avoid the direct contact by introducing isolating films between both metals, or employing a non-conducting adapter, when the chemistry of

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the involved systems makes this possible. In some cases, the less noble metal area can be increased thus reducing the current density on it. 11.5.2.2. Dealloying The structure of an alloy is characterized by the existence of small particles, called grains, of the component metals. This type of localized corrosion is produced because of the selective dissolution of one of the alloy components in an aggressive medium. Brass, which is an alloy of copper and zinc, is characteristically affected by dealloying corrosion because of the dissolution of zinc, which leaves a porous residue of copper and corrosion products. In Figure 11.24 a brass valve that has been affected by dealloying is shown.

Figure 11-24: Preferential dissolution of zinc in a brass valve (https://corrosiondoctors.org/Forms-selective/dezinc-valve.htm).

11.5.2.4. Intergranular corrosion This type of attack is produced on the alloy grain boundaries and is due to local inhomogeneities. Because of its nature it is very difficult to detect it by visual inspection and is usually assessed by post-mortem studies, performed after the metal part has failed and there are reasons to believe that intergranular corrosion has happened. In these cases, after replacement of the part, a sample is cut from it and, after acid leaching, submitted to metallographic microscopy. Figure 11-25 shows a micrograph of a sample of stainless steel affected by intergranular corrosion as is evidenced by the accumulation of darker corrosion products on grain borders.

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Figure 11-25: Microscopic image of an intergranular corrosion case.

Stainless steels are alloys whose main components are iron, carbon, nickel and chromium and whose passive film stability is mainly due to the latter. Careful study by scanning electron microscopy showed that, in this case, chromium carbide had accumulated on grain borders and, as a consequence, local galvanic pairs were formed which fostered localized corrosion. This type of attack is frequently linked to an inadequate thermal treatment of the alloy. In this respect, it is very important to emphasize that the specific techniques employed in the production of metal parts should be known and must be specified by the purveyors including not only thermal treatments applied on the raw material and during manufacturing, but also the applied machining methods, employed surface conditioning techniques and the performed chemical treatment. 11.5.2.5.- Pitting corrosion When in the metal environment significant concentrations are present of some anions like Cl-, S2-, S2O32- or F-, which affect the oxide film stability, a very localized attack is generated in which the metal is affected by the formation of pits. Figure 11-26 shows, schematically, anodic polarization curves taken at different chloride concentrations on steel electrodes.

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Figure 11-26: Chloride concentration effect on pitting.

As seen from this graph, chloride ions affect the potential range of the passive zone. When no chlorides are present passivity breakdown occurs at ȟȌ1 with Fe3+ dissolution and oxygen evolution at higher potential values. When chloride ions are present passivity breaks down at lower potentials, current density increases, and pit formation is observed; the potential value at which this happens is called the critical pit potential. In the case of Figure 11-26 at chloride concentrations c2 and c3 critical pit potentials are ȟȌ2 and ȟȌ3, respectively. The first step in the mechanism of pit formation is the displacement of oxygen atoms in the oxide film by chloride anions, that react with the metal

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forming salts that are hydrolysed. The hydrolysis reaction lowers pH at very restricted areas where local corrosion is enhanced. Figure 11-27 shows, schematically, this mechanism.

Figure 11-27: Schematic description of the mechanism of pit formation.

When a salt whose ions do not interfere with the oxide layer stability is added, the impact of aggressive ions is reduced. Figure 11-28 shows the behaviour of polarization curves obtained for an electrode of 18-8 stainless steel in NaCl 0.1 M with different additions of Na2SO4 (Leckie and Uhlig, 1966)

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Figure 11-28: Polarization curves of 18-8 stainless steel in NaCl 0.1 M with different additions of Na2SO4. (1) Na2SO4 1 M; (2) Na2SO4 0.15 M + NaCl 0.1 M; (3) Na2SO4 0.1 M + NaCl 0.1 M; (4) Na2SO4 0.05 M + NaCl 0.1 M; (5) Na2SO4 0.0125 M + NaCl 0.1 M; (6) NaCl 0.1 M,

Figure 11-29 shows a case of pitting corrosion on aluminium.

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Figure 11-29: Pitting corrosion case on an aluminium plate.

Chloride ions exist in a wide variety of natural waters, a fact that makes pitting corrosion a very frequent problem in ocean coasts and causes serious damage to iron and aluminium alloys in this environment. A number of catastrophic failures have occurred for this reason, such as the Champlain Towers building collapse in Miami in 2021. 11.5.2.6.- Corrosion coupled with mechanical efforts. When a metal part is exposed to mechanical efforts in a corrosive environment, corrosion rates are enhanced and the attack on the metal is occurs preferably on those parts of the piece that are affected by the mechanical requirement. There are several kinds of this type of corrosion: fretting corrosion, cavitation-erosion corrosion and stress corrosion. Fretting corrosion is frequently observed in pieces in which specific zones are continuously exposed to a slight relative slip with another surface. The slip is usually oscillatory, but it also happens in other situations as when a cylinder moves slightly faster than another which is in contact with the first. Figure 11-30 shows a bearing affected by this type of corrosion.

