CHAPTER 1. FUNDAMENTAL CONCEPTS OF ELECTROANALYTICAL CHEMISTRY

Transcription

1 CHAPTER 1. FUNDAMENTAL CONCEPTS OF ELECTROANALYTICAL CHEMISTRY I. Introduction What is electrochemistry? Fundamental process: charge transfer at an interface FIGURE 1-1. Electrochemistry as an Interdisciplinary Science Advantages of Electrochemical Methods: 1. High sensitivity 2. Generally specific for a particular form of an element (influence by the element's color is avoided) 3. Comparatively inexpensive 4. Rapid 5. Suitable for continuous analysis (e.g., automatic analyzer, flow-injection analysis) 6. Wide scope (aqueous solution, nonaqueous solution, molten salt) 7. Respond to the activity of a chemical species rather than to the concentration. II. Electrochemical Cells and Reactions i/. Electrochemical Reaction - Transport of charge across interface Movement of ions Movement of electrons 1

2 ii/. Electrochemical Cell - Collections of interfaces a) Electrode - A phase through which charge is carried by electronic movement. Electrodes can be metals or semiconductors, and they can be solid or liquid. Anode - oxidation occur Cathode - reduction occur b) Electrolyte - A phase through which charge is carried by the movement of ions. e.g. Zn Zn 2+, Cl AgCl Ag In this notation, a slash represents a phase boundary and a comma separates two components in the same phase. By convention, the anode and information with respect to the solution with which it is in contact are always listed on the left. FIGURE 1-2. Bard Fig a (p. 3). When a gaseous phase is involved, it is written adjacent to its corresponding conducting element. e.g.,pt H 2 H +, Cl AgCl Ag FIGURE 1-2. Bard Fig b (p. 3). c) Salt Bridge - A tube filled with a solution that is saturated with potassium chloride, or sometimes, some other electrolyte. The purpose of the salt bridge is to isolate the contents of the two halves of the cell while maintaining electrical contact between them. e.g., Zn ZnSO 4 (1 M) CuSO 4 (1 M) Cu Isolation is necessary to prevent direct reaction between copper ions and the zinc electrode. The two vertical lines inserted between the zinc and copper sulfate solutions represent the two junctions at either end of the salt bridge. FIGURE 1-3. Skoog Fig (p. 463). iii/. Galvanic and Electrolytic Cells a) Galvanic Cell: Chemical energy electrical energy (reactions occur spontaneously when connected) e.g., FIGURE 1-4. Bard Fig a (p. 15). Zn ZnSO 4 (1 M) CuSO 4 (1 M) Cu E = V Zn (s) + Cu 2+ Zn 2+ + Cu (s) 2

3 b) Electrolytic Cell: Reactions are affected by the imposition of an external voltage greater than the reversible potential of the cell. e.g., FIGURE 1-4. Bard Fig b (p. 15). The cell could be made electrolytic by connecting the negative terminal of a DC power supply to the zinc electrode and the positive terminal to the copper electrode. If the output of this supply was made somewhat greater than 1.1 V, the two electrode reactions would be reversed and the net cell reaction would become Cu(s) + Zn 2+ Cu 2+ + Zn(s) A cell in which reversing the direction of the current simply reverses the reactions at the two electrodes is termed a chemically reversible cell. The fundamental nature of the reactions and behavior of a single electrode are independent of whether this electrode is part of a galvanic or electrolytic cell. Cathodic current - electrons cross the interface from electrode to a species in solution Anodic current - electrons flow from a solution species into electrode electrolytic cell galvanic cell cathode + Anode + III. Potentials and Thermodynamics of Cells i/. Free Energy and Cell Electromotive Force Consider a generalized half-reaction pp + qq ne rr + ss +... The equilibrium constant K for this reaction is given by K = [(a R ) er (a S ) es...]/[(a P ) ep (a Q ) eq...] where (a) e indicates equilibrium activity Define a second quantity Q such that Q = [(a R ) ir (a S ) is...]/[(a P ) ip (a Q ) iq...] where (a) i indicates instantaneous activity The quantity Q is not a constant, but changes continuously until equilibrium is reached. From thermodynamics, ΔG = nfe cell = RT In(Q) RT In(K) where F is the faraday (96,485 coulombs per mole of electrons) When all substances are at unit activity, ΔG = RT In(K) = nfe cell where E cell is the standard electromotive force (emf) of the cell reaction. i.e, E cell = (RT/nF) In(K) E cell = E cell (RT/nF) In(Q) = E cell (0.0592/n) log(q) (Nernst equation) 3

