Download Parabolic Rate Law Derivation

Derivation of the Parabolic Rate Law
In oxidation processes, parabolic kinetics occurs when the mass gain or oxide growth on a
sample is proportional to the square root of time. In general, parabolic kinetics indicates that
diffusion of reactants (such as Fe, O, or electrons) through a growing oxide scale is ratedetermining (Kofstad 1966). If the diffusion of Fe ions is rate-determining, the oxidation rate is
proportional to the flux of cations through the oxide scale:
dx
∝ j Fe
dt
(1)
j Fe = c Fe v Fe
(2)
The cation flux, j Fe, can be written as
where cFe is the concentration in particles cm-3 and v Fe is the velocity of Fe ions in cm s-1 , giving
j Fe units of particles cm-2 s-1 . The velocity of a particle or ion in the scale is proportional to the
force, F, on the particle:
v Fe = B Fe F
(3)
where BFe is the mobility of the ion. In the case of metallic oxidation, the driving force results
from a chemical potential gradient across the oxide scale. Writing the chemical potential as µFe,
this force is written as
F=
dµ Fe
dx
(4)
for an oxide scale with thickness x. Combining equations 3 and 4 yields
j Fe = c Fe BFe
dµ Fe
dx
(5)
from the relationship
µ Fe = µ oFe + k T ln a Fe
(6)
we can write
dµ Fe
d ln aFe k T da Fe
= kT
=
dx
dx
a Fe dx
(7)
In an ideal system, the concentration, cFe, is equivalent to activity, aFe. Substituting and
combining equations, we get
j Fe = c Fe B Fe
dc
k T dc Fe
= BFe k T Fe
cFe dx
dx
(8)
As shown in equation 1,
dx
= [constant] j Fe
dt
(9)
so that a combination of equations 1 and 8 gives
dc
dx
= [constant]B Fek T Fe
dt
dx
(10)
If we assume that the potential is fixed at each boundary of the oxide scale, we can replace
dcFe/dx in equation 10 with ∆cFe/x. We then introduce the parabolic rate constant k’, and set
k ' = [constant ]BFe kT∆c Fe
(11)
Combining equations 10 and 11 then gives
dx k '
=
dt x
(12)
xdx = k ' dt
(13)
Equation 12 can be rewritten as
Upon integration of equation 13,
x= x
t =t
x= 0
t =0
∫ xdx = ∫ k' dt
(14)
we arrive at the parabolic rate law:
x 2 = 2 k 't
(15)