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Figure 11-30: Case of fretting corrosion on a bearing. Photograph courtesy of Philippe Gachet (SKF).

Cavitation – erosion corrosion is frequent in pump rotors that move liquids with suspended particles (erosion) and/or bubbles which produce mechanical impacts (cavitation). Figure 11.31 displays an example of this phenomenon.

Figure 11-31: Cavitation corrosion in centrifugal pumps rotors. Photograph courtesy of Prof. Güner (Ankatra University).

Stress corrosion has been responsible for a good share of the catastrophes produced because of corrosion effects: planes have crashed due to stress corrosion of bolts in a wing, gas transmission pipelines have exploded due to leakages through cracks produced by stress corrosion, bridges have fallen down because of stress corrosion cracking at critical structural parts.

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Generally, stress corrosion occurs when stresses are applied on a susceptible metal piece which is in contact with an aggressive environment. Stresses might be due to the specific work conditions on the affected piece or to residual stresses that exist within it and had been induced by the processes employed in its manufacturing or during installation that include: x x x x x

Thermal treatments. Machining techniques. Mechanical deformations. Inadequate soldering procedures. Applied chemical treatments.

Stress corrosion caused by specific work conditions depends on three items: x Chemical composition of the aggressive medium. x Metallurgical characteristics of the metal or alloy. x Mechanical stresses. Each of these items includes a large number of variables and, consequently, there is not a single mechanism that explains stress corrosion. A specific case of catastrophic consequences due to stress corrosion was the collapse, in 1967, of the Silver Bridge on the Ohio river which caused the death of 46 people. In this case stress corrosion advanced on the basis of a cable, probably because of accumulation of rainwater with low concentrations of ammonium nitrate. The material was carbon steel (0.7% C) and the internal fracture of the collapsed cable had just 2.5 mm long. Figure 11-32 shows a photograph of the bridge at the year of its inauguration and another photograph after the collapse.

Figure 11-32 (a) Silver Bridge after inauguration in 1955.

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Figure 11-32 (b) Silver Bridge after its collapse in 1967. Photographs from https://en.wikipedia.org/wiki/Silver_Bridge.

11.5.2.7. Hydrogen damage When corrosion occurs in acid media or when a buried or submerged structure is under cathodic protection, different deleterious phenomena can affect the metal. These are globally named as “hydrogen damage” being the two main cases hydrogen embrittlement and blistering. The cause of these events is found in the fact that hydrogen evolution proceeds, as mentioned in Chapter 5, with an adsorbed hydrogen atom as intermediary. The adsorbed atoms, depending on the chemical nature and metallurgical characteristics of the metal, might migrate into the solid metal and form hydrides (embrittlement) or can accumulate in local voids and recombine forming hydrogen molecules: if the gas pressure in the void increases beyond a certain limit, blisters can be formed. Figure 11-33 shows a sketch of this latter mechanism.

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Figure 11-33: Blister formation due to hydrogen atoms diffusion in the metal.

Figures 11-34 and 11-35 show, respectively, embrittlement of a steel plate and blistering in aluminium.

Figure 11-34: Hydride embrittlement of a steel plate. https://upload.wikimedia.org/wikipedia/commons/9/99/Steel-with-HydrogenInduced-Cracks-01.jpg

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194

Figure 11-35: Blisters formation on aluminium. https://www.researchgate.net/figure/Blistering-in-internalsurface_fig1_327233969/download

11.6. Protection methods Protection methods can be classified in four main groups: x x x x

Coatings application Use of inhibitors and passivators Cathodic protection Anodic protection

11.6.1. Coatings Protection of metallic surfaces by application of coatings can be performed through the use of three types of coverings: x Metallic x Inorganic x Organic When resorting to this type of protection method it must be kept in mind that the purpose of any coating is to be a protective barrier that impedes that the aggressive medium be in touch with the metallic substrate. However, every coating has a certain porosity degree and, also, defects that can leave very small areas without protection. These coating defects increase with time and continuous use and may generate sites where differential aeration corrosion might occur at a very limited zone, thus generating potentially