4 ii/. Calculation of Cell Potentials Sign convention: E cell = E right -E left = E cathode -E anode Consider the hypothetical cell, i.e., Zn ZnSO 4 (a 2+ Zn = 1.00) CuSO 4 (a 2+ Cu = 1.00) Cu anode: oxidation of elemental Zn to Zn(II); cathode: reduction of Cu(II) to metallic Cu. E cell = ( 0.763) V = V The positive sign for the cell potential indicates the reaction Zn(s) + Cu 2+ Zn 2+ + Cu(s) occurs spontaneously and that this is a galvanic cell. Alternatively, consider Cu CuSO 4 (a Cu 2+ = 1.00) ZnSO 4 (a Zn 2+ = 1.00) Zn E cell = (+0.337) V = V The negative sign indicates nonspontaneity of the reaction Cu(s) + Zn 2+ Cu 2+ + Zn(s) External E > V would be required to cause this reaction to occur. e.g., Cr 2 O H + + 6e 2Cr H 2 O; E 1 = 1.33 V (1) Cr +3 + e Cr +2 ; E 2 = 0.41 V (2) Cr 2 O H + + 7e Cr +3 + Cr H 2 O; E 3 =? V (3) Reaction (3) is composed of Reactions (1) and (2) as given by Equation (4) Rxn. (3) = Rxn. (1) + Rxn. (2) (4) Hence, ΔG 3 = ΔG 1 + ΔG 2 n 3 FE 3 = n 1 FE 1 n 2 FE 2 and E 3 = (n 1 FE 1 + n 2 FE 2 )/n 3 F = 1.08 V e.g., IO 3 + 3H2O + 6e I + 6OH ; E 5 = 0.26 V (5) H 3 IO e IO 3 + 3OH ; E 6 = 0.70 V (6) H 2 O H + + OH ; K w = 1.0 x (7) H 3 IO H + + 8e I + 6H 2 O; E 8 =? V (8) Reaction (8) is composed of Reactions (5) - (7) as given by Equation (9) Rxn. (8) = Rxn. (5) + Rxn. (6) 9 Rxn. (7) (9) Hence, ΔG 8 = ΔG 5 + Δ G6 9 ΔG 7 n 8 FE 8 = n 5 FE 5 + n 6 FE 6 9 (2.3RT) log(k w ) and E8 = [n5e5 + n6e6 9 (2.3RT/F) log(k w )]/n 8 = 1.31 V where 2.3RT = 1364 cal/mol 4

