The parabolic rate law

If we postulate diat die chemical potentials of all species are equal in two phases in contact at any interface, dieii Einstein s mobility equation may be simply applied, in Pick s modified form, to describe die rate of a reaction occun ing dirough a solid product which separates die two 

When die conceiiuation, c, and diffusion coefficient, D of die diffusing species in die reaction product are constant, dieii die rate of growdi of die product in diickness will be given by die simple equation 

A parabolic rate law will also be obtained if part or even all, of the diffusion through the product layer is by grain boundary diffusion rather than diffusion through the volume of each grain. The volume diffusion coefficient is quite simply defined as the phenomenological coefficient in Fick s laws. The grain boundary diffusion must be described by a product, DbS, where S is the grain 

If we consider the reaction of oxygen with a soHd (such as in the oxidation of the surface of a metal), the oxide layer on the surface thickens as the reaction proceeds. The rate of the reaction can be described in terms of the thickness of the layer, x, by the rate law 

However, as x increases, the rate of the reaction decreases because the oxygen must diffuse through the layer of metal oxide. Therefore, the rate is proportional to 1 /x, so the rate law becomes 

This equation can be integrated between the limits of x = 0 at i = 0 and some other thickness, x, at a later time, t. The result after integration can be 

Because this equation has the form of an equation for a parabola, this rate law is referred to as the parabolic rate law. It is interesting to examine the units on k for this case. If the thickness of the product layer is measured in cm and the time is in sec, k = cm / sec. If we consider the weight of the 

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In certain circumstances even the parabolic rate law may be observed under conditions in which the oxide is porous and permeated by the oxidising environment". In these cases it has been shown that it is diffusion of one or other of the reactants through the fluid phase which is rate controlling. More usually however the porous oxide is thought to grow on the surface of a lower oxide which is itself growing at a parabolic rate. The overall rate of growth is then said to be paralinear - and may be described by the sum of linear and parabolic relationships (see equations 1.197 and 1.198). 

If the PBR is less than unity, the oxide will be non-protective and oxidation will follow a linear rate law, governed by surface reaction kinetics. However, if the PBR is greater than unity, then a protective oxide scale may form and oxidation will follow a reaction rate law governed by the speed of transport of metal or environmental species through the scale. Then the degree of conversion of metal to oxide will be dependent upon the time for which the reaction is allowed to proceed. For a diffusion-controlled process, integration of Pick s First Law of Diffusion with respect to time yields the classic Tammann relationship commonly referred to as the Parabolic Rate Law ... 

Because the rate law in this form is a quadratic equation, the rate law is known as the parabolic rate law. When we solve this equation for the thickness of the product layer, x, we obtain... 

At high temperature and a higher partial pressure of oxygen (1 < P(02) < 20 torr), the rate of growth of the FeO layer follows the parabolic rate law. The rate of formation of FeO is determined by the rate of diffusion of Fe2+, but the rate of diffusion of O2- determines the rate at which the thickness of Fe203 increases. 

Among the theories proposed, essentially two main mechanisms can be distinguished these are that the rate-determining step is a transport step (e.g., a transport of a reactant or a weathering product through a layer of the surface of the mineral) or that the dissolution reaction is controlled by a surface reaction. The rate equation corresponding to a transport-controlled reaction is known as the parabolic rate law when... 

Nielsen, A. E. (1981), "Theory of Electrolyte Crystal Growth. The Parabolic Rate Law", Pure Appl. Chem. 53, 2025-2039. 

Figure 2. The rate constants k2 for some electrolytes following the parabolic rate law, vg = k2

The parabolic-rate law for the growth of thick product layers on metals was first reported by Tammann (1920), and a theoretical interpretation in terms of ambipolar diffusion of reactants through the product layer was advanced later by Wagner (1936, 1975). Wagner s model can be described qualitatively as follows when a metal is... 

Although the parabolic rate law has the same form as the mean square displacement (see Section 4.3.1), its physical background is quite different. Parabolic growth is always observed in a one dimensional experiment when due to a gradient-driven flux and where the boundaries are kept at constant potentials. 

Comparison of the Parabolic Rate Law Constants and the Energies, Entropies, and Free ... 

Table VIII lists the same quantities for the parabolic rate law for columbium, tantalum, titanium, zirconium, and iron. The K values show that the resistance to oxidation increases in the series columbium, tantalum, zirconium, titanium, iron. This is borne out by a comparison of the AF values for these oxidations. Although the oxidation of tantalum needs the highest energy of activation E, it is actually not as resistant to oxidation as, for example, iron, since the entropy difference for the tantalum oxidation is not sufficiently negative to make the overall potential barrier as high as that existing for the oxidation of iron. These...