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serious problems. For such a reason, it is very important to make an adequate coating selection and maintenance program. Metallic coatings: This technique consists of depositing, on the metal surface to be protected, metal layers of adjustable width. These layers can be applied by different methods the most common being immersion in the liquid metal (zinc or aluminium on steel) or electrodeposition. In the latter case the piece to be protected is immersed in a solution of specific composition and an adequate potential difference is applied with the piece to be protected being the cathode of the circuit, Metallic coatings can be classified in two types, noble or sacrificial. In the first case the coating metal is thermodynamically more stable and simply acts as a barrier for corrosion. Sacrificial coatings are made of metals that are less stable than the base metal and, besides acting as a barrier, provide the metal base an additional protection since in points at which the barrier effect fails, coating and the base metal will form a galvanic pair in which the metal acts as cathode. Examples of noble metal coatings are nickel and chromium layers deposited on steels. These coatings are very compact and show very low porosity degree, which is practically zero for thickness greater than 0.1 mm. Zinc and aluminium are classical examples of sacrificial coatings. Inorganic coatings: The most common are glass coatings of suitable expansion coefficient, which are fused on metals and include vitreous and porcelain enamels as well as glass linings. These coatings are applied on previously treated surfaces, usually with acids to induce high porosity on the surface, and then the glass or enamel is applied and the piece is taken to an oven at high temperature. Other type of inorganic coating is obtained by in situ reactions that lead to the formation of a protective layer. An example of this procedure is phosphate coatings on steel that are obtained by spraying on the metal surface an acid solution of zinc orthophosphate (ZnH2PO4 + H3PO4), that generates a porous network of iron phosphate crystals. Although phosphate coatings are not efficient corrosion barriers its use is widely employed because they are a very good basis for paints application. Oxide coatings on aluminium can be produced at room temperature by electrochemical oxidation in an adequate electrolyte at current densities about 10 mA.cm-2. The obtained Al2O3 coating is immediately hydrated by

196

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treatment with steam or hot water for several minutes which leads to an efficient sealing of the oxide layer Organic coatings: This kind of coatings include paints, plastic or polymeric linings (rubber, neoprene, polyethylene, etc.). It is, by far, the most commonly used coating and, in fact, during 2006 total sales of paints, lacquers and varnishes was 21 billion dollars in the U.S.

11.6.2. Inhibitors and passivators An inhibitor is a chemical compound that, at very low concentration, sharply reduces corrosion rate. These products can be classified as organic inhibitors and vapour phase inhibitors. Passivators are inorganic oxidizing compounds (chromates, nitrites, molibdates, tungstates) that passivate the metal by shifting the corrosion potential to values in the passive zone. Organic inhibitors frequently employed in acid media are molecules that adsorb on the metal surface, usually in a monolayer, impeding metal dissolution and hydrogen evolution. These are molecules, with a not very high molecular mass, containing chemical groups that efficiently adsorb on the metal surface. Some of the most popular inhibitors are Į-nitroso ȕnaphthol, p-nitroaniline, piperidine, cyclohexylamine and benzotriazole. Vapour phase inhibitors are compounds of low vapour pressure that work through an adsorption mechanism similar to that of organic inhibitors. Dicyclohexylammonium nitrite is frequently used. Passivators are ions whose thermodynamic electrode potential allows compact oxide formation with a fast reaction rate. In the case of dichromate, the occurring reaction is: ‫ݎܥ‬ଶ ܱ଻ଶି ൅ 8‫ ܪ‬ା ൅ 2‫ ݁ܨ‬՜ 2‫ ݎܥ‬ଷା ൅ 4‫ܪ‬ଶ ܱ ൅ ‫݁ܨ‬ଶ ܱଷ

(11.71)

Some ions that, thermodynamically, could be employed as passivators do not have an adequate reaction rate and cannot be used. This is the case of sulphates, nitrates or perchlorates. In order to be efficient, a passivator must be applied at concentrations that are above a certain critical value, since at lower concentrations the passive film is not completely formed and, thus, localized corrosion occurs, specially under the form of pits. Critical concentrations for the aforementioned ions are between 10-4 and 10-3 M.

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197

Some organic compounds, like sodium benzoate or sodium cinnamate, which do not have oxidizing properties are used as passivators in neutral or alkaline solutions. The mechanism in this case seems to be the formation of a film that permits dissolved oxygen molecules to pass through it, forming the oxide film. This assumption is supported by the fact that a 0.0005 M solution of sodium benzoate is a good inhibitor on steel, when immersed in water that contains dissolved oxygen, but does not induce metal passivation in deoxygenated water.

