Electrochemical Corrosion. A. Senthil Kumar Roll No M.Tech Energy Systems IIT Bombay

Transcription

1 Electrochemical Corrosion A. Senthil Kumar Roll No M.Tech Energy Systems IIT Bombay August 2008

2 Contents 1. Review of the Electrochemical Basis of Corrosion 2. Quantitative Corrosion Theory 3. Polarization Resistance 4. Calculation of Corrosion Rate from Corrosion Current 5. IR Compensation 6. Current and Voltage Conventions 7. Chemical and Electrochemical Potential 8. Nernst Equation 9. Cell Potential at Equilibrium 10. Potential of a Galvanic Cell 11. Sign of a Cell Potential 12. Electron Transfer 13. Current Density 14. Symmetry Coefficient 15. Exchange Current Density 16. Current Equations 17. Butler-Volmer Equation 18. The Tafel Equation 19. MassTransfer 20. Limiting Current 21. Units and Symbols 2

3 22. Silver / Silver Chloride Reference Electrode 23. References 3

4 Review of the Electrochemical Basis of Corrosion Most metal corrosion occurs via electrochemical reactions at the interface between the metal and an electrolyte solution. A thin film of moisture on a metal surface forms the electrolyte for atmospheric corrosion. Wet concrete is the electrolyte for reinforcing rod corrosion in bridges. Although most corrosion takes place in water, corrosion in non-aqueous systems is not unknown. Corrosion normally occurs at a rate determined by equilibrium between opposing electrochemical reactions. The first is the anodic reaction, in which a metal is oxidized, releasing electrons into the metal. The other is the cathodic reaction, in which a solution species (often O2 or H + ) is reduced, removing electrons from the metal. When these two reactions are in equilibrium, the flow of electrons from each reaction is balanced, and no net electron flow (electrical current) occurs. The two reactions can take place on one metal or on two dissimilar metals (or metal sites) that are electrically connected. Figure 1-1 diagrams this process. The vertical axis is potential and the horizontal axis is the logarithm of absolute current. The theoretical current for the anodic and cathodic reactions are shown as straight lines. The curved line is the total current -- the sum of the anodic and cathodic currents. This is the current that you measure when you sweep the potential of the metal with your potentiostat. The sharp point in the curve is actually the point where the current changes signs as the reaction changes from anodic to cathodic, or vice versa. The sharp point is due to the use of a logarithmic axis. The use of a log axis is necessary because of the wide range of current values that must be displayed during a corrosion experiment. Because of the phenomenon of passivity, it is not uncommon for the current to change by six orders of magnitude during a corrosion experiment. Figure 1-1 Corrosion Process Showing Anodic and Cathodic Current Components 4

5 The potential of the metal is the means by which the anodic and cathodic reactions are kept in balance. Refer to Figure 1-1. Notice that the current from each half reaction depends on the electrochemical potential of the metal. Suppose the anodic reaction releases too many electrons into the metal. Excess electrons shift the potential of the metal more negative, which slows the anodic reaction and speeds up the cathodic reaction. This counteracts the initial perturbation of the system. The equilibrium potential assumed by the metal in the absence of electrical connections to the metal is called the Open Circuit Potential, Eoc. In most electrochemical corrosion experiments, the first step is the measurement of Eoc. The value of either the anodic or cathodic current at Eoc is called the Corrosion Current, Icorr. If we could measure Icorr, we could use it to calculate the corrosion rate of the metal. Unfortunately, Icorr cannot be measured directly. However, it can be estimated using electrochemical techniques. In any real system, Icorr and Corrosion Rate are a function of many system variables including type of metal, solution composition, temperature, solution movement, metal history, and many others. The above description of the corrosion process does not say anything about the state of the metal surface. In practice, many metals form an oxide layer on their surface as they corrode. If the oxide layer inhibits further corrosion, the metal is said to passivate. In some cases, local areas of the passive film break down allowing significant metal corrosion to occur in a small area. This phenomena is called pitting corrosion or simply pitting. Because corrosion occurs via electrochemical reactions, electrochemical techniques are ideal for the study of the corrosion processes. In electrochemical studies, a metal sample with a surface area of a few square centimeters is used to model the metal in a corroding system. The metal sample is immersed in a solution typical of the metal's environment in the system being studied. Additional electrodes are immersed in the solution, and all the electrodes are connected to a device called a potentiostat. A potentiostat allows you to change the potential of the metal sample in a controlled manner and measure the current the flows as a function of potential. Both controlled potential (potentiostatic) and controlled current (galvanostatic) polarization is useful. When the polarization is done potentiostatically, current is measured, and when it is done galvanostatically, potential is measured. This discussion will concentrate on controlled potential methods, which are much more common than galvanostatic methods. With the exception of Open Circuit Potential vs. Time, Electrochemical Noise, Galvanic Corrosion, and a few others, potentiostatic mode is used to perturb the equilibrium corrosion process. When the potential of a metal sample in solution is forced away from Eoc, it is referred to as polarizing the sample. The response (current) of the metal sample is measured as it is polarized. The response is used to develop a model of the sample's corrosion behavior. Suppose we use the potentiostat to force the potential to an anodic region (towards positive potentials from Eoc). In Figure 1-1, we are moving towards the top of the graph. This will increase the rate of the anodic reaction (corrosion) and decrease the rate of the cathodic reaction. Since the anodic and cathodic reactions are no longer 5