5 iii/. Half-Reactions and Reduction Potentials The overall chemical reaction taking place in a cell is made up of two independent halfreactions, which describe the real chemical changes at the two electrodes. Each half reaction responds to the interfacial potential difference at the corresponding electrode. Since no method exists for determining the absolute potential of a single electrode, most of the time only one of these reactions will be focused on, and the electrode at which it occurs is called the working (or indicator) electrode. To focus on it, one standardizes the other half of the cell by using an electrode made up of phases having constant composition. Just as the overall cell reaction comprises two independent half-reactions, the cell potential could also be broken into two individual electrode potentials. In order to develop a useful list of relative half-cell or electrode potentials, it is necessary to have a carefully defined reference electrode. The internationally accepted primary reference is the standard hydrogen electrode (SHE), or normal hydrogen electrode (NHE): Pt/H 2 (a = 1)/H + (a = 1) The electrode potential of the half reaction: 2H + + 2e H 2 has been assigned values of zero at all temperatures. Sign Convention by IUPAC: Chosen to write half-reactions as reductions. Negative E values for those which are more powerful reducing agents than hydrogen, and positive E values for those which are less powerful. iv/. Standard Electrode Potential The standard potential, E, is often defined as the potential of a half-cell reaction (vs. SHE) when all reactants and products exist at unit activity. e.g., Cu e Cu(s) E = V 2H + + 2e H 2 (g) E = V Cd e Cd(s) E = V Zn e Zn(s) E = V Decreasing strengths as oxidizing agents: Cu 2+ > H + > Cd 2+ > Zn 2+. The sign of the electrode potential will indicate whether or not the reduction is spontaneous with respect to the standard hydrogen electrode. e.g., Cu 2+ + H 2 (g) 2H + + Cu(s) proceeds toward right. Zn 2+ + H 2 (g) 2H + + Zn(s) does not occur and equilibrium proceeds toward left. In general, when the potential of an electrode is moved from its equilibrium (or its zerocurrent) value toward more negative potentials, the substance that will be reduced first (assuming all possible electrode reactions are rapid) is the oxidant in the couple with the least negative (or most positive) E. FIGURE 1-5. "Bard" Fig (p. 4). e.g., FIGURE 1-6. "Bard" Fig (p. 5). Critical potentials at which redox processes occur relate to standard potentials E for A/A & A/A +. 5

6 v/. Calculation of Half-Cell Potentials a) In the absence of precipitation or complex-forming reagents e.g., EXAMPLE 1-7. Skoog Ex (p. 472). b) In the presence of precipitation e.g., EXAMPLE 1-8. Skoog Ex (p. 473). Or, AgI(s) + e Ag(s) + I E = V where E AgI = log(k sp ) E = E AgI log(a I ) c) In the presence complex-forming reagents e.g., Ag + + 2S 2 O 2 3 Ag(S 2 O 3 ) 3 2 Formation constant, K f = (a Ag(S2 O 3 ) 2 3 )/[(a Ag +) (a S2 O 3 2 ) 2 ] Ag(S 2 O 3 ) e Ag(s) + 2S 2 O 2 3 E Ag(S2 O 3 ) 2 3 = log(1/k f ) vi/. Formal Potentials O + ne R E = E + (RT/nF)In{(O)/(R)} = E + (RT/nF)In{γ O [O] /γ R [R]} where ( ) are in activities and [ ] are in concentrations, and γis activity coefficient. i..e., E = E ' + (RT/nF)In{[O]/[R]} where E ' is defined as the formal potential. The formal potential lacks the thermodynamic significance of the standard potential, but is often experimentally useful, and can often be directly measured. 6

7 vii/. Prediction of Reactions a) Spontaneous Reactions The extent to which an electrochemical reaction has occurred at equilibrium is related to the value of the equilibrium constant for the reaction. e.g., Ce +4 + Fe +2 Ce +3 + Fe +3 with Pt Ce +4, Ce +3 Fe +3, Fe +2 Pt for which K eq = ([Ce +3 ] [Fe +3 ])/([Ce +4 ] [Fe +2 ]) E R = E +3 Fe, Fe +2 ' + (2.3RT/F) log([fe +3 ]/[Fe +2 ]) E L = E +4 Ce, Ce +3 ' + (2.3RT/F) log([ce +4 ]/[Ce +3 ]) Thus, E cell = E R -E L = E +3 Fe, Fe +2 ' E +4 Ce, Ce +3 ' + (2.3RT/F) log(q) where Q = ([Ce +3 ] [Fe +3 ])/([Ce +4 ] [Fe +2 ]) is the quotient of actual concentrations Assuming that Q 1 and that the medium is 1 M HClO 4, E cell = E +3 Fe, Fe +2 ' E +4 Ce, Ce +3 ' = 0.77 V 1.70 V = 0.93 V Electrical current would spontaneously flow from right to left, i.e., the spontaneous reaction is the oxidation of Fe +2 at the right hand electrode (i.e., the anode) and the reduction of Ce +4 at the left hand electrode (i.e., the cathode). Equilibrium concentrations will eventually be established by continued electrolysis. At equilibrium, electrolysis has ceased and the values of E cell and I are equal to zero. i.e., 0 = E +3 Fe, Fe +2 ' E +4 Ce, Ce +3 ' + (2.3RT/F) log(k eq ) Solving for K eq, K eq = exp[(e +3 Fe, Fe +2 ' E +4 Ce, Ce +3 ')/(RT/F)] The large magnitude of K eq is assurance that the reaction of Ce +4 with Fe +2 proceeds extensively to the right. Conversely, for the reaction Ce +3 + Fe +3 Ce +4 + Fe +2 K eq << 1 and the reaction does not proceed to an appreciable extent to the right. General Rule: The oxidized form of any species in an electrochemical half-reaction can spontaneously oxidize the reduced form of a couple having a lesser value of the standard reduction potential. 7