11.6.3. Cathodic protection The most efficient way of corrosion control consists in establishing conditions by which the metal part or structure to be protected is transformed into the cathode of an electrochemical reaction at an electrode potential in the immunity zone. This condition might be attained by two alternative ways: a) Cathodic protection by impressed current (CPIC) b) Cathodic protection by sacrificial anode (CPSA) In the first case a circuit is built in which an electrical current is applied, with the metal part to be protected being the cathode of the circuit. In the second case a metal or alloy, that is less noble than the metal to be protected, is short-circuited with it, thus creating a galvanic cell in which the cathode is the structure or part to be protected. In both cases energy is consumed: in CPIC electrical energy is taken from a power source while, in CPSA chemical reaction energy is transformed into electricity. The most important limitation to cathodic protection is due to the fact that a conducting medium must exist and, as a consequence, this technique is widely used in buried or submerged structures but, in general, is not applicable to atmospheric corrosion protection. Since cathodic protection establishes the metal potential in the immunity zone, any type of corrosion is impeded but a wrong design might produce hydrogen damage if hydrogen evolution on the protected metal is large. General aspects of CPIC The circuit employed in this type of cathodic protection is schematically shown in Figure 11-36. In this example a buried pipe is the protected structure.

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Figure 11-36: Sketch of CPIC circuit.

In this case the applied potential difference must be large enough to bring the pipe electrode potential into the immunity zone of the Pourbaix diagram. However, some considerations are necessary when designing this kind of circuit. In the first place it is important that the electric potential and current distribution lines on the protected structure be as homogeneous as possible. Thus, the geometry of the system must be taken into consideration and equations (10.1) and (10.3), with the particular boundary conditions of the case, must be solved. For instance, when cathodic protection is applied to a buried tank an acceptable current distribution is obtained when the anodes are positioned as shown in Figure 11-37

Figure 11-37: Sketch of CPIC on a buried tank with symmetrical positioned anodes.

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In the case of a pipeline, anodes and power sources should be distributed along it as shown in Figure 11-38

Figure 11-38: Sketch of the CPIC system in a sector of a pipeline.

When a pipeline which is not painted or protected by other non-conducting coating is under CPIC the electrical potential varies along the pipeline according to Revie and Uhlig (Revie and Uhlig, 2008): ȟȲ௫ െ ȟȲ௖௢௥௥ ൌ ሺȟȲ௢ െ ȟȲ௖௢௥௥ ሻ exp ൤ቀ

ିଶగ௥ோಽ ଵ/ଶ ௞

ቁ

‫ݔ‬൨

(11.72)

where ȟƘx is the electrode potential at point x, measured as the distance to the point of contact of the power source, ȟƘo is the electrode potential at the contact point, r the pipe radius, RL the resistance per unit length and k an empirical parameter. Obviously, ȟƘx - ȟƘcorr tends to zero at large distances from the contact point and, then, it is necessary to repeat the installation at regular intervals. It is important to consider the nature of the reactions occurring at the electrodes. According to the pH value of the conductive medium the most common reactions on the cathode are hydrogen evolution or oxygen reduction. In the anode, electrode reactions depend on the nature of the

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employed material: on an inert anode oxygen evolution is usually produced but, if the anode material can be oxidized other reactions occur which imply dissolution of the employed metal. Current density values adequate for a good protection depend on the difference between the corrosion potential and the equilibrium potential of the metal to be protected. For carbon steel buried structures, a reasonable value is usually 0.1 A.m2, while in the case of hot water or sea water 0.15 A.m2 are recommended. If the medium is fresh water 0.05 A.m2 are acceptable. Further aspects that must be considered are overprotection, which leads to high values of hydrogen production and possible hydrogen damage, and anode materials selection in which the characteristics of the anodic reactions, the time during which anodes can be employed at the adequate current density and economic costs must be considered. General aspects of CPSA This technique employs anodes that are thermodynamically more reactive than the protected metal in a circuit as the one of Figure 11-39

Figure 11-39: CPSA basic circuit.

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Usually, sacrificial anodes are made with magnesium or aluminium alloys with which the potential difference of the galvanic cell is in the 1 – 1.5 V; however, its use is limited because of the restricted area that can be protected by a single anode, and usually it is necessary to employ several units as shown, for the case of a ship hull and rudder, in Figure 11-40

Figure 11-40: Distribution of sacrificial anodes in the hull and rudder of a ship.

CPSA is particularly useful in the case of movable structures, like ships, or devices that must be protected internally, like water tanks or storage water heaters as shown in Figure 11-41

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Figure 11-41: CPSA for a domestic storage water heater.

11.6.3. Anodic protection This type of protection technique can be effectively used on metals and alloys with a passivation zone that, besides an ample electrode potential range, must show very low passivation current density values. The procedure, originally proposed by Edeleanu (Edeleanu, 1954) is based on the application of an anodic potential which makes that the protected metal electrode potential be within the passivation zone. Figure 11-42 shows the basic circuit

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Figure 11-42: Anodic protection of a steel tank. (1) Tank; (2) Tank connection to positive pole of the power source; (3) Reference electrode; (4) Counterelectrode; (5) Potentiostat; (6) Voltmeter; (7) Ammeter.

