Electrochemistry for Engineers 0581.5271 Electrochemistry for Engineers LECTURE 3 Lecturer: Dr. Brian Rosen Office: 128 Wolfson Office Hours: Sun 16:00

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

Polarization under Activation Control jac

Mixed Control

Mass Transport at Electrodes

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 Ji is the flux of species i(mol sec-1 cm-2) at a distance x from the surface. Di is the diffusion coefficient (cm2/sec) is the concentration gradient at a distance x, is the potential gradient zi and Ci 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.

Meaning of Terms • 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.

1st Fick’s law: How does diffusion change the concentration profile with respect to space where: DO is a diffusion coefficient of the species “O”, in cm2/s, usually ca. 10-5 cm2/sec. We are seeking a current vs. time relationship!

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 id All lead to a definition of chronoamperometry, measurements of currents vs. time, under potential step conditions. Under the boundary conditions: The initial condition = the homogeneity condition (before the experiment starts at t = 0): The semi-finite condition (regions sufficiently distant from the electrode are unperturbed by the experiment): The surface condition after the potential transition:

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

The Cottrell Experiment A: surface area in cm2; 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 potential step experiment: Only “O” is initially present t <0, E = E1, no reaction initiation (well above EO) t ≥0, E = E2, reaction occurs E2 is sufficiently negative (in case of reduction) that C(0,t)=0 A reversible reaction: O+ne R

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

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

The Laplace transformation is linear in that: 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(a2t) erfc(at1/2) f(s) A/s 1/(s+a) a/(s2+ a2) s/(s2 + a2) a/(s2 - a2) s/(s2 - a2) 1/s2 1/sn 1/s1/2 1/s3/2 e-x, where  = (s/k)1/2 e-x / e-x /s e-x /s 

Solutions of Partial Differential Equations There is a need for solving the diffusion equation (PDE): 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

Solutions of Partial Differential Equations (whiteboard)

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.

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

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

Concentration Profile Solution

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

The Cottrell equation for a reversible Faradaic reaction! 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 t1/2 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!

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

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 CO (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”, id

NEW Boundary condition Generalized Case (points 1,2,3, or 4) for a step of any size Fast kinetics, therefore CO(0,t) and CR(0,t) obey the Nernst Equation Initial condition Semi-infinite NEW Boundary condition (requires knowing CR (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

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

…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)

in their standard concentrations! The reduction of O still occurs Important!!! Why does cathodic current flow above E0? Recall: At E0, 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 E0 but the oxidation of R occurs faster giving a net anodic current. Here, a net cathodic current is seen, even above E0, since R does not exist initially

Reversibility Reversible systems in electrochemistry are always able to maintain their equilibrium concentrations (according to the Nernst equation) at the surface As the electrode is swept to more potentials potentials, systems with slow kinetics cannot “keep up” the maintenance of equilibrium concentration at the surface and there is a lag in the current response

Double Potential Step

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