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Electrochemical Kinetics

R.A. Cottis

Kinetics of activation controlled systems

An activation controlled system or process is one where the rates of the electrochemical processes are controlled by the charge transfer across the metal solution interface (hence the alternate term charge transfer control). For a simple activation controlled metal dissolution (anodic) process, equation (1), with the rate of the reverse, metal ion reduction reaction (2) considered to be small, the current density is given by Tafel’s Law1 (3). −→ ne + M M +n (1)

Mne + M +n →− (2)

density current exchangeanodic

density currentanodic

reactionanodic for tcoefficien Tafel β

below) (see reactionanodic for potential mequilibriu where

lnβ

,0

0

,0

0

====

+=

a

a

a

a

a

aaa

i

i

E

i

iEE

(3)

As people generally find it easier to work with logarithms to base 10, it is common to express (3) as

)(mV/decade reactionanodic for tcoefficien Tafel 2.303β where

log

logβ303.2

,0

0

,0

0

==

+=

+=

aa

a

aaa

a

aaa

b

i

ibE

i

iEE

(4)

This can also be written in terms of the dependence of current density on potential as:

1 The Tafel coefficient, β, is related to the kinetics of the rate-determining step (rds) of the reaction, and typically has a value of the form RT/αnF, where α is a constant related to the symmetry of the effects of a change in potential on the forward and reverse reaction rates, and is typically in the region of 0.5; R is the Gas Constant; T is the absolute temperature; n is the number of electrons involved in the rds and F is Faraday’s constant. RT/F is approximately 59 mV at room temperature, so values of Tafel coefficients (b or 2.303 β) are typically in the region of 60 to 120 mV per decade, although higher or lower values are possible.




2

βa

aa,a

E-E i = i

0

0 exp (5)

If the potential is changed then

βa

aa i = dE

id (6)

which defines the charge transfer resistance, Rct, for the process

a

a

aact, i

id

dER

β== (7)

If we consider the reverse of the above reaction −← ne + M M +n (8)

this will also have a rate that is dependent on the potential, and in this case it will also depend on the metal ion concentration. As this is a cathodic reaction, the rate will increase as the potential decreases according to2:

reactionanodic reverse the fordensity current

reactionanodic reverse for tcoefficien Tafel where

ln

,

,

,0

,,

0

=

=β

β+=

ca

ca

a

cacaa

i

i

iEE

(9)

The Tafel slope for this reaction, βa,c, will usually be similar to that for the forward reaction, but it is not normally identical to it. If we combine equations (3) and (9), we get a relationship for the net anodic current, known as the Butler-Volmer equation:

−

−=

ca

a

a

aacaanet

E-EE-E i = iii

,

00

,0, expexpββ

(10)

and the net current density for the reaction will be zero when ia = ia,c = i0,a. The potential at which this

occurs is 0aE , the equilibrium potential for the reaction. The current density i0,a is the rate at which the

forward and reverse reactions are occurring at the equilibrium potential, and is know as the exchange current density. For an irreversible cathodic reaction, Red ne + Ox →− (11)

the cathodic current density is given by

2 Note that we use the term ‘reverse anodic reaction’ in (9), rather then ‘cathodic reaction’, in order to distinguish it from the main cathodic reaction in the corrosion process.




3

βc

csc,c

)E-(E- C i = i

0

0 exp (12)

where Cs is the surface concentration of the species being reduced3, often oxygen or hydrogen ions in corrosion systems and βc is the cathodic Tafel slope4. If the cathodic reaction is water reduction, then Cs is the concentration of water, which is essentially unchangeable. From (12), if Cs does not change with potential, i.e. for activation control, we obtain

βc

cc i- = dE

id (13)

where we can define, as before,

i

id

dE- R

c

c

ccct,

β== (14)

with the minus sign since the cathodic current increases as the potential decreases5.

As for the anodic reaction, we can also consider the reverse of the cathodic reaction, and we find that 0cE is

the equilibrium potential for this reaction, and i0,c is the exchange current density for the cathodic reaction. For the corroding interface the net current (assuming that the rates of the two reverse reactions are negligible) is given by: i- i = i canet (15)

so

dEid

- dE

id =

dEid canet (16)

or

R

1 +

R

1 =

R

1

cct,act,totct, (17)

Thus the charge transfer resistances for the two reactions are added in parallel to get the overall resistance of the interface. For a system at its corrosion potential, ia = ic = icorr so, substituting in (17) from (7) and (14) we get

ββ cacorr

totct,

1 +

1i =

R

1 (18)

3 Note that this assumes that only one molecule/ion of the species being reduced is involved in the rate-determining step of the reaction (that elementary step in the reaction sequence that limits the overall rate of the reaction). This is usually the case, if only because is it very difficult to get two molecules together on the electrode surface. 4 Note that we have taken βc to be positive. Arguably βc should be negative (since the actual slope of E versus log|i| is negative), and the term inside the brackets in (12) therefore does not have the leading minus sign. Both conventions are used; it is usually easy to see which is being used, but be aware of the possible confusion. 5 As with the question of the Tafel slope, two conventions can be used for cathodic currents: we can take the current as positive, but recognise that it actually has a negative sign when measured in the conventional way (i.e. we take the actual current to be –ic) or we can take ic to be negative. This document takes ic as positive, as the ‘negativeness’ of the cathodic current is made more explicit.




4

where Rct,tot is the slope of the E - i curve at Ecorr. This is the standard Stern-Geary equation used for polarisation resistance measurements. Here the term charge-transfer resistance, Rct, is used instead of polarisation resistance, Rp. Strictly the polarisation resistance describes the resistance as measured between the metal and a reference electrode in the solution, and includes any resistance in the system, including the resistance of the solution and the effects of changes in surface concentration of reactants (see below), as well as the charge-transfer resistance. Note, however, that the derivation of the Stern-Geary equation actually applies to the charge-transfer resistance, and any other resistance in circuit will lead to errors. The definition here is based on the slope of the polarisation curve at Ecorr, and should, in principle, apply for any measurement. However, the amplitude of the potential sweep used to measure polarisation resistance should be kept as low as possible to avoid causing changes to the corroding metal under test. When using ac methods to obtain corrosion rates the same equation is used, with the resistance determined from the ac response of the system. In this case non-linearity of the current - voltage curve is important. This is because the current throughout the ac cycle is used in the calculation of the resistance, rather than the slope at Ecorr.

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