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Electrochemistry for Engineers LECTURE 3 Lecturer: Dr. Brian Rosen Office: 128 Wolfson Office Hours: Sun 16:00

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What Have We Done So Far?? Thermodynamics of electrode potentials Use of 2 electrodes connected in Galvanic and Electrolytic configurations Effect of kinetic parameters on electrode polarization (I-V) relationships – Effect of exchange current density – Effect of barrier symmetry

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Polarization under Activation Control j ac

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Mixed Control

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Mass Transport at Electrodes

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General Statement of Mass Transport to a Planar Electrode Mass transfer to an electrode is governed by the Nernst-Planck equation. For one- dimensional mass transfer of a species i (or j) along the x-axis: where J i is the flux of species i(mol sec -1 cm -2 ) at a distance x from the surface. D i is the diffusion coefficient (cm 2 /sec) is the concentration gradient at a distance x, is the potential gradient z i and C i are the charge and concentration of species i, respectively, v(x) is the velocity (cm/sec) with which a volume element in solution moves along the axis.

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1st term: diffusion. Movement of a species under the influence of a gradient of chemical potential (i.e., a concentration gradient). 2nd term: migration. Movement of a charged body under the influence of an electric field (a gradient of potential). 3rd term: convection. Stirring or hydrodynamic transport, but also natural convection (convection caused by density gradients) and forced convection. Meaning of Terms

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1st Fick’s law: How does diffusion change the concentration profile with respect to space where: D O is a diffusion coefficient of the species “O”, in cm 2 /s, usually ca cm 2 /sec. We are seeking a current vs. time relationship!

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2nd Fick’s law: How does diffusion affect the concentration grad with respect to time? Is controlled by mass transport only! This yields the diffusion-limited current i d All lead to a definition of chronoamperometry, measurements of currents vs. time, under potential step conditions. Under the boundary conditions: 1.The initial condition = the homogeneity condition (before the experiment starts at t = 0): 2.The semi-finite condition (regions sufficiently distant from the electrode are unperturbed by the experiment): 3.The surface condition after the potential transition:

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Forms of the Laplacian Operator for Different Geometries Type Linear Rectangular Spherical Cylindrical (axial) Variables x x,y,z r Example Planar disk electrode Cube-shaped electrode Hanging drop electrode Wire (rod) electrode Generalizing for Any Geometry Laplacian Where 2 is the Laplacian operator

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The Cottrell Experiment A potential step experiment: Only “O” is initially present t <0, E = E 1, no reaction initiation (well above E O ) t ≥0, E = E 2, reaction occurs E 2 is sufficiently negative (in case of reduction) that C(0,t)=0 A: surface area in cm 2 ; n, a number of electrons per molecule (ion), dimensionless; F, the Faraday constant. (The flux equation gives the current in Amperes.) Flux, and electric current, may be entirely controlled by mass transport of the species “O” to the electrode surface. A reversible reaction: O+ne R

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Boundary Conditions for Cottrell Experiment 1.The initial condition C O (x,t) = C O * for t<0 2.The semi-finite condition (regions sufficiently distant from the electrode are unperturbed by the experiment): 1.The surface condition after the potential transition: C O (0,t) = 0 for t<0 t > 0

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The Laplace transform in t of the function F(t) is symbolized by L{F(t)}, f(s), or and is defined by: Laplace Transformation The existence of the transform is conditional: (a)F(t) must be bound at all interior points on the interval 0 ≥ t < ∞; (b)that it has an infinite number of discontinuities; and (c)that is to be of an exponential order In practical applications, (a) and (c) do occasionally offer obstacles, but (b) rarely does. 0 ∞

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The Table gives a short list of commonly encountered functions and their transforms (p. 771). The Laplace transformation is linear in that: Table: Laplace Transforms of Common Functions F(t) A(constant) e -at sin at cos at sinh at cosh at t t (n-1)/ (n-1)! ( t) -1/2 2(t/ ) 1/2 erfc[x/2(kt) 1/2 ] exp(a 2 t) erfc(at 1/2 ) f(s) A/s 1/(s+a) a/(s 2 + a 2 ) s/(s 2 + a 2 ) a/(s 2 - a 2 ) s/(s 2 - a 2 ) 1/s 2 1/s n 1/s 1/2 1/s 3/2 e - x, where = (s/k) 1/2 e - x / e - x /s e - x /s

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There is a need for solving the diffusion equation (PDE): Solutions of Partial Differential Equations The solution requires an initial condition (t = 0), and two boundary conditions in x. Typically one takes C (x, 0) = C* for the initial state, and the semi-infinite condition: “This requires only one additional boundary condition to define a problem completely.” Without, a partial solution can be obtained. for t > 0

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Solutions of Partial Differential Equations (whiteboard)

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Transforming (Laplace) the PDE on the variable t, we obtain We then obtain (ODE): and, after transformation of the semi-infinite condition hence, B’(s) must be zero for the second boundary condition.