6 balanced, a net current will flow from the electronic circuit into the metal sample. The sign of this current is positive by convention. If we take the potential far enough from Eoc, the current from the cathodic reaction will be negligible, and the measured current will be a measure of the anodic reaction alone. In Figure 1-1, notice that the curves for the cell current and the anodic current lie on top of each other at very positive potentials. Conversely, at strongly negative potentials, cathodic current dominates the cell current. In certain cases as we vary the potential, we will first passivate the metal, then cause pitting corrosion to occur. With the astute use of a potentiostat, an experiment in which the current is measured versus potential or time may allow us to determine Icorr at Ecorr, the tendency for passivation to occur, or the potential range over which pitting will occur. 6

7 Quantitative Corrosion Theory As seen earlier, Icorr cannot be measured directly. In many cases, you can estimate it from current versus voltage data. You can measure a log current versus potential curve over a range of about one half volt. The voltage scan is centered on Eoc. You then fit the measured data to a theoretical model of the corrosion process. The model we will use for the corrosion process assumes that the rates of both the anodic and cathodic processes are controlled by the kinetics of the electron transfer reaction at the metal surface. This is generally the case for corrosion reactions. An electrochemical reaction under kinetic control obeys Equation 1-1, the Tafel equation. I = I 0 exp( 2.303(E-E o )/b) Equation 1-1 In this equation, I is the current resulting from the reaction I 0 is a reaction dependent constant called the Exchange Current E is the electrode potential E o is the equilibrium potential (constant for a given reaction) b is the reaction's Tafel Constant (constant for a given reaction). Beta has units of volts/decade. The Tafel equation describes the behavior of one isolated reaction. In a corrosion system, we have two opposing reactions anodic and cathodic. The Tafel equations for both the anodic and cathodic reactions in a corrosion system can be combined to generate the Butler-Volmer equation (Equation 1-2). I = Icorr (exp(2.303(e-ecorr)/ba) - exp(-2.303(e-ecorr)/bc)) Eq 1-2 where I is the measured cell current in amps Icorr is the corrosion current in amps E is the electrode potential Ecorr is the corrosion potential in volts ba is the anodic Beta Tafel Constant in volts/decade bc is the cathodic Beta Tafel Constant in volts/decade What does Butler-Volmer Equation predict about the current versus voltage curve? At Ecorr, each exponential term equals one. The cell current is therefore zero, as you would expect. Near Ecorr, both exponential terms contribute to the overall current. Finally, as the potential is driven far from Ecorr by the potentiostat, one exponential term predominates and the other term can be ignored. When this occurs, a plot of log current versus potential becomes a straight line. 7

8 A log I versus E plot is called a Tafel Plot. The Tafel Plot in Figure 1-1 was generated directly from the Butler-Volmer equation. Notice the linear sections of the cell current curve. In practice, many corrosion systems are kinetically controlled and thus obey Equation 1.2. A log current versus potential curve that is linear on both sides of Ecorr is indicative of kinetic control for the system being studied. However, there can be complications, such as: 1. Concentration polarization, where the rate of a reaction is controlled by the rate at which reactants arrive at the metal surface. Often cathodic reactions show concentration polarization at higher currents, when diffusion of oxygen or hydrogen ion is not fast enough to sustain the kinetically controlled rate. 2. Oxide formation, which may or may not lead to passivation, can alter the surface of the sample being tested. The original surface and the altered surface may have different values for the constants in Equation Other effects that alter the surface, such as preferential dissolution of one alloy component, can also cause problems. 4. A mixed control process where more than one cathodic, or anodic, reaction occurs simultaneously may complicate the model. An example of mixed control is the simultaneous reduction of oxygen and hydrogen ion. 5. Finally, potential drop as a result of cell current flowing through the resistance of your cell solution causes errors in the kinetic model. This last effect, if it is not too severe, may be correctable via IR compensation in the potentiostat. In most cases, complications like those listed above will cause non-linearities in the Tafel plot. The results derived from a Tafel Plot that does not have a well-defined linear region should be used with caution. Classic Tafel analysis is performed by extrapolating the linear portions of a log current versus potential plot back to their intersection. See Figure 1-2. The value of either the anodic or the cathodic current at the intersection is Icorr. Figure 1-2 Classic Tafel Analysis 8