The viability of anodic protection depends on the characteristics of the polarization curve of the metal to be protected, as shown in Figure 11-43

 Figure 11-43: Potential range within which anodic protection is effective.

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Critical parameters are: x The potential range of the passive zone (οȲ௣ െ οȲ௧௥ ) x Critical passivation current density (݅௖௣ ) x Passivation current density ip According to these criteria anodic protection is much more efficient on titanium and its alloys than if applied to iron alloys and, among these, with better results on austenitic than ferritic steels. Anodic protection is widely used on tanks with very aggressive media, under the condition that chlorides must be absent; for this reason, the most frequent application is on tanks containing sulfuric acid. Considering the example of a 316 stainless steel tank at 25o C and pH = 0 the critical parameters have the following values: οȲ௣ = 0,27 V; οȲ௧௥ ൌ 1,2 V; ݅௖௣ = 2,5 A/m2; ݅௣ = 0,6 mA/m2. Thus, a tank with an internal surface of 100 m2 will need to apply, initially, a current of 250 A which, once attained the passivation zone potential range of 0.5 – 0.8 V, must be maintained at 0.06 A. In order to reduce energy consumption in the initial phase anodic protection is applied in successive partial areas.

References Cohen, Morris. 1959. The formation and properties of passive films on iron. Can. J. Chem. 286-291 Edeleanu, C. 1954. Corrosion Control by Anodic Protection Metallurgia 113-116 Flade, Friedrich. 1911. Beiträge zur Kentnnis der Passivität. Z. Physik. Chem. 513-559 Franck,Ulrich. 1949. Über das anodische Verhalten des Eisens in Schwefelsäure Z. Naturforsch. 378-391 King, Peter and Uhlig, Herbert. 1959. Passivity in the Iron-Chromium binary Alloys J. Phys. Chem. 2026-2032 Leckie, H. and Uhlig, Herbert. 1966 Environmental factors affecting the critical potential for pitting in 18-8 stainless steel. J.Electrochem.Soc. 1262-1267 Pourbaix, Marcel. Atlas of electrochemical equilibria in aqueous solutions. 1974. Houston. NACE – CEBELCOR Revie, Winston and Uhlig, Herbert. 2008. Corrosion and corrosion control 4th edition New York. John Wiley, page 257

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Rocha, Hans and Lennartz, Gustav. 1955. Die Aktivierungspotentiale von Eisen-Chrom-Legierungen und ihre Beziehungen zu der chemischen Beständigkeit in Schwefelsäure. Arch. Eisenhüttenw. 117-123 Sato, Norio. 1978. Passivity of metals in Proceedings of the Fourth International Symposium on Passivity (R.P. Frankenthal y J. Kruger Eds.) The Electrochemical Society 29-45 Stern, Milton and Geary, Arthur. 1957. Electrochemical Polarization: I. A Theoretical Analysis of the Shape of Polarization curves. J. Electrochem. Soc. 56-63

Bibliography Revie, Winston and Uhlig, Herbert. 2008. Corrosion and corrosion control 4th edition New York. John Wiley McCafferty, Edward. 2010. Introduction to corrosion science Cham. Springer Pedeferri, Pietro. 2018. Corrosion science and engineering. Cham. Springer,

CHAPTER TWELVE INDUSTRIAL PRODUCTION OF MATERIALS

12.1. General aspects The production of a compound P by means of an electrochemical reaction at a given rate demands, as mentioned in Chapter 1, a certain amount of energy per unit mass of P. Thus, the production process can be depicted, for the general case, as: ‫ ܣ‬൅ ‫ ܤ‬൅ |‫ܫ‬. ȟȲ| ՞ ܲ ൅ ܳ

(12.1)

where |IƅȲ| corresponds to the power needed for achieving the desired reaction rate. Most electrochemical processes which are carried on at industrial scale take place in tank reactors whose adequate design is a critical factor in attaining economic efficiency. It is necessary, then, that reactor characteristics and operating conditions are defined with the goal of optimising overpotential value on each electrode as well as the solution electrical resistance. Two main factors determine the electrical resistance of a cell: solution conductivity and the distance between electrodes. Solution conductivity is optimized, in every case, by adjusting temperature and reactant concentrations and, when possible, introducing a supporting electrolyte, which must be carefully chosen since an inadequate choice might interfere with the electrode reactions or even chemically react with reactants or products. By its side, the distance between electrodes can be reduced up to a limiting value which will depend on the characteristics of the occurring reactions. In effect, if the products of one of the electrodes, say the anode, can react at the cathode it will be necessary to introduce some type of ionic conducting separator. Also, if at one of the electrodes gas evolution occurs, electrode distance diminishment is limited because of the need to leave space for bubbles formation and evolution. By the same token, if one of the products