8 b) ph Dependency The extent of an electrochemical reaction will also be a function of solution ph when H + or OH - are consumed or produced by the reaction. e.g., HA + e P + H + where HA is a weak acid HA H + + A where K a = [(H + )(A )]/(HA) and (HA) tot = (HA) + (A ) E = E HA,P + (2.303RT/F) log{(ha)/[(p)(h + )]} (HA) = α HA (HA) tot i.e., α HA = (HA)/[(HA) + (A )] 1/α HA = 1 + (A )/(HA) = 1 + K a /(H + ) α HA = (H + )/[K a + (H + )] (HA) = {(H + )/[K a + (H + )]}(HA) tot E = E HA,P + (2.303RT/F) log{1/[k a + (H + )]} + (2.303RT/F)log{(HA)tot/(P)} Assume (P) = (HA) tot, For [H + ] >> K a, E = E HA,P + (2.303RT/F) log{1/(h + )} = E HA,P + (2.303RT/F) ph For [H + ] << K a, E = E HA,P (2.303RT/F) log{k a } = E HA,P + (2.303RT/F) pk a Plot E vs ph. At the intersection point, ph = pk a. FIGURE 1-9. E vs ph. viii/. Liquid Junction Potential Many cells are never at equilibrium because they feature different electrolytes around the two electrodes. There is somewhere an interface between the solutions, and at that point mass transport processes attempt to mix the solutes until the two electrolytes are identical. Three types of liquid junctions: 1. Two solutions of the same electrolyte at different concentrations; e.g., FIGURE "Bard" Fig a (p. 63). 2. Two solutions at the same concentrations with different electrolytes having an ion in common; e.g., FIGURE "Bard" Fig b (p. 63). 3. Two electrolytes not satisfying conditions 1 or 2; e.g., FIGURE "Bard" Fig c (p. 63). 8

9 For example, since H + has a much larger mobility than Cl, it penetrates the dilute phase initially at a higher rate. This process gives a positive charge to the dilute phase and a negative charge to the concentrated one, with the result that a boundary potential difference develops, known as the liquid junction potential, E j. The electric field then retards the movement of H + and speeds up the passage of Cl, until the two cross the boundary at equal rates. Thus, there is a detectable steady-state potential, which is not due to an equilibrium process. A familiar method for minimizing E j is to replace the junction, for example, HCl(C 1 ) NaCl(C 2 ) with a system featuring a concentrated solution in an intermediate salt bridge, where the solution in the bridge has ions of nearly equal mobility. Such a system is HCl(C 1 ) KCl(C) NaCl(C 2 ) As C increases, E j falls markedly because ionic transport at the two junctions is dominated more and more extensively by the massive amounts of KCl. The series junctions become more similar in magnitude and have opposite polarities; hence they tend to cancel. IV. Currents in Electrochemical Cells Electricity is transported within a cell by the migration of ions. With small currents, Ohm s law is usually obeyed, E = I R S where E = potential difference in volts responsible for ionic movement I = current in amperes R S = resistance of the electrolyte in ohms (A)Faradaic and Nonfaradaic Currents Both faradaic and nonfaradaic processes occur when electrode reactions take place. i/. Nonfaradaic process - Charges do not transfer across interface, but external current can flow (at least transiently), since each electrode surface behaves as one plate of a capacitor. This current, called the charging current, creates an excess (or a deficiency) of negative charge at the electrode surface. 9