9 Polarization Resistance Equation 1-2 can be further simplified by restricting the potential to be very near to Ecorr. Close to Ecorr, the current versus voltage curve approximates a straight line. The slope of this line has the units of resistance (ohms). The slope is, therefore, called the Polarization Resistance, Rp. An Rp value can be combined with an estimate of the Beta coefficients to yield an estimate of the corrosion current. If we approximate the exponential terms in Equation 1-2 with the first two terms of a power series expansion (e x = 1+x +x 2 /2...) and simplify, we get one form of the Stern-Geary equation: Icorr = (1/ Rp) b a b c / (2.303 (b a + b c ) ) Equation 1-3 In a Polarization Resistance experiment, you record a current versus voltage curve as the cell voltage is swept over a small range of potential that is very near to Eoc (generally ± 10 mv). A numerical fit of the curve yields a value for the Polarization Resistance, Rp. Polarization Resistance data does not provide any information about the values for the Beta coefficients. Therefore, to use Equation 1-3 you must provide Beta values. These can be obtained from a Tafel Plot or estimated from your experience with the system you are testing. Calculation of Corrosion Rate from Corrosion Current The numerical result obtained by fitting corrosion data to a model is generally the corrosion current. We are interested in corrosion rates in the more useful units of rate of penetration, such as millimeters per year. How is corrosion current used to generate a corrosion rate? Assume an electrolytic dissolution reaction involving a chemical species, S: S --> S n+ + ne - You can relate current flow to mass via Faraday's Law. where Q = n F M Equation 1-4 Q n F M is the charge in coulombs resulting from the reaction of species S is the number of electrons transferred per molecule or atom of S is Faraday's constant = 96,486.7 coulombs/mole is the number of moles of species S reacting A more useful form of Equation 1-4 requires the concept of equivalent weight. The equivalent weight (EW) is the mass of species S that will react with one Faraday of charge. For an atomic species, EW = AW/n (where AW is the atomic weight of the species). Recalling that M = W/AW and substituting into Equation 1-4 we get: 9

10 W = EW Q / F Equation 1-5 where W is the mass of species S that has reacted. In cases where the corrosion occurs uniformly across a metal surface, the corrosion rate can be calculated in units of distance per year. Be careful - this calculation is only valid for uniform corrosion; it dramatically underestimates the problem when localized corrosion occurs. For a complex alloy that undergoes uniform dissolution, the equivalent weight is a weighted average of the equivalent weights of the alloy components. Mole fraction, not mass fraction, is used as the weighting factor. If the dissolution is not uniform, you may have to measure the corrosion products to calculate EW. Conversion from a weight loss to a corrosion rate (CR) is straightforward. We need to know the density, d, and the sample area, A. Charge is given by Q = I T, where T is the time in seconds and I is a current. We can substitute in the value of Faraday's constant. Modifying Equation 1-5: CR = Icorr K EW / d A Equation 1-6 CR The corrosion rate. Its units are given by the choice of K (see Table 1-1 below) Icorr The corrosion current in amps K A constant that defines the units for the corrosion rate EW The equivalent weight in grams/equivalent d Density in grams/cm 3 A Sample area in cm 2 Table 1-1 shows the value of K used in Equation 1-6 for corrosion rates in the units of your choice. Table 1-1 Corrosion Rate Constants Corrosion Rate Units K Units mm/year (mmpy) 3272 mm/(amp-cm-year) millinches/year (mpy) 1.288x10 5 milliinches(amp-cm-year) See ASTM Standard G 102, Standard Practice for Calculation of Corrosion Rates and Related Information from Electrochemical Measurements, for further information. 10

11 Chemical and Electrochemical Potential Chemical potential Defined as the increase in free energy of a system with the addition of dn i mole of component i; temperature, pressure and number of moles of all other components being constant. Since we are ulimately concerned with ions, at least two kinds of ions must be added. Since dg is a perfect differential Chemical potential can be expressed in terms of standard potential; Electrochemical potential Since an ionic species has a different chemical potential in different phases, as two adjacent phases come to equilibrium with respect to a given species which exists in both species, there is a transfer of charge between the phases until the difference in chemical potential of the charged species is balanced by the electrical energy of the species with charge z i in phase a with potential φ. Expressed interms of the standard chemical potential; 11

12 Nernst Equation Electrode Potential Given a metal electrode in a solution that contains ions of that metal, a potential difference between the metal and the solution will develop. This equilibrium is expressed as The potential difference between the metal and the solution cannot be measured directly. You can measure the difference in potential between two electrodes in the solution. This involves two potential differences, one at each interface between electrode and solution. This potential difference can be measured by comparison with a reference electrode where the potential difference between the two phases is known. The ultimate reference is the normal hydrogen electrode, the potential of which is defined as zero. Another type involves an inert metal in contact with a gas in equilibrium with ions in solution, such as the hydrogen or chlorine electrodes. Equilibrium is established between the ions, adsorbed gas and the metal which determines the potential of the electrode. The normal hydrogen electrode consists of hydrogen ion activity of 1 and 1atm. H 2 at 25 C. In some cases the reduced and oxidized forms are in solution, e.g., Fe 3+ and Fe 2+. The inert metal serves as an electronic conductor between the two species. This is called a redox electrode. The potential difference between the two phases, metal and solution, is represented by φ and is given by φ M and φ S are called the inner or Galvani potentials of the metal and solution phases. Each of these systems reaches equilibrium by the transfer of charge across the interface between the two phases. The Galvani potential has two distinct components. One is the Volta potential ψ which is the long range coulombic forces near the electrode and the 12

13 surface potential χ which is determined by short range effects of adsorbed ions and oriented water molecules. Thus the Galvani or inner potential is expressed as The Volta potential can be measured directly. The surface potential cannot. Thus the Galvani potential can only be measured relative to a reference electrode. 13