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Then to solve, we must take the reverse transform: Final evaluation of depends on the third boundary condition. and

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Application of Laplace transformation and the first 2 boundary conditions to the 2nd Fick’s equation yields (see above): Inversion using inverse Laplace transform gives the concentration profile, to be developed! can be inverted to obtain the concentration profile for species O. Transforming the surface condition gives (evaluate A’(s)): Using the surface condition, the function A(s) can be evaluated, and: Which implies that: A=-C o */s

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Concentration Profile Solution

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The flux at the electrode surface is proportional to current: Which is transformed to: Obtaining Current vs. Time

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Substitution of yields (at x = 0): and inversion produces the current-time response: The Cottrell equation for a reversible Faradaic reaction! The Cottrell equation predicts: an inverted t 1/2 current-time relationship at t approaching zero the current goes to infinity current measured is always proportional to the bulk concentration of “Ox”. A planar electrode! R is initially absent!

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Potential Pulse E vs. t Concentration Grad C O (x,t) vs. x Current Response i vs. t Qualitative Understanding of Cottrell Experiment

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Sampled Current Voltammetry Cottrell Experiment only valid for point 5 Cottrell equation is only valid if we assume the overpotential,E-E°, was negative enough to keep our boundary condition C O (0,t)=0 Only at large overpotentials can we assume that the surface concentration will be zero because of the large rate of reaction! When the surface concentration is zero, the current is limited only by diffusion This is the “diffusion limiting current”, i d

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DO NOT CONFUSE “Fast kinetics” – Electron transfer is rapid, therefore surface concentration of O and R can be assumed to obey the Nernst equation “Fast Rate of Reaction” – Reaction occurs quickly due to large driving force a high overpotentials. Therefore, the surface concentration of reactant can be assumed as zero. For example, in a reduction reaction one can have fast kinetics at but a low rate of reaction because the potential is just below E 0

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Generalized Case (points 1,2,3, or 4) for a step of any size Initial conditionSemi-infinite NEW Boundary condition (requires knowing C R (0,t) since boundary condition of Cottrell does not apply) …since a gain in R is a loss in O according to O + ne R Fast kinetics, therefore C O (0,t) and C R (0,t) obey the Nernst Equation

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Laplace transformation of Ficks second law on “O” and “R” in consideration of initial and semi-infinite conditions gives: Application of the new boundary condition shows that B(s) = -A(s)ξ Where ξ =, the root-ratio of the diffusion coefficients of the reduced ‘R’ and oxidized ‘O’ species

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…upon reverse transformation, we get: The difference between the GENERALIZED expression and the COTTRELL expression (at large overpotentials) is a factor of 1/(1+ξϴ)

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Why does cathodic current flow above E 0 ? Recall: At E 0, the oxidation and reduction reaction rates are equal assuming O and R are present in their standard concentrations! The reduction of O still occurs ABOVE E 0 but the oxidation of R occurs faster giving a net anodic current. Here, a net cathodic current is seen, even above E 0, since R does not exist initially Important!!!

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Reversibility

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Double Potential Step

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but…. the interface of the electrode and the electrolyte form a capacitor! Therefore, experimentally measured current includes non-electrochemical (non-Faradaic) currents!!!

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The Electrical Double Layer

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Review: RC Circuits When a capacitor is charged in a circuit, it acquires a potential difference between the plates. A typical RC circuit: S chg is on at t = 0, the charge on the capacitor is time dependent: q(t) chg = Q o (1- e -t/RC ) = Q o (1 - e -t/ )..where (Q o = EC)

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A RC circuit for charging a capacitor in a single “charge” step: The charge - time profile: q(t) = Q o (1-e -t/RC ), at R =10 , C = 20 F, Q 0 = 1 mC, E = 50 V The Time Constant,

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At t = , the capacitor is charged to 0.63 (63%) of its final value (Q o ), q(t) = Q o (1-e -1 ) = 0.63Q o. At a time equal to 3 , the capacitor is charged to 95% of Q o. Upon charging to the voltage E after the switch (S chg ) is on, the current in the resistor R decreases, and eventually stops. When the capacitor is discharged, the current in the resistors starts at E/R and then decreases to zero: q(t) dis = Q o e -t/RC = Q o e -t/

