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

2. 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

3. Polarization under Activation Control
jac

4. Mixed Control

5. Mass Transport at Electrodes

6. 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.

7. 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.

8. 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 cm2/sec.
We are seeking a current vs. time relationship!

9. 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:

10. Forms of the Laplacian Operator for Different Geometries
Generalizing for Any Geometry
Where 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

11. 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

12. 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

13. 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.

14. 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 

15. 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

16. Solutions of Partial Differential Equations (whiteboard)

17. 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.

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

19. 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!

20. Concentration Profile Solution

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

22. 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!

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

24. 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

25. 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

26. 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

27. …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)

28. 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

30. 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

31. Double Potential Step

32. 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!!!