Parabolic law

In a study of tarnishing the parabolic law, Eq. VII-30, is obeyed, with kj = 0. The film thickness y, measured after a given constant elapsed time, is determined in a... 

This rate expression is known as the parabolic law. It is obeyed by oxidation of Ni, Ti, Cu, and Cr and by halogenation of silver. The product coat retards both diffusion and heat transfer. 

The behavior type may change with temperature range. For instance, oxidation of zinc above 350°C (662°F) obeys the parabolic law, but at lower temperature the product coat seems to develop cracks and the logarithmic law, nw = kt, is obsei ved. 

In part the parabolic law may also apply to multilayer oxide systems where the cation diffusion coefficient is much higher in the lower oxide tlran in the higher oxide, which, growing as a thin layer, undergoes plastic deformation at high temperatures, thus retaining the overall oxide layer as impervious to enuy of tire gas. 

The sulphation of cobalt oxide, CoO, follows the parabolic law up to 700°C and above 850°C, proceeding by outward diffusion of cobalt and oxygen ions through a sulphate layer which is coherent up to about 700°C. The mechanism... 

A protective oxide layer forms a continuous barrier between the reactants (oxygen and metal), which inhibits the reaction. The simplest assumption that can be made about the effectiveness of this barrier is that its protecting power is directly proportional to its thickness. Mathematically, AX/At = k2/X, which on integration gives the parabolic law,... 

If Q is the volume of the oxide per metal atom, the rate of growth, dA /d/, is equal to / Q. Thus from equation 1.179 we derive the parabolic law... 

Wagner s theory of the parabolic law involves the following assumptions ... 

The volume ratio (see Section 1.9) for cuprous oxide on copper is 1 7, so that an initially protective film is to be expected. Such a film must grow by a diffusion process and should obey a parabolic law. This has been found to apply for copper in many conditions, but other relationships have been noted. Thus in the very early stages of oxidation a linear growth law has been observed (e.g. at 1 000°C) . 

At 180-290°C it was found that the parabolic law first applied but subsequently changed to a logarithmic relationship of the type... 

At medium and high temperatures copper ultimately follows the parabolic law " . It has been shown " using radioactive tracers that the diffusion of copper ions in cuprous oxide is the rate-determining step at 8(X)-1 000°C, and there is considerable evidence in favour of the view that metal moves outwards through the film by means of vacant sites in the oxide lattice . 

The possible employment of beryllium in nuclear engineering and in the aircraft industry has encouraged considerable investigation into its oxidation characteristics. In particular, behaviour in carbon dioxide up to temperatures of 1 000°C has been extensively studied and it has been shown that up to a temperature of 600°C the formation of beryllium oxide follows a parabolic law but with continued exposure break-away oxidation occurs in a similar fashion to that described for zirconium. The presence of moisture in the carbon dioxide enhances the break-away reaction . It has been suggested that film growth proceeds by cation diffusion and that oxidation takes place at the oxide/air interface. ... 

The literature on the oxidation of nickel-copper alloys is not extensive and emphasis tends to be placed on the copper-rich materials. The nickel-rich alloys oxidise according to a parabolic law and at a rate similar to that for nickel Corronil (Ni-30Cu) exhibited a parabolic rate behaviour below 850°C but a more complex behaviour involving two parabolic stages above 900°C. Electron diffraction examination of the oxide films formed on a range of nickel-copper alloys showed the structures of the films to be the same as for the bulk oxides of the component metals and on all the alloys examined only copper oxide was formed below 500°C and only nickel oxide above 700°C . 

This expression, the parabolic law [38,77,468], has been shown to be obeyed in the oxidation of metals, where the reactant is in the form of a thin sheet. Variations in behaviour are apparent when diffusion in the barrier layer is inhomogeneous as a consequence of cracking or due to the development of more than a single product layer. Alternative rate relations may be applicable, e.g. 

Kinetic expressions for appropriate models of nucleation and diffusion-controlled growth processes can be developed by the methods described in Sect. 3.1, with the necessary modification that, here, interface advance obeys the parabolic law [i.e. is proportional to (Dt),/2]. (This contrasts with the linear rate of interface advance characteristic of decomposition reactions.) Such an analysis has been provided by Hulbert [77], who considers the possibilities that nucleation is (i) instantaneous (0 = 0), (ii) constant (0 = 1) and (iii) deceleratory (0 < 0 < 1), for nuclei which grow in one, two or three dimensions (X = 1, 2 or 3, respectively). All expressions found are of the general form... 

This deceleratory reaction obeyed the parabolic law [eqn. (10)] attributed to diffusion control in one dimension, normal to the main crystal face. E and A values (92—145 kJ mole-1 and 109—10,s s-1, respectively) for reaction at 490—520 K varied significantly with prevailing water vapour pressure and a plot of rate coefficient against PH2o (most unusually) showed a double minimum. These workers [1269] also studied the decomposition of Pb2Cl2C03 at 565—615 K, which also obeyed the parabolic law at 565 K in nitrogen but at higher temperatures obeyed the Jander equation [eqn. (14)]. Values of E and A systematically increased... 

The integrated form is x = Kt -i- C, or x = D Vt, where D is the Diffusion Coefficient. Eiquation 4.5.3. is called the Parabolic Law of Diffusion. If the growth of a phase can be fitted to this equation, then it is likely that the primary reaction mechanism involves simple diffusion. 

Solution of the differential equation obtained by combining Eqs (5.8.20) and (5.8.21) yields the Wagner parabolic law of film growth,... 

There is specificity of the antioxidant action in the presence of heterogeneous catalyst. The kinetics of ionol retarding action on the oxidation of fuel T-6 catalyzed by the copper powder and homogeneous catalyst copper oleate was studied in Ref. [12]. Copper oleate appeared to be very active homogeneous catalyst it was found to catalyze the autoxidation of T-6 in such small concentration as 10 6 mol L-1 (T = 398 K). The kinetics of autoxidation catalyzed by copper salt obeys the parabolic law (see Chapter 4) ... 

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