Parabolic reaction law