14 Electrode potentials and activity: the Nernst equation If we have the case of a metal in equilibrium with its ions in solution, at equilibrium the electrochemical potential of the common species is equal in both phases. Suppose we had a copper electrode in a solution that contained cupric ions. For the system at equilibrium For each phase the electrochemical potential consists of the chemical potential and a potential term. The chemical potential can further be separated into the standard chemical potential (unit activity) and an activity term. So the electrochemical potential for species i is given by R is the gas constant joule/mole K, T is absolute temperature (K), F is the Faraday constant 96,485 coul/mole. This equation can be substituted for each phase in the equilibrium equation for copper. Then if we want the potential difference between the two phases we get or which is known as the Nernst equation. The logarithmic term can be generalized as ln of products of activities of reaction products (raised to powers of stoichiometric coefficients) divided by products of activities of reactants (raised to powers of stoichiometric coefficients). This equation only applies to systems in equilibrium. At equilibrium there is no net current. 14

15 Applied to the general form and replacing the difference in galvani potentials with E gives The van t Hoff reaction equation expresses free energy change for a chemical reaction as where p and r indicate products and reactants. For the reaction M n+ +ne=m this can be written The free energy change is related to electrode potential by and This means that reduction of one mole of M n+ to M requires passage of n Faradays, nf coulombs. Passage of charge nf through a potential difference of E volts constitutes work of nfe joules. This work, done at constant temperature and pressure is equal to the the decrease in free energy of the system, - G. Relation Between Free Energy and Potential of a Galvanic Cell Given a cell with emf E (in volts), if it is operated reversibly at constant temperature and pressure the electrical work done on passage of an infinitesimal quantity of electricity dq (in coulombs) is the product dqe (in volt-coulombs or joules). If - G (joules /mole) is the decrease in free energy due to the passage of n faradays of electricity nf (in coulombs), the change in free energy due to the passage of dq coulombs is - G dq/nf which is equivalent to the electrical work or - G dq/nf=edq. Thus 15

16 For the general process aa+bb+...=ll+mm+...the change in free energy with the passage of n faradays of charge is G=lm L +mm M +...-am A -bm B -..., where m i is the partial molar free energy or chemical potential, and since then or which is called the Van't Hoff isotherm. At equilibrium G=0 and where K is called the thermodynamic equilibrium constant. Combining the above two equations we have Using the expression, the above equation becomes 16

17 This can separated into two parts representing the two electrode reactions which comprise the cell or. The existence of an open circuit potential indicates that although each phase within the cell is in a state of equilibrium with adjacent phases, the cell as a whole is not at a state of equilibrium. If the components of the two half-cells were brought together there would be a spontaneous chemical reaction. If the reaction was allowed to go to completion there would be a decrease in free energy, i.e., a negative G. If the two terminals are connected with an electronic conductor a current is produced, electrons are transferred, species are oxidized and reduced. The potential difference between the two terminals is proportional to the free energy change. For a spontaneous reaction (a galvanic cell or energy producer) the cell potential is positive. Thus we write The van t Hoff isotherm expresses the energy in terms of the standard free energy change and activities. Activities of the initial states (reactants) are taken as negative and the final states (products) are negative. Substitution gives The state of equilibrium between adjacent phases means that any species that exists in both phases can move between the two phases without a change in energy. When two phases are brought together, before equilibrium, a given species has a different chemical 17

18 potential in each phase, being in a different chemical environment. The state of equilibrium is reached by the transfer of charged particles across the interface until the difference in chemical potential is balanced by a potential difference between the two phases. We then say that the electrochemical potential of species i, is the same in the two phases P and Q. or Thus the electrochemical potential of species in a given phase is given by Given the relation the electrochemical potential of a species in a given phase can also be written as, If two phases are in equilibrium we can set the RHS of the above equation for one phase equal to the same expression for the other phase, and written as the sum of the expression for each species that occupies both phases. If we then solve for the difference in potential in each phase we have. which gives us the potential change at the interface. Since the cell potential is the sum of the interface potentials, the cell potential is given by,. 18

19 Sign of a Cell Potential The anode is defined as the electrode at which oxidation takes place, and the cathode is where reduction takes place. An easy way to remember this is that both anode and oxidation start with vowels, and cathode and reduction start with consonants. Let's take the cell which we can represent by Zn ZnSO 4 aq KCl aq CuSO 4 aq Cu In this cell, Zn is oxidized and Cu2+ is reduced. The overall reaction is On the left side we have the oxidation of Zn Zn + Cu 2+ = Zn 2+ + Cu. Zn = Zn 2+ + e - The electrons released result in a negatively charged Zn electrode. On the right we have Cu 2+ + e - = Cu The electrons are drawn from the electrode to reduce the cupric ions, thus leaving a positive charge on the Cu electrode. If the two electrodes are connected by an external circuit we have a galvanic cell, and the flow of electrons will be from the Zn electrode to the Cu electrode. The Zn electrode is the anode, as oxidation is taking place there, and the anode is negatively charged. The Cu electrode is the cathode, where we have reduction taking place. The cathode is positively charged. This is the reverse of an electrolytic cell where an applied potential causes the anode to be positive and the cathode to be negative. 19