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At , the charge on the capacitor is reduced to 0.37 of the initial value; the discharging curve: q(t) = Q o e -t/RC : For charging (back to):

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(a) charge vs. time, and (b) current in time. Circuit parameters are: R =10 , C= 20 F. [Q 0 = 20 C, = 10 x 20 F = 0.2 ms]. a b = E/R. e -t/RC i (total) = i (faradaic) + i (non-Faradaic) From Cottrell From DL charging Therefore, in a POTENTIAL STEP EXPERIMENT

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i (total) = i (faradaic) + i (non-Faradaic) From Cottrell From DL charging Therefore, in a POTENTIAL STEP EXPERIMENT = E/R. e -t/RC Since the non-faradaic current contributes to our experimental signal, we must figure out how to model C, the capacitance, of an electrochemical system! To do this, we must be able to model the interface! Which model should we use?

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electric charge resides on a surface of an electronic conductor, e.g. metal (in an electrochemical cell) The Helmholtz model: a counterion resides at the surface 2 sheets of charge densities ( ) of opposite sign, separated by a distance of molecular order gives the “double layer” (dl) I. The Helmholtz Model is the stored charge density, is the dielectric constant of the medium, 0 is permittivity of the free space, V is the voltage drop between the plates. Charge density in units of charge/area

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I.The Helmholtz Model VISUALIZED METAL (M) SOLUTION (S)

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predictions (incorrect): no bulk concentration dependence of C d : The differential capacity (at constant parameters): I.Helmholtz Model - Problems

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Interplay of two tendencies at the metal solution interface: - the tendency of the electrostatic interactions between ions and the charged plane - the tendency of thermal motions The result: the idea of a diffuse layer used a statistical mechanical approach to the diffuse dl description The model did much better (than the Helmholtz (H) model) but did not yield a satisfactory description of real system II. Gouy and Chapman

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a solution sub-divided into laminae, parallel to the electrode and of thickness dx, all laminae are in thermal equilibrium with each other ions of any species i are not at the same energy in the various laminae, because the electrostatic potential f varies Components of the G-C theory

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Hence: the number concentrations of species in two laminae has a ratio determined by a Boltzmann factor take a reference lamina far from the electrode, where every ion is at its bulk concentration n i 0 then the population in any other lamina is:

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II. Gouy and Chapman VISUALIZED METAL (M) SOLUTION (S)

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is the potential at a given point with respect to the bulk solution, e is the electronic charge, k is the Boltzmann constant, T is the absolute temperature T, and the z i is the ion charge. The total charge per unit volume in any lamina is then: i runs over all ionic species. (x) is related to the potential at distance x by the Poisson equation: By combining these equations, one obtains the Poisson-Boltzmann equation: Substituting the property:

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=,, gives: Separate and integrate: gives: far from electrode the field is zero

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Half angle relation: Symmetric electrolytes, n i 0 (anions) = n i 0 (cations) = n 0, the number concentration of each ions: “2:2”, “1:1”, z:z Substituting:, and using the number concentration of each ion in the bulk, n 0. Hyperbolic cosine: Anions Cations

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A square root, both sides: : the electric field or the field strength; the gradient of potential at a distance x.

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After separating d and dx and integrating over 0 and 0 x Where is a constant at constant T; For dilute aqueous solutions at 25 °C, ɛɛ 0 = 78.4, Boltzman K is x J/K : = (3.29 x 10 7 )zC* 1/2 where C is in mol/L and is in cm -1. At small 0 (<50/z mV at 25 °C): = 0 e - x 1/ : the characteristic thickness of the diffuse double layer

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At small 0 (<50/z mV at 25 °C): = 0 e - x 1/ : the characteristic thickness of the diffuse double layer

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One More Step! How to get Capacitance? Electric field = = 0 for all surfaces Except the electrode itself. On the surface = (dɸ/dx) at x=0 (notes to be posted)

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Recall from earlier:

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Still not completely correct Guoy ChapmanReal Systems

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III. The Stern Model Stern combined the H and G-C theory and offered the most complete model for electrical dl without surface phenomena: chemisorption (specific adsorption), surface oxidation, reconstruction, relaxation, phase transitions, etc..

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The inner layer: the compact Helmholtz or Stern layer; the inner Helmholtz plane (IHP), at the distance x 1. Specifically adsorbed ions. Solvated ions can approach the metal at the distance x 2 (the locus of the center of such non specifically adsorbed ions is called OHP).

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