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is deposited on an electrode it will form layers that increase electrode thickness and thus limits the initial anode – cathode distance. Activation overpotential is optimized adjusting two variables: physico – chemical characteristics of the electrodes and their area. Good properties of the electrode material will yield low Tafel slopes and high exchange currents while, if the electrode area is increased, current densities are lower and, consequently, lower overpotentials are needed. However, these variables cannot be freely adjusted: the best electrocatalyst might be very expensive and area increase implies capital costs that, from certain size on, reduce process rentability. Finally, mass transfer overpotential can be reduced by improving the hydrodynamic conditions of the system but also, in this case, associated costs, such as pumping energy, and efficiency losses due to lower residence time of reactants, affect the economic output and must be considered. The above mentioned features and the fact that production volumes (as pointed out in Chapter 1) require large operating currents and, consequently, large electrode surfaces, determine that, instead of building a single reactor of large size, in many cases it is a better solution to resort to the use of a large number of small reactors (cells) which are interconnected. According to the type of interconnection between cells, reactors are called unipolar or bipolar. In unipolar reactors all cathodes are connected to the negative pole of the power source and all anodes are connected to the positive pole, as shown in Figure 12-1.

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Figure 12-1: Connections of electrodes in a unipolar electrolyser.

As seen in this Figure, cathodes and anodes are connected in parallel to the corresponding pole and, consequently, the electric potential difference between poles is of the order of some volts, while current intensity is the sum of the currents of all the cathodes. Thus, in a process in which an electrolyser that contains 50 cells and in each of them the potential difference is 4 V and the current intensity is 1.5 A, the electric potential difference between poles is 4 V and the total current in the connection bars is 75 A with a power demand of 300 W. In bipolar reactors, cells are connected in series, as shown in Figure 12-2, and in each cell the potential difference is the same. In this case the electrodes, excepting the terminal ones, have a real bipolar behaviour: one side of the electrode acts as an anode while on the other side the cathodic reaction takes place.

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Figure 12-2: Electrode connections in a bipolar electrolyser. E: bipolar electrode, (1) Inflow of reactants, (2) Outflow of anodic products, (3) Outflow of cathodic products.

In a bipolar reactor current flow between the power source poles is the same as that in each cell, but the potential difference between poles is the sum of the potential difference of all the cells. Thus, if the process described in the previous case is carried out employing a bipolar electrode total current in the system is 1.5 A but the potential difference between poles is 200 V and, power demand is, again, 300 W. Deciding for one of these reactor types depends on different aspects. The first one is given by the characteristics of the electrochemical reactions that occur in each electrode, since in the bipolar arrangement the composition of each side of the electrode is, in general, different. Thus, when the occurring process implies that the structure or even the nature of the anode is not compatible with that of the cathode it is not possible to employ the bipolar design as, for example, in electrorefining processes, as discussed below for the case of copper. However, bipolar reactors have significant advantages since the low current operation allows more compact designs and is preferred in some cases, particularly when electrolysis products include gases. A general look on the role that electrochemical processes have in modern industry as a route for producing substances shows that there is a significant number of cases in which commodities (aluminium, chlorine, copper), as

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well as several high value products, are electrochemically obtained. Table 12-1 lists some of the most important cases. Product Aluminium Chloride – NaOH Hydrogen Copper .999 Potassium chlorate Potassium perchlorate Hydrogen peroxide Nickel.99 Silver.99 Zinc .99 Adiponitrile Anthraquinone Perfluorocarbons Ethylene glycol Nitrobenzene Propylene oxide

Raw material Alumina Sodium chloride Water Low purity copper Potassium chloride Potassium chloride Water Low purity nickel Low purity silver Low purity zinc Acrylonitrile Anthracene Alkyl compounds Formaldehyde p – aminophenol Propylene

Table 12-1: Some industrially relevant electrochemical processes. In this chapter the chlor – alkali, hydrogen, copper and aluminium production processes will be described in some detail.

12.2. The chlor – alkali process The world production of chlorine and caustic soda is performed, in more than 90% of its volume, by electrolysis at temperatures near 90o C, from concentrated solutions of sodium chloride, following the reaction: 2ܰܽ‫ ݈ܥ‬൅ 2‫ܪ‬ଶ ܱ ൅ |‫ܫ‬. ȟȲ| ՞ ‫݈ܥ‬ଶ ൅ ‫ܪ‬ଶ ൅ 2ܱܰܽ‫ܪ‬

(12.2)

Installed production capacity in 2020 was about 90 million tons of chlorine and actual production that year reached 80 million tons (López, 2018) whose final use was distributed in the production of PVC (30%), different solvents (24%), inorganic compounds (13%), paper mills (5%), water treatment (5%) and a wide variety of other applications (10%).