20 This distinction can be clarified by picturing the process in this manner. Looking at the Cu electrode, if the external circuit were closed and current were passed, the result would be that Cu ions migrated toward the electrode where they would be reduced to Cu metal as electrons flowed through the circuit from the zinc to the copper electrode. The copper electrode is an electron source and is thus negatively charged. The cell is not in a state of equilibrium in this case. If the circuit is opened so that no current can flow the cell can then reach a state of equilibrium. Now if a cupric ion arrives at the surface of the Cu electrode and is reduced to copper metal by the transfer of two electrons to the ion, since there is no flow of electrons into the copper electrode from the external circuit, the only source of the electrons is from the copper metal in the electrode. Such a process would result in the formation of a new copper ion from one or two of the copper atoms in the electrode. So if we simply imagine the copper electrode as having some adsorbed cupric ions on the surface, this is consistent with the spontaneous tendency of the cupric ions to move toward the copper electrode and the result is a positively charged electrode. The opposite process would be happening at the zinc electrode where the spontaneous tendency is for zinc ions to migrate away from the surface of the electrode, leaving the excess electrons on the zinc metal. IUPAC Conventions 1) nfe=- G 2) Represented so that emf of cell is R with respect to L. 3) Electrode potential = cell potential with electrode on R vs standard hydrogen electrode on L. Example: 1/2 H 2 + AgCl = Ag + HCl(m) H 2, Pt aq. HCl (m) AgCl Ag /p> If molality of HCl is < 9.1 mole kg -1, G is negative and emf is positive (Ag is positive of Pt). Example: H 2, Pt H + Fe 3+, Fe 2+ Pt 1/2 H 2 + Fe 3+ = H + + Fe 2+ > 20

21 Example: For an electrode directly reversible to its cations the reduced species if the metal itself and its activity is one. Thus for a Zn electrode at equilibrium with Zn 2+ ions in solution,. 21

22 Charge Transfer Kinetics 22

23 Electron Transfer In an electrolytic cell, electron transfer between a metal electrode and species in solution near the electrode depend on the energy of electrons in the electrode and the energy of the lowest unoccupied orbitals or the highest occupied orbitals or the species in solution. The energy level of electrons in the metal is referred to as the Fermi level. As the charge on the electrode becomes more negative the Fermi level increases. When the energy of electrons in the metal equals or exceeds that of the lowest unoccupied orbital, electrons can transfer from the electrode to the species in solution. The species is reduced in the process. For example Cu +2 could be reduced to Cu +1. If the potential of the electrode is made more positive it will reach the point where the energy of electrons in the metal is lower than that of electrons in the highest occupied orbitals of the dissolved species. In this case electrons can transfer from the species in solution to the metal. These processes are shown in Figure 1. Figure 1. In an electrolytic cell the electrode potential is controlled. In the illustration below (Figure 2.) the relation between the energy of electrons in the electrode and lowest unoccupied orbital of the species in solution are shown. If they were at the same energy, electron transfer would be taking place but at equal rates in both directions. The greater the difference the faster the rate of reduction of the dissolved species. 23

24 Figure 2. The figure below (Figure 3.) shows the situation where the Fermi level falls below the energy of electrons in the highest occupied orbitals in the solute. Electrons are passed to the electrode and the solute is oxidized in the process. If the potential of the electrode were not being controlled this process would cause the Fermi level to increase. In the case of the controlled potential of an electrolytic cell the electrons passed to the electrode are drained off as current through the cell. The opposite process is taking place at the counter electrode. Figure 3. If both the oxidized and reduced species are in solution in the presence of an electrode in a galvanic cell where the potential of the electrode is not controlled, the Fermi level is 24

25 determined by the relative concentrations of oxidized and reduced species in solution. This is the so-called redox electrode. This situation is illustrated in Figure 4. Figure 4. In the case of a redox electrode there is no net current beyond the initial charging. The redox potential of the electrode is measured against a reference electrode. There is no net chemical change involved either. If the two species in solution are ferric and ferrous ions there is a constant exchange of electrons between the two but there is no change in the concentration of either. The metal electrode functions as a sort of middle-man for electron exchange. In the case of an electrolytic cell with an applied potential sufficient to cause reduction or oxidation of solute species, a current flows through the cell and can be measured. The current is a measure of the rate of the reaction. Obviously the current is a function of the surface area of the electrode-solution interface. At the same potential a larger electrode will produce more current. This is a linear relationship. The current is also a function of the concentration of the species being reduced or oxidized. 25

26 Current Density The rate of an electrochemical reaction is measured as current density, current per area. Common units are amps per square meter A m -2 or milliamps per square centimeter ma cm -2. This is related to mol m -2 s -1 by Faradays laws which say that with the passage of 96,485 coul of charge across the interface we have transformed one mole the substance if it was one electron per particle process. If it involved more than one electron, n electrons, then 1/n moles would have been reduced or oxidized. For a two electron process the passage of one mole of electrons would have reduced or oxidized 1/2 mole of material. So the reaction rate is given by where F is the Faraday constant, 96,485 C mol -1. The Faraday constant is the charge on an electron multiplied by Avogadro's number coul mol -1 = coul mol -1 For a typical current density of 10 ma cm -2 the rate is 10/96485 = mol cm -2 s -1. In an experiment the current is measured rather than the current density. The current is not only a function of the rate of the reaction but of the area of the electrode / solution interface as well. If a metal electrode is only partially submerged in the solution you can increase the current by lowering it further into the solution. (Most of the charge accumulates at the part in contact with the electrolyte rather than the air due to the double layer formation.) The larger the area of contact the more product will be formed per unit time but the current density will not change (provided you have not exceeded the limits of your power supply) (Figure 1). So while the current is a function of the mechanical arrangement it is the current density that is a characteristic of the reaction aside from the specifics of the particular apparatus. 26