10 a) Capacitance and Charge of an Electrode FIGURE "Bard" Fig (p. 7). q/e = C where q = charge on the capacitor (in coulombs, C) E = potential across the capacitor (in volts, V) C = capacitance (in farads, F) During this charging process, the charging current will flow. The electrode-solution interface has been shown experimentally to behave like a capacitor. At a given electrode, q M = q S where q M = charge on the metal q S = charge in the solution FIGURE "Bard" Fig (p. 8). charge density, σ M = q M /A (unit: C/cm 2 ) The charge on the metal q M represents an excess or deficiency of electrons and resides in a very thin layer (< 0.1 A ) on the metal surface. The charge in solution q S is made up of an excess of either cations or anions in the vicinity of the electrode surface. b) Electrical Double Layer The charge in the solution adjacent to the electrode surface is termed an electrical double layer. FIGURE "Bard" Fig (p. 9). 1. Helmholtz or Stern layer (inner layer) - Contains solvent molecules and sometimes other species (ions or molecules) that are said to be specifically adsorbed. Inner Helmholtz Plane (IHP) - the locus of the electrical centers of the specifically adsorbed ions. Outer Helmoltz Plane (OHP) - the locus of centers of these nearest solvated ions. 2. Diffuse Layer - Contains non-specifically adsorbed ions. σ S =σ i + σ d = σ M where σ i = total charge density from specifically adsorbed ions in the inner layer σ d = excess charge density in the diffuse layer The thickness of the diffuse layer depends on the total ionic concentration in the solution. FIGURE "Bard " Fig (p. 10). The structure of the double layer can affect the rates of electrode processes. Consider an electroactive species that is not specifically adsorbed. This species can only approach the electrode to the OHP, and the total potential it experiences is less than the potential between the electrode and the solution by an amount φ 2 - φ S, which is the potential drop across the diffuse layer. 10

11 ii/. Faradaic process -Charge transfer across interface, governed by Faraday's law, i.e., amount of chemical reaction caused by flow of current amount of electricity passed. a) Reversibility 1. Chemical Reversibility - Reversing the cell current merely reverses a spontaneous cell reaction and no new reactions appear. e.g., Pt/H 2 /H +, Cl /AgCl/Ag shorted electrodes H 2 + 2AgCl 2Ag + 2H + + 2Cl reverse current 2Ag + 2H + + 2Cl H 2 + 2AgCl e.g., Zn/H +, SO 2 4 /Pt shorted electrodes Zn Zn e (Zn electrode) 2H + + 2e H 2 (Pt electrode) reverse current 2H + + 2e H 2 (Zn electrode) 2H 2 O O 2 + 4H + + 4e (Pt electrode) 2. Thermodynamic Reversibility - An infinitesimal reversal in a driving force causes the process to reverse its direction. It must essentially be always at equilibrium. NOTE: A cell that is chemically irreversible cannot behave reversibly in a thermodynamic sense. A chemical reversible cell may or may not operate in a manner approaching thermodynamic reversibility. 3. Practical Reversibility The faradaic process may proceed at various rate. If the process is so rapid that the oxidized and reduced species are in equilibrium, then the reaction is termed reversible and the Nernst Equation applies. O + ne R E eq = E (RT/nF) In{(R)/(O)} where (O) and (R) are activities of O and R respectively. Since all actual processes occur at finite rates, they cannot proceed with strict thermodynamic reversibility. However, they may in practice be carried out in such a manner that thermodynamic equations apply to any desired accuracy. Under these circumstances, one might term the processes reversible. In electrochemistry, if an electrode system follows the Nernst equation or an equation derived from it, the electrode reaction is often said to be reversible (or nernstain). Reversibility, so defined, actually depends on the relative rates of the electrode process and the rapidity of the electrochemical measurement: a particular system behave reversibly when measurements are made slowly, but irreversibly if the measurement involves short time. i.e., Whether a process appears reversible or not depends on one s ability to detect the signs of disequilibrium. In turn, that ability depends on the time domain of the possible measurements, the rate of change of the force driving the observed process, and the speed with which the system can reestablish equilibrium. If the perturbation applied to the system is small enough, or the system can attain equilibrium rapidly enough compared to the measuring time, thermodynamic relations will apply. 11