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Caustic soda (NaOH) production in 2020 was 66 million tons which were applied to the chemical industry (67%), paper industry (22%), surfactants (10%) and textiles (1%) Chlor – alkali industrial reactors in use can be classified according to the employed technology in three groups: diaphragm cells, mercury cells and membrane cells. Bipolar reactors are the almost exclusive choice for plants employing diaphragm and membrane technologies, while mercury technology relies on unipolar reactors.

12.2.1. Diaphragm cells This is the oldest technology and the main reactions occurring at the electrodes are: 2‫ ି ݈ܥ‬՜ ‫݈ܥ‬ଶ ൅ 2݁ ି

(12.3)

at the anode and 2‫ܪ‬ଶ ܱ ൅ 2݁ ି ՜ 2ܱ‫ ି ܪ‬൅ ‫ܪ‬ଶ

(12.4)

at the cathode. In order to avoid mixing of anolyte and catholyte a separator diaphragm made of asbestos is employed, as shown in Figure 12-3

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Figure 12-3: Schematic description of chlor-alkali diaphragm cell.

Faradaic efficiency of both electrode reactions is diminished because of the existence of different parallel reactions that, in the anode case, include chlorine dissolution, hypochlorite formation, followed by a dismutation reaction, and oxygen evolution: ‫݈ܥ‬ଶ ሺ݃ሻ ՜ ‫݈ܥ‬ଶ ሺ‫݈݊݋ݏ‬ሻ

(12.5)

‫݈ܥ‬ଶ ሺ‫݈݊݋ݏ‬ሻ ൅ ‫ܪ‬ଶ ܱ ՜ ‫ ܱ݈ܥܪ‬൅ ‫݈ܥܪ‬

(12.6)

‫ ܱ݈ܥܪ‬՜ ‫ ܪ‬ା ൅ ‫ିܱ݈ܥ‬

(12.7)

2‫ ܱ݈ܥܪ‬൅ ‫ ିܱ݈ܥ‬՜ ‫ܱ݈ܥ‬ଷି ൅ 2‫ ܪ‬ା ൅ 2‫ି ݈ܥ‬

(12.8)

2‫ܪ‬ଶ ܱ ՜ ܱଶ ൅ 4‫ ܪ‬ା ൅ 4݁ ି

(12.9)

Regarding oxygen evolution it is important to note that, although the thermodynamic electrode potential of reaction (12.9) is more favourable than that of reaction (12.3), the difference of Tafel slopes determines that, at high current density values, chlorine evolution is favoured and the impact of oxygen evolution diminishes, as described in Figure 12-4:

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Figure 12-4: Overpotential and current density for chlorine and oxygen evolution.

Clearly, anode electrocatalytic properties will have a paramount importance on the impact of oxygen evolution on faradaic efficiency as is exposed in Figure 12-5 where oxygen content in the anode gas is described as a function of current density for two different anode materials.

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Figure 12-5: Oxygen content in produced chlorine as a function of current density for different anode materials.

Anode materials employed in diaphragm cells evolved from the original use of steel (on which a magnetite film is formed) to the employment of different types of graphite anodes, which were dominant in the industry for several decades, until the development of dimensionally stable anodes (DSA) that are made of titanium covered by a mixture of RuO2 and TiO2. Graphite anodes, that during operation undergo mass losses, show higher overpotential values, as depicted in Figure 12-6, and for these reasons DSA, in spite of its cost, are almost the exclusive choice in present plants.

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Figure 12-6: Overpotential as a function of current density for steel, graphite and DSA anodes.

With respect to faradaic efficiency at the cathode, hydrogen evolution is affected by two reactions that are produced as a consequence of diffusion, through the diaphragm, of chlorine containing ions: ‫ ିܱ݈ܥ‬൅ ‫ܪ‬ଶ ܱ ൅ 2݁ ି ՜ ‫ ି ݈ܥ‬൅ 2ܱ‫ି ܪ‬

(12.10)

‫ܱ݈ܥ‬ଷି ൅ 3‫ܪ‬ଶ ܱ ൅ 6݁ ି ՜ ‫ ି ݈ܥ‬൅ 6ܱ‫ି ܪ‬

(12.11)

and

Originally, cathodes were made of steel whose activation overpotential compares favourably with other metals as shown in Figure 12-7 where it is seen that for a current density of 50 mA.cm-2 overpotential on steel is about 150 mV, considerably below corresponding values for cobalt or nickel.

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Figure 12-7: Polarization curves for hydrogen evolution on steel, nickel and cobalt cathodes.