27 Figure 1. If you have a measured current and you want to know the current density you have to divide the current by the area of the electrode. The area of the electrode if often somewhat different than the area measured by macroscopic means. The surface area of an apparently smooth electrode is usually greater than the mechanical measurement due to microscopic roughness. A typical method is to measure the current in a standard solution of an electroactive substance (e.g., potassiun ferricyanide) where the current density is known for a certain concentration and applied potential. This gives you an electrochemical measurement based on a reaction at the interface. By dividing the measured current by the known current density you can calculate the active area. 27

28 Symmetry Coefficient The physical meaning of the symmetry coefficient (or charge transfer coefficient) is a difficult subject. I have approached it here by simply setting it to various values and plotting the results. In Figure 1 the solid red curve is a plot of current density with α=0.5. Cathodic current density is plotted downward from the zero current line in this plot. At α=0.5 the curve is symmetrical in that the anodic and cathodic portions are equivalent. The dotted blue curve is the result of the same equation but with α=0.6. The dashed green curve has α=0.7. Examination of the plots show that for a reduction reaction and α >0.5 a negative overpotential results in a greater current density than an equivalent positive overpotential. Figure 1. This can be interpreted as meaning that for α>0.5 the activation energy for the reduction process is decreased while the activation energy for the oxidation process is increased aside from the effect of the potential difference between the electrode and the solution. 28

29 Exchange Current Density For a redox reaction written as a reduction at the equilibrium potential the electrode/solution interface still has electron transfer processes going on in both directions. The cathodic current is balanced by the anodic current. When the potential is set more negative the cathodic current is greater than the anodic current. This ongoing current in both directions is called the exchange current density. Written as a reduction, cathodic current is positive. The net current density is the difference between the cathodic and anodic current density. Figure 1 is a graphic representation of the relation between the exchange current i 0 and the net current. In the figure i c is cathodic current, i a is anodic current. Figure 1. A system with a high exchange current density has fast kinetics and can respond rapidly to a potential change. Figure 2 allows comparison of current density plots that vary only in exchange current density. Examination of the plots shows that at a given nonzero overpotential the magnitude of the current is greater with the higher the exchange current density. Figure 2. 29

30 Current Equations The current in an electrolytic cell is a measure of the rate of the electrochemical reaction and is proportional to the concentration of the electroactive species. Taking the reversible reduction of an oxidized species to a reduced species, both in solution and not accompanied by purely chemical reactions with k 1 and k 2 standing for the forward and backward rate constants, we have the rates of each reaction expressed as where c O and c R represent the concentrations of the oxidized species and reduced species. These rates can be expressed in terms of current density i by multiplying by the Faraday constant F and the number of electrons transferred n. The Faraday is 96,485 coul mol -1. So the rate equations can be written and where k 1 and k 2 are a function of potential. If the forward and backward current densities are equal there is no net current but there are varying rates of electron transfer in both directions referred to as the exchange current. If the exchange current density is high the net and measurable current density represents a slight bias in the ongoing exchange of electrons in both directions, especially if the potential is near the equilibrium potential. So the net current density is the difference between the cathodic and anodic currents We have written the reaction as a reduction and in this case a cathodic current density is positive. The current density is an exponential function of the potential. Written in terms of a potential E versus a reference electrode the current density expression becomes. 30

31 where the rate constants k f and k b are independent of potential. and The term α is called the symmetry coefficient and usually is 0.5±0.2. R is the gas constant and T is the absolute temperature. At a certain potential the forward and backward rates are equal and there is no net current. The system is then at equilibrium. The potential E in this case is the equilibrium potential E eq. Of course we could substitute a single (heterogeneous) rate constant k 0 for the forward and backward rate constants. So now we can write a general current density equation as where E 0 is the standard potential for the system. If we separate the cathodic and anodic current densities and set them equal, as they would be at equilibrium, we have This can be rearranged to give. When this is solved for E eq we have. which is our old friend the Nernst equation, derived from a current density equation. This applies to the special case where there is no net current and the system is at equilibrium. 31

32 For a system not at equilibrium we can write The term (E-E 0 ) is called the overpotential represented by η. A plot of this equation is given in Figure 1.. Figure 1. The curve intersects the zero current density line at E eq. At this point we still have the exchange current density. If we plot the cathodic and anodic current densities separately we can see the exchange currents (Figure 2.). Figure 2. In a state of equilibrium between a metal electrode and a solution containing an electroactive solute there is no net current and the potential difference between the electrode and bulk solution phase can be called φ eq. Yet there will be electron transfer processes taking place in opposite directions at equal rates. This is referred to as the equilibrium exchange current density. 32