12 b) Mass Transfer The movement of material from one location in solution to another, arises from differences in electrical or chemical potential at the two locations, or from movement of a volume element of solution. rate, v mt = I/nFA 1. Migration - Movement of a charged body under the influence of an electric field (a gradient of electrical potential). 2. Diffusion - Movement of a species under the influence of a gradient of chemical potential (i.e., a concentration gradient). 3. Convection - Fluid flow occurs because of natural convection (convection caused by density gradients) and forced convection (e.g., by stirring). (B) Effects of Currents on Cell Potentials i/. Ohmic Potential (IR Drop) To develop a current in either a galvanic or an electrolytic cell, a driving force in the form of a potential is required to overcome the resistance of the ions to movement toward the anode and the cathode. This force follows Ohm s law and is generally referred to as the ohmic potential, or the IR drop. The net effect of IR drop is to increase the potential required to operate an electrolytic cell and to decrease the measured potential of a galvanic cell. i.e., E cell = E cathode E anode IR S (10) where IR S = ohmic potential drop in the solution (sometimes called ohmic polarization) ii/. Polarization The departure of the electrode potential (or cell potential) from the reversible (i.e., nernstian or equilibrium) value upon passage of faradaic current, i.e., the magnitude of the current is less than if it was behaving reversibly. The current is limited by the rate of one (or more) of the steps in the electrode process. 1. Activation Polarization - If charge transfer is the slow (limiting) step. 2. Concentration Polarization - If slow movement of depolarizer or product is responsible. FIGURE "Bard " Fig (p.32). 12

13 a) Sources of Polarization Three regions of a half-cell where polarization can occur. These include 1.The electrode itself (charge-transfer polarization), 2.A surface film of solution immediately adjacent to the electrode (reaction polarization, here the rate of formation or decomposition of an intermediate limits the current, e.g., coupled chemical reactions, adsorption, desorption, or crystallization), and 3.The bulk of the solution (concentration polarization). FIGURE Bard Fig (p. 21). b) Ideal Polarized and Nonpolarized Electrodes 1. Ideally Polarized Electrode (IPE) An electrode at which no charge-transfer occurs across the electrode solution, regardless of the potential imposed from an outside source of voltage. i.e., A very large change in potential upon passage of an infinitesimal current, characterized by a horizontal region of an I-E curve. FIGURE "Bard " Fig a (p.20). Only nonfaradiac processes occur at an ideally polarized electrode. No real electrode can behave in this manner at all potentials; but certain systems approach this behavior over a limited range of potentials. e.g., Hg in deoxygenated KCl solution a range of > 2 V. At V vs NHE, 2Hg Hg e Hg 2 2+ Hg 2 Cl 2 At 2.12 V vs NHE, Hg K + + e K(Hg) H 2 O + e 1/2H 2 + OH (thermodynamically favorable but kinetically slow) A cell consisting of an IPE and an ideal reversible electrode can be approximated by an electrical circuit of a resistor, R S, representing the solution resistance, and a capacitor C d representing the double layer. FIGURE "Bard" Fig (p. 11). 2. Ideal Nonpolarized Electrode (or Ideal Depolarized Electrode) - Potential does not change upon passage of current, i.e., an electrode of fixed potential (e.g., SCE constructed with a large-area Hg pool). FIGURE "Bard " Fig b (p.20). 13