However, because of the high pH values in the catholyte, steel electrodes undergo loss of material as HFeO2- during plant stops and its life span is affected. For this reason, and since nickel has better chemical stability in highly alkaline solutions, significant research efforts have been devoted in the search of modified nickel cathodes with better catalytic properties, and scores of patents on this issue have been generated. Asahi Kasei electrolysers employ a nickel coated steel cathode on which a mixture of NiO and Cr2O3 is plasma sprayed (Houda et al., 1998) and Dow has patented nickel cathodes covered by ruthenium and palladium oxides (Tsou, 1999). Electrode geometry is very important for process optimization: anode – cathode distance must be minimized, area/volume quotient should be maximized, and electrolyte flow must be adequate. Thus, there are different design concepts applied to the unit cells in the reactor. For instance, Dow has developed diaphragm cells with “digital” geometry (electrodes resemble fingers) depicted in Figure 12-8 for the case of steel cathodes.

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Figure 12-8: Cutaway view of the Dow diaphragm cell: (1) Cathode space, (2) Steel perforated plate, (3) Asbestos diaphragm, (4) DSA anode, (5) Copper contact plate, (6) Titanium contact plate.

These cells are connected as in Figure 12-2 in units that contain scores or hundreds of cells. This units, in turn, are connected in series. Figure 12-9 shows a sector of De Nora diaphragm cells plant.

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Figure 12-9: Partial view of a De Nora diaphragm cell plant. https://www.denora.com/products/applications/chloralkali-process/diaphragmcell.html

12.2.2. Mercury cells This type of cell employs mercury cathodes on which, because of the high hydrogen overpotential on mercury, the cathodic reaction is sodium reduction, forming sodium amalgam: ܰܽା ൅ ݁ ି ՞ ܰܽ െ ‫݃ܪ‬ሺ݈ܽ݉ܽ݃ܽ݉ሻ

(12.12)

The obtained amalgam is transferred to a tank where it reacts with water: 2 ܰܽ െ ‫݃ܪ‬ሺ݈ܽ݉ܽ݃ܽ݉ሻ ൅ ‫ܪ‬ଶ ܱ ՞ 2ܱܰܽ‫ ܪ‬൅ ‫ܪ‬ଶ ൅ 2‫݃ܪ‬

12.13)

and the recovered mercury is pumped back to the electrochemical cell. Figure 12-10 gives a sketch of the process.

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Figure 12-10: Sketch of the chlor – alkali mercury cell process. (1) Concentrated brine, (2) Dilute brine, (3) Amalgam flow to decomposition reactor, (4) Clean mercury recycled, (5) NaOH 50%, (6) Water.

12.2.3. Membrane cells This technology was introduced in the seventies and its fundamental innovation is the replacement of the asbestos diaphragm by an ionic exchange membrane through which Na+ ions can be transferred, but Cl- ions are impeded to move from anolyte to catholyte. Figure 12-11 gives a schematic description of its operation.

Figure 12-11: Operation of a membrane cell. (1) Brine income, (2) Brine recirculation, (3) Water, (4) NaOH 33%, (5) NaOH recirculation, M: Membrane. Gray zones indicate concentration gradients.

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As it is easily seen, membrane cells have a very similar functioning scheme to that of diaphragm cells, but with important differences: x Membranes, which are perfluoro sulphonates and/or perfluoro carboxylates allow Na+ ions to pass through them, but anion transfer from the anolyte to the catholyte is blocked. Thus, faradaic efficiency is higher. x Membranes electrical resistance is considerably lower than that of electrolyte impregnated diaphragms and the potential drop due to ohmic effects is considerably diminished. x By its nature, membrane thickness can be very low; thus, distance between electrodes can be reduced significantly with respect to the case of diaphragm cells and further lowering of the ohmic overpotential is achieved. x A drawback of these cells is the need to purify the brine that enters the cell, since membranes are severely affected by small concentrations of Mg2+ or Ca2+ x As seen from Figures 12-3 and 12-11 liquid circulation in the cathode compartment is different from that of diaphragm cells: although in both cases brine input is placed at the anode compartment, electrolyte flow in diaphragm cells occurs from anode to cathode by pressure difference, while in membrane cells there is no brine outlet at the cathode, and it is necessary to inject water to sustain the cathodic reaction. x The membrane is easily poisoned by other cations which makes inconvenient the use of nickel at the cathode. For this reason, hydrogen evolution takes place on titanium mesh electrodes on which platinum crystallites are deposited. Chlorine evolution is produced on DSA similar to those employed in diaphragm cells. Membrane technology allows more compact designs for production units. Figure 12-12 exhibits a membrane cells plant sector

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Figure 12-12: Partial view of a chlor-alkali production plant with membrane cells technology. Photograph courtesy of Thyssenkrupp AG.

Main aspects of the three discussed technologies are summarized in Table 12-1

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 Electricdemand (kWh/tonCl2) NaOH 50%, O2 in Cl2