33 The exchange current density is is function of the electrode material and the electroactive species in solution. According to activated complex theory the current density is an exponential function of the potential difference. For the reduction of the electron acceptor species A to the donator D, A + + e - =D, at equilibrium the reduction and oxidation current densities are equal which can also be written If the concentrations of A and D are equal we can designate both the anodic and cathodic current densities by the equilibrium exchange current density i 0. In the non-equilibrium case we can write. or it can be simplified by substituting i 0 in the equation where η is the difference between the non-equilibrium potential difference and the equilibrium potential difference. This is called the overpotential η. This is known as the Butler-Volmer equation. Given the hyperbolic sine function the current equation can also be written as. 33

34 Butler-Volmer Equation High-Field Approximation As the overpotential becomes more negative the reverse or anodic current becomes less significant so that the total or net current density approaches the condition of purely cathodic current. In Figure 1 the solid red line is a plot of the net current density and the dotted blue line is a plot of cathodic current density alone. It is apparent that they tend to merge at high overpotential. Figure 1. Table 1 demonstrates quantitatively how small the difference between the total and cathodic becomes at an overpotential of 1.5V for a system with α=.5, i 0 =1x10-5 A cm -2, 25 C. 34

35 Table 1. 35

36 Low-field approximation At very low overpotential the Butler-Volmer equation reduces to The linear low-field approximation is plotted along with the total current density in Figure 2. The dotted blue line is the plot of the linear approximation.. Figure 2. 36

37 The Tafel Equation In the case of a large cathodic current density the current as a function of overpotential η can be written Solving for η gives This corresponds to an early empirical equation by Tafel Below (Figure 1) is a plot of η vs. log of anodic current with α=0.5. The linear plot is of the Tafel equation. The slope b is 2.3RT/αnF. The intercept at η=0 gives the exchange current density i 0. This is known as a Tafel plot. With this we can calculate α and i 0. Figure 1. 37

38 Mass Transport Given the reduction reaction at a cathode, we know that at a given applied potential the rate and thus the current density of the reaction is proportional to the concentration of the species being reduced, O. If the reduced product R remains in solution and the reaction is reversible a simultaneous anodic current density will be a function of the concentration of R at the electrode surface. If we assume an initial concentration of species O in the solution, c O, and an initial concentration of the product R as zero (c R = 0) and an initial current density of zero, we have the bulk concentration of O in the bulk of the solution, c O (bulk), at the electrode surface. There is no net current and no net reaction. If we step the potential to a more negative potential the reduction reaction will be initiated. After the initial (and brief) charging current required to charge the electrode, the rate will be a function of the potential and the concentration of O. Immediately the concentration of O at the electrode surface c O (x=0) will begin will begin to decrease as O is reduced to R. Thus the forward rate will decrease with time as c O (x=0) decreases. Also the anodic component of the current will increase with the appearance of R at the electrode as the reaction proceeds. At this time a concentration gradient will have developed for both O and R and diffusion begins. For now we will ignore the contribution of migration where mass transfer of charged species is a function of the electric field or potential gradient. The concentration gradient of species O shortly after the potential step is shown in Figure 1. It is assumed that the solution is quiescent (not stirred). Figure 1. Provided that the potential is only so negative a condition will be reached where rate of the reduction reaction (electron transfer) matches the flux of O by diffusion to the electrode surface. This is not a steady state as the concentration gradient profile will 38

39 change over time. As the gradient decreases the flux will decrease. Thus the current density will decrease with time. As the electrode potential is shifted more negative a potential will be reached where any O that arrives at the the electrode surface x=0 will immediately be reduced. The reaction rate will then be limited entirely by the flux of O to the electrode. The current density is diffusion controlled at this point. Shifting the potential more negative will not increase the current unless it initiates another reaction. In this case as well the concentration gradient profile will change with time (Figure 2). Figure 2. Here too the current density will decrease with time as the concentration gradient decreases. Again, we are assuming no convection. If the solution is stirred or is flowing the depth of the diffusion region will be limited relatively constant. 39

40 Limiting Current If our dissolved reactant is being consumed or transformed by electron transfer at the electrode the concentration near the electrode is diminished. A concentration gradient dc/dx will form. Given a reasonably high exchange current density the reaction rate and thus the current may become limited by rate at which the reactant arrives at the electrode by diffusion. The diffusion limited current density is given by where F is the Faraday constant in coul mol -1, D is the diffusion coefficient in cm 2 sec -1, and dc/dx is in mol cm -4. The current density i is in coul cm -2 sec -1. (coul mol -1 )(cm 2 sec -1 )((mol cm -3 )(cm -1 )) = coul cm -2 sec -1. In the presence of convection, stirring for example, the bulk concentration will be maintained up to the hydrodynamic stagnant layer at the surface of the electrode. It is in this stagnant layer that the concentration gradient exists. Although there is no sharp distinction between the stagnant and moving regions an approximation is used to give a definite linear quantity to this layer called the Nernst diffusion layer and symbolized by d. This is illustrated in Figure 1. Figure 1. If we assume this layer has a constant value as determined by the mechanics of the apparatus then we can see that the concentration gradient will be at a maximum when the concentration at the electrode (x=0) is zero. At this point the rate of diffusion has reached a maximum. This is the maximum diffusion limited current density. 40