Many reactions in actual soil-water systems are controlled by mass transfer or diffusion of reactants to the surface minerals or mass transfer of products away from the surface and to the bulk water. Such reactions are often described by the parabolic rate law (Stumm and Wollast, 1990). The reaction is given by... 

Fig. 19. Examples of electrolytes which follow the parabolic rate law. , Ag2C204 O, CaC204--H20 0> CaC03 0, BaS04 , AgCl. (Reproduced from ref. 47 by courtesy of North-Holland Publishing Co.)...

Comparison of kA and kh for aqueous systems at 25°C calculated from the data given by Nielsen [39] for the parabolic rate law... 

In summary, the parabolic rate law may be derived using eqn. (152) to determine the fluxes, with the adsorption layer composition determined by the equilibrium relationship (161) with [A](s) = [B](s) for a salt AB. Because the integration step is considered to be the rate-limiting process, it is of considerable interest to evaluate If the rate constant for the removal of water from the anion is much larger than for the cation, the dehydration of the cation will be rate-limiting. In this instance... 

FOLLOWING A SHORT introduction dealing with the relationship between diffusion process and field transport phenomena in tarnishing layers on metals and alloys, the mechanism of oxidation of iron is discussed. Epitaxy plays an important role on the gradient of the concentration of lattice defects and, therefore, on the validity of the parabolic rate law. Classical examples of metal oxidation with a parabolic rate law are presented and the various reasons for the deviation observed are elucidated on the systems Iron in CO/CO2 and CU2O in <>2. In addition, the oxidation of alloys with interrupted oxide-metal interfaces is treated. Finally, attention is focussed on the difficulties in explaining the low temperature-oxidation mechanism. 

The oxidation of iron at high temperatures, where several iron oxide phases form, obeys the parabolic rate law whereas in CO-CO2mixtures above 900°C, it obeys a linear rate law with the exclusive formation of an FeO layer (18). This result is understandable if one considers the high defect concentration in FeO of approximately 10%, which ensures high diffusion velocity. The linear rate constant - exhibits the following de-... 

With these relations it is possible to evaluate the experimental results of Moore and Lee (40). In addition to the parabolic rate law (Fig. 8) the layer constant k j is found to be proportional to the logarithm of the oxygen pressure (Fig. 9). 

Assiuning that and cr are average values for the layer and do not vary with composition, gives a result that is the parabolic rate law ... 

Theories of oxidation have been developed by Wagner and by Mott ° . In general the logarithmic rate law applies to very thin oxide layers which form protective coatings and the parabolic rate law to thick oxide layers. More recent reviews of the subject have been given by Grimley , Kubaschewski and Hopkins and by Wyn Roberts . ... 

Ammonium diuranate, identified as UO3.NH3.2H2O, decomposes in two stages [120]. The first stage, completed at about 500 K, was identified as a onedimensional diffiision process, by its conformity to the parabolic rate law (a = kt) with Ej = 42 4 kJ mol. The product, UO3.2H2O, is dehydrated at 500 to 600 K to P-UO3. This reaction is fitted by the contracting volume equation, with E, = 75 8 kJ mol and there is an increase of surface area. UO3 decomposes to U3O8... 

Sulfidation tests of the different alloys were carried out over the temperature range 750-950°C. Plots of the square of mass gain vs. time in different H2-H2S atmospheres at 900°C are shown in Fig. la,b,c.The kinetics of Ni36Al and Ni45Al follow the parabolic rate law with rather high corrosion rates. In gas mixtures containing less than 0.5 vol.% H2S no sulfidation but alumina growth was observed. 

The kinetics on the Ni25Al alloy deviates from the parabolic rate law at 900°C, in atmospheres containing less than 10 vol.% H2S a fast initial stage is followed by a slower final stage (Fig. la). 

The influence of temperature on the sulfidation kinetics of Ni36Al at a constant sulfur partial pressure of 6.4X10-7 bar is shown in Fig. 3a. The sulfidation kinetic follows the parabolic rate law over the whole temperature range and the sulfidation rate increases with increasing temperature. From the Arrhenius plot of the parabolic rate constants for sulfidation of Ni36Al in Fig. 3b we calculate an apparent activation energy of 58kJ/mol. 

The sulfidation kinetics of the investigated nickel aluminides except Ni25Al approximately follows the parabolic rate law in H2-H2S atmospheres between 750 and 950 °C. 

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