The Current-Potential Relationship

Physical Elearochemistry Fundamentals, Techniques and Applications. Eliezer Gileadi Copyright 2011 WILEY-VCH Verlag GmbH Co. KGaA, Weinheim ISBN 978-3-527-31970-1 

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Figure 1.62b shows the result of raising the potential of a corroding metal. As the potential is raised above B, the current/potential relationship is defined by the line BD, the continuation of the local cell anodic polarisation curve, AB. The corrosion rate of an anodically polarised metal can very seldom be related quantitatively by Faraday s law to the external current flowing, Instead, the measured corrosion rate will usually exceed... 

The current-potential relationship ABCDE, as obtained potentiosta-tically, has allowed a study of the passive phenomena in greater detail and the operational definition of the passive state with greater preciseness. Bonhoeffer, Vetter and many others have made extensive potentiostatic studies of iron which indicate that the metal has a thin film, composed of one or more oxides of iron, on its surface when in the passive state . Similar studies have been made with stainless steel, nickel, chromium and other metals... 

In this section we consider experiments in which the current is controlled by the rate of electron transfer (i.e., reactions with sufficiently fast mass transport). The current-potential relationship for such reactions is different from those discussed (above) for mass transport-controlled reactions. 

The current-potential relationship indicates that the rate determining step for the Kolbe reaction in aqueous solution is most probably an irreversible 1 e-transfer to the carboxylate with simultaneous bond breaking leading to the alkyl radical and carbon dioxide [8]. However, also other rate determining steps have been proposed [10]. When the acyloxy radical is assumed as intermediate it would be very shortlived and decompose with a half life of t 10" to carbon dioxide and an alkyl radical [89]. From the thermochemical data it has been concluded that the rate of carbon dioxide elimination effects the product distribution. Olefin formation is assumed to be due to reaction of the carboxylate radical with the alkyl radical and the higher olefin ratio for propionate and butyrate is argued to be the result of the slower decarboxylation of these carboxylates [90]. 

The current-potential relationship predieted by Eqs. (49) and (50) differs strongly from the Butler-Volmer law. For y 1 the eurrent density is proportional to the eleetro-static driving force. Further, the shape of the eurrent-potential curves depends on the ratio C1/C2 the curve is symmetrical only when the two bulk concentrations are equal (see Fig. 19), otherwise it can be quite unsymmetrieal, so that the interface can have rectifying properties. Obviously, these current-potential eurves are quite different from those obtained from the lattice-gas model. 

The two-step charge transfer [cf. Eqs. (7) and (8)] with formation of a significant amount of monovalent aluminum ion is indicated by experimental evidence. As early as 1857, Wholer and Buff discovered that aluminum dissolves with a current efficiency larger than 100% if calculated on the basis of three electrons per atom.22 The anomalous overall valency (between 1 and 3) is likely to result from some monovalent ions going away from the M/O interface, before they are further oxidized electrochemically, and reacting chemically with water further away in the oxide or at the O/S interface.23,24 If such a mechanism was operative with activation-controlled kinetics,25 the current-potential relationship should be given by the Butler-Volmer equation... 

The electrochemical detection of pH can be carried out by voltammetry (amper-ometry) or potentiometry. Voltammetry is the measurement of the current potential relationship in an electrochemical cell. In voltammetry, the potential is applied to the electrochemical cell to force electrochemical reactions at the electrode-electrolyte interface. In potentiometry, the potential is measured between a pH electrode and a reference electrode of an electrochemical cell in response to the activity of an electrolyte in a solution under the condition of zero current. Since no current passes through the cell while the potential is measured, potentiometry is an equilibrium method. 

Modeling the Mn-Al system is particularly difficult because the kinetics of the Mn and A1 deposition reactions can not be measured directly. Although it is possible to estimate the current-potential relationships for both Mn and A1 from electrodeposit composition, no examination along these lines appears in the literature. A close ex-... 

Denote the forward and backward rate constants of this reaction by ka and kb- When the reaction proceeds under stationary conditions, the rates of the chemical and of the electron-transfer reaction are equal. Derive the current-potential relationship for this case. Assume that the concentrations of A and of the oxidized species are constant. 

Because of the different potential distributions for different sets of conditions the apparent value of Tafel slope, about 60 mV, may have contributions from the various processes. The exact value may vary due to several factors which have different effects on the current-potential relationship 1) relative potential drops in the space charge layer and the Helmholtz layer 2) increase in surface area during the course of anodization due to formation of PS 3) change of the dissolution valence with potential 4) electron injection into the conduction band and 5) potential drops in the bulk semiconductor and electrolyte. 

