PDF BASIC POTENTIAL STEP METHODS - Seoul National University

Ch. 5

BASIC POTENTIAL

STEP METHODS

5.1 OVERVIEW OF STEP EXPERIMENTS

▪ Potentiostat

control the potential difference between the working and reference electrodes by forcing the current across the working and counter electrodes.

5.1 OVERVIEW OF STEP EXPERIMENTS

▪ Basic potential step experiment

▪ For example, consider anthracene in deoxygenated dimethylformamide (DMF).

E₁ region: faradaic processes do not occur E₂ region (more negative potential):

the kinetics for reduction of anthracene become so rapid that no anthracene can coexist with the electrode

its surface concentration goes nearly to zero.

"mass-transfer-limited" region

5.1 OVERVIEW OF STEP EXPERIMENTS

▪ At E₂

Current flows to maintain the fully reduced condition at the electrode surface.

The initial reduction has created a concentration gradient that in turn produces a continuing flux of anthracene to the electrode surface.

Since this arriving material cannot coexist with the electrode at E₂, it must be eliminated by reduction.

The flux of anthracene, hence the current as well, is proportional to the concentration gradient at the electrode surface.

5.1 OVERVIEW OF STEP EXPERIMENTS

▪ Note, however, that the continued anthracene flux causes the zone of anthracene depletion to thicken

thus the slope of the concentration profile at the surface declines with time, and so does the current.

called chronoamperometry, because current is recorded as a function of time.

5.2 POTENTIAL STEP UNDER DIFFUSION CONTROL

▪ Consider an experiment involving an instantaneous change in potential from a value where no electrolysis occurs to a value in the mass-transfer-controlled region for

reduction of anthracene

Develop a quantitative treatment of such an experiment to derive the current-time response

▪ Conditions:

1) Assume a planar electrode (e.g., a platinum disk) and an unstirred solution.

2) Consider the general reaction О + ne R 3) Apply a sufficiently negative potential

the surface concentration of О becomes effectively zero.

5.2 POTENTIAL STEP UNDER DIFFUSION CONTROL

(a) Solution of the Diffusion Equation

▪ Use the linear diffusion equation

▪ Boundary conditions:

▪ Laplace transformation is required

A.1.2 Introduction to the Laplace Transformation

▪ Using Laplace Transformation, PDEs can be transformed into ODEs, which are then solved conventionally or by further application of transform techniques.

This method is extremely convenient, but it is restricted almost entirely to linear differential equations.

▪ The Laplace transform in t of the function F(t) is symbolized by L{F(t)}, (s), or (s), and is defined by

A.1.2 Introduction to the Laplace Transformation

▪ Table A. 1.1 gives a short list of some commonly encountered functions and their transforms.

A.1.2 Introduction to the Laplace Transformation

▪ The Laplace transformation is linear in that

where a and b are constants.

▪ The value of the transformation for solving differential equations issues from its conversion of derivatives with respect to the transformation variable into algebraic expressions in s.

▪ For example,

A.1.2 Introduction to the Laplace Transformation

▪ A proof rests upon integration by parts:

▪ One can show similarly that

▪ The transformation is oblivious to differential operators other than those involving t:

because variables other than t are regarded as constants for purposes of conversion.

A.1.2 Introduction to the Laplace Transformation

▪ Other useful properties involve the transforms of integrals and the effect of multiplication by an exponential:

5.2 POTENTIAL STEP UNDER DIFFUSION CONTROL

▪ Consider an experiment involving an instantaneous change in potential from a value where no electrolysis occurs to a value in the mass-transfer-controlled region for

reduction of anthracene

Develop a quantitative treatment of such an experiment to derive the current-time response

▪ Conditions:

1) Assume a planar electrode (e.g., a platinum disk) and an unstirred solution.

2) Consider the general reaction О + ne R 3) Apply a sufficiently negative potential

the surface concentration of О becomes effectively zero.

5.2 POTENTIAL STEP UNDER DIFFUSION CONTROL

(a) Solution of the Diffusion Equation

▪ Use the linear diffusion equation

▪ Boundary conditions:

▪ Laplace transformation:

5.2 POTENTIAL STEP UNDER DIFFUSION CONTROL

▪ Solve the ODE using the Laplace transformation:

s s a

s s a

s s a

s a s s

sY Y

s s b

Y ( ) ( ) ( )

) )(

(

) 0 ( ' )

0 ) (

(

2

α β + γ

+ −

= + +

−

+ +

= −

b x

Y dx a

x Y

d ( ) −

₂

( ) = −

2 2

s s b

Y a Y

sY s

Y

s

²

( ) − ( 0 ) − ' ( 0 ) −

²

( ) = −

)

2

( a

e b e

x

Y = α

^−ax

+ β

^ax

+

s = 0

₂

a

= b γ

2

) ( )

) (

( a

b a

s s a

s s s

Y +

+ −

= α + β

) ) (

( )

( )

)(

(

) 0 ( ' )

0

2

(

a s s

s s a s

s s a

s a s

sY Y

s

b α β + γ

+ −

= + +

−

+ +

−

5.2 POTENTIAL STEP UNDER DIFFUSION CONTROL

▪ Semi-infinite boundary condition:

▪ Third boundary condition (surface concentration) to evaluate A(s):

)

2

( )

( a

e b e

x C x

Y = = α

^−ax

+ β

^ax

+

5.2 POTENTIAL STEP UNDER DIFFUSION CONTROL

▪ The flux at the electrode surface is proportional to the current

▪ Laplace transformation:

= Co*/s x s

^1/2

/ D

_O^1/2

5.2 POTENTIAL STEP UNDER DIFFUSION CONTROL

▪ Inversion of the Laplace transformation:

known as the Cottrell equation

Note that the current is an inverse t^1/2 function for the mass-transfer limiting case if the potential step is applied.

5.2 POTENTIAL STEP UNDER DIFFUSION CONTROL

(b) Concentration Profile

▪ Inversion of the Laplace transformation:

5.2 POTENTIAL STEP UNDER DIFFUSION CONTROL

▪ Error function: the integrated normal error curve

5.2 POTENTIAL STEP UNDER DIFFUSION CONTROL

▪ The thickness of the diffusion layer depends significantly on the time scale of the experiment

5.2 POTENTIAL STEP UNDER DIFFUSION CONTROL

Semi-Infinite Spherical Diffusion

▪ If the electrode in the step experiment is spherical rather than planar (e.g., a hanging mercury drop)

use the spherical diffusion equation of Fick's second law

▪ Boundary conditions:

: where r is the radial distance from the electrode center

: where r₀ is the radius of the electrode.

5.2 POTENTIAL STEP UNDER DIFFUSION CONTROL

(a) Solution of the Diffusion Equation

▪ Replace C_O(r,t) by Y_O(r,t)/r

2 2

( , ) )

, (

r t r D Y

t t r Y

∂

= ∂

∂

∂

_O

O O

▪ Solve the equation and Relate to the current

the diffusion current for the spherical case is just that for the linear situation plus a constant term.

5.2 POTENTIAL STEP UNDER DIFFUSION CONTROL

▪ For a planar electrode,

▪ In the spherical case,