41 The Butler -Volmer equation seems to indicate that with increasing field strength the current will increase without limit. Of course this is not right as the reaction soon becomes limited by the rate at which the reactant arrives at the electrode. The limiting current density can be written as which is obviously at a maximum when the concentration at x=0 is zero. In this case the concentration goes from the bulk concentration to zero. The relationship between the Butler-Volmer equation curve and the effect of diffusion limited current is shown in Figure 2. Figure 2. In practice it is not easy to maintain a diffusion layer of constant thickness over time. The rotating disk electrode provides control over this. 41

42 Units and Symbols There are a number of rate constants used in electrochemical literature. They vary in that they do or do not contain a potential term, etc. The letter k is used with various subscript modifiers. If you wonder about the units of the constant you determine them by dimensional analysis. Suppose you see the expression i = nfak f C O Right away we can see that the letter i stands for current rather than current density because the area A is included on the right. So i is in amps or coul sec -1. We solve for k f getting i/(nfac O ). We know that n is a unitless number, F is the Faraday in coul mol -1. In terms of centimeters we can use cm 2 for the area A and mol cm -3 for the the concentration C O of the oxidized species. Assembling the units we have (coul sec -1 )/(coul mol -1 cm 2 mol cm -3 ) = cm sec -1. This can be interpreted as passage of a volume per area as cm 3 cm -2 sec -1. This is related to flux as in Fick's first law which is in mol cm -2 sec -1. The diffusion coefficient D is in cm 2 sec -1. Some current density equations contain a symmetry coefficient symbolized with α and then some use β. These depend on whether the reaction is written as a reduction or oxidation. If we write then we might see and If written as an oxidation. then we have 42

43 and. 43

44 The Silver / Silver Chloride Reference Electrode An Electrochemical Cell ( A little background.) The simplest cell consists of two electrodes in an electrolyte solution. The electrodes are usually made of metal, where current is carried by movement of electrons, the electrolyte solution consists of ions in a polar solvent, typically water. The ions are typically those produced by dissolution of an ionic compound such as potassium chloride (KCl). Potassium chloride dissolves in water producing potassium ions (K + ) and chloride ions (Cl - ). The potassium ions carry a positive charge, the chloride ions carry a negative charge. (Positive ions are called cations, negative ions are called anions.) Electric current consists of the net movement of charged particles in a particular direction. In metals the charge carrier is the negative electron. In an electrolyte solution the charge carriers are ions, both negative and positive. In the presence of an electric field (a potential gradient), electrons in a metal will move toward the more positive potential. In the electrolyte, current (movement of charge) is carried by the ions. Cations will move toward the more negative potential, anions toward the more positive potential. In the absence of a current there is no potential difference across a conducting phase; the entire phase is at the same potential. (In this example each electrode is a phase, the electrolyte solution is another phase. The air above the cell is a phase and the container (a glass beaker) is another phase.) Measurement of Potential Differences in an Electrochemical Cell To measure the potential difference between two electrodes in a cell we can connect a voltmeter between the two electrodes. The appropriate term for the point of attachment at the top of each electrode is the terminal. If the two electrodes were identical, platinum for instance, there would be no potential difference between the two terminals. But there would be a difference in potential between an electrode and the liquid phase, the electrolyte solution. Since the potential is the same across each phase (in the absence of current) the potential difference between the phases takes place in a very narrow region, the interface, between the phases. Figure 1 is a diagram of the cell with a voltmeter attached. 44

45 If the electrodes are of different materials we may have a significant difference in potential between the terminals. This depends on the specific composition of the electrolyte as well. In this case the difference between the potential of one electrode and the solution would be different than the potential difference between the other electrode and the solution. In both cases there are two interfacial regions at which the potential differences occur. In one case they are the same, but opposite in direction; in the other they differ. In both cases we can read only a single value with the voltmeter. If the two interfacial potential differences are equal but opposite in direction, as they would be in the case of identical electrodes, the voltmeter would register zero volts. So the voltmeter gives us the sum of two potential differences. An important point is that we cannot know from such a measurement what the magnitude or direction of the potential difference is at either of the two interfaces. Reference Electrodes If it were possible to set up an electrode-solution combination that would have a predictable and stable potential difference between the electrode and the solution and combine that with another electrode-solution interface this would enable us to calculate the second potential difference. We would know the potential difference between the two terminals and the potential difference between one electrode and the solution. Taking the difference would give us the potential between the other electrode and the solution. It is possible to set up such a system with a predictable potential difference. This is called a reference electrode and the potential difference is referred to as an electrode potential or half-cell potential. This usually requires two separate solutions and a porous junction between them. The junction should prevent mixing of the two solutions but allow ions to pass through. There may be a small potential difference across the junction, but this can be kept small and constant. In order to minimize the junction potential it is common to use a salt bridge, often a concentrated solution of potassium chloride, connected to each solution chamber by a junction. Beside being able to establish a reproducible potential difference a reference system must also be able to pass a small amount of current without a significant change in potential. This requires what is called a high exchange current. This will be discussed in more detail later. 45