The current-potential relationship is now revealed at an atomic scale to be correlated with the change of surface state of the underpotential deposition. 

Erdey-Gruz and Volmer (2) derived the current-potential relationship in 1930 using the Arrhenius equation (1889) for the reaction rate constant and introduced the transfer coefficient. They also formulated the nucleation model of electrochemical crystal growth. 

In this chapter we derive the Butler-Vohner equation for the current-potential relationship, describe techniques for the study of electrode processes, discuss the influence of mass transport on electrode kinetics, and present atomistic aspects of electrodeposition of metals. 

Equations (6.4) and (6.5) can be used to derive the current-potential relationship for a general electrochemical equation ... 

For example, if Qi = 50 tF/cm and R = 2 fi, t = 4.6 X 10 " s (0.46 ms). Thus, in the galvanostatic transient technique, the duration of the input current density pulse is on the order of milliseconds. From a series of measurements of for a set of i values, one can construct the current-potential relationship for an electrochemical process. For example. Figure 6.20 shows the current-potential relationship for the electrodeposition of copper from acid CUSO4 solution. 

In electroless deposition of a metal M, the current-potential relationships for the partial reaction may be written as linear functions ... 

The commensurable kinetics of an electrode reaction bring about a hindrance or resistance to the transfer of charge at the interface and the current—potential relationship for the interface is more complex than the one predicted by Ohm s law for conducting phases. 

Similar reasoning may hold if the electroactive species to be determined or studied otherwise can be adsorbed at the electrode surface. Also, in this case, the current—potential relationship is mathematically more complex and shapes and magnitudes (peak heights, limiting currents) of polarograms are more severely affected if the technique is faster (see also Sect. 6). As a consequence, calibration curves may become non-linear and even horizontal (i.e. the quantity monitored is independent of... 

If a Butler-Volmer formalism is assumed (see Eq. 1.101) for the sake of simplicity, the expression of the current-potential relationship can be rewritten as... 

This expression of the current-potential relationship is totally general. For each particular situation, the expressions of the rate constants (through a given kinetic model) and of the limiting currents and mass transport coefficients should be provided to analyze the influence of the different factors that can control the global rate. 

Unfortunately, a large number of substances reduced at the dropping-mercury electrode do not behave reversibly in other words, they do not behave according to equilibrium thermodynamics. For a totally irreversible process (one in which the kinetics for the backreaction are essentially equal to zero), the current-potential relationship takes the form... 

However, since this corrosion reaction is short-circuited on the corroding surface, no current will flow in any external measuring circuit. Consequently, a direct electrochemical measurement of the corrosion current (convertible to corrosion rate by the application of Faraday s law) cannot be made. Despite this limitation, electrochemical techniques can be used to decouple the two half-reactions, thereby enabling each to be separately and quantitatively studied. This involves the determination of the current-potential relationships for each half-reaction. Subsequently, the behavior under electrochemically unperturbed (open-circuit or natural corrosion) conditions can be reconstructed by extrapolation of these relationships to Ecorr-... 

The current-potential relationship for a reversible redox system (note the very high value of the exchange current density) is displayed in Fig. 3. As can be seen in Fig. 3(b) there is no -> Tafel region. On the other hand, in the case of an irreversible system, the Tafel region may spread over some hundreds of millivolts if the stirring rate is high enough (see there). 

The current-potential relationship of the totally - irreversible electrode reaction Ox + ne - Red in the techniques mentioned above is I = IiKexp(-af)/ (1+ Kexp(-asteady-state voltammetry, a. is a - transfer coefficient, ks is -> standard rate constant, t is a drop life-time, S is a -> diffusion layer thickness, and

logarithmic analysis of this wave is also a straight line E = Eff + 2.303 x (RT/anF) logzc + 2.303 x (RT/anF) log [(fi, - I) /I -The slope of this line is 0.059/a V. It can be used for the determination of transfer coefficients, if the number of electrons is known. The half-wave potential depends on the drop life-time, or the rotation rate, or the microelectrode radius, and this relationship can be used for the determination of the standard rate constant, if the formal potential is known. 

With continuing increase in anode potential, the current-potential relationship deviates from the linear relationship. As the potential continues to increase, the increase in the current slows down until it reaches a maximum and then decreases to a minimum point, and finally increases to the limiting current plateau. This is a transition from kinetics of electrochemical reaction domain to mass transport domain. This region may have a very different shape depending on electrolytes, potential scan rate, and other factors. Details of this region are discussed elsewhere [7,8]. 

When the effective resistance is low, the interphase is said to be nonpolarizable. In this case a significant current can be passed with only minimal change of the potential across the interphase. The current-potential relationship for the two cases is shown schematically in Fig. 3A. 

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