14 iii/. Overpotential (or Overvoltage) - The extent of polarization, i.e., deviation of the potential from the thermodynamic or equilibrium value. η = E E eq where actual electrode potential, E < equilibrium potential, E eq i.e., η is always negative. E cell = E cathode E anode + η cathode + η anode IR S where η cathode = cathodic overpotential η anode = anodic overpotential e.g., High overpotential associated with the formation of H 2 on P V. Classification of Electrochemical Methods FIGURE Skoog Fig (p. 485). These methods are divided into interfacial methods and bulk methods. Interfacial methods can be divided into two major categories: 1. Those involving no net current flow, one measures the equilibrium thermodynamic potential of a system essentially without causing electrolysis or current drain on the system. (potentiometry) 2. A voltage or current is applied to an electrode and the resultant current flow through, or voltage change of, the system is monitored. (amperometry, coulometry, conductometry) In each category of techniques, they are further divided into direct and indirect methods of analysis. Technique Direct Indirect a) Potentiometry Classical (E at I = 0) Potentiometric Titrations (E vs Vol. at I = 0) b) Amperometry Chronopotentiometry Amperometric Titrations (I vs Vol.) (E vs t at controlled I) Chronoamperometry (I vs t at controlled E) D.C. Polarography (I vs E) Derivative D.C. Polarography (δi/δt vs E) Task Polarography (I vs E at t max ) Normal Pulse Polarography (I vs E) Differential Pulse Polarography (I vs E) Linear-sweep Voltammetry (I vs E) Cyclic Voltammetry (I vs E) Stripping Voltammetry (I vs E) A.C. Voltammetry (I vs E) c) Coulometry Electrogravimetry Constant current Coulometric Titrations (wt. of deposit at constant E & I) (t at constant I) Constant Potential Coulometry (Q at constant E) d) Conductometry Classical (κ = 1/ρ) Conductometric Titrations (κ vs Vol.) Oscillometry or High-frequency Titrations (relative κ = 1/R S vs Vol.) 14

15 FIGURE 1-1a. FIGURE 1-1b. 15

16 FIGURE 1-2. Bard et al, Electrochemical Methods, 1st Ed. FIGURE 1-3. Skoog et al, Principles of Instrumental Analysis, 4th Ed. 16

17 FIGURE 1-4. Bard et al, Electrochemical Methods, 1st Ed. FIGURE 1-5. Bard et al, Electrochemical Methods, 1st Ed. 17

18 Fe 3+ is easiest to reduce! Sn 2+ is easiest to oxidize! FIGURE 1-6. Bard et al, Electrochemical Methods, 1st Ed. EXAMPLE 1-7. Skoog et al, Principles of Instrumental Analysis, 4th Ed. 18

19 EXAMPLE 1-8. Skoog et al, Principles of Instrumental Analysis, 4th Ed. FIGURE

20 FIGURE Bard et al, Electrochemical Methods, 1st Ed. FIGURE Bard et al, Electrochemical Methods, 1st Ed. 20

21 FIGURE Bard et al, Electrochemical Methods, 1st Ed. FIGURE Bard et al, Electrochemical Methods, 1st Ed. 21

22 FIGURE Bard et al, Electrochemical Methods, 1st Ed. When I = I l, η conc FIGURE Bard et al, Electrochemical Methods, 1st Ed. 22

23 FIGURE Bard et al, Electrochemical Methods, 1st Ed. FIGURE Bard et al, Electrochemical Methods, 1st Ed. 23

24 FIGURE Bard et al, Electrochemical Methods, 1st Ed. FIGURE Skoog et al, Principles of Instrumental Analysis, 4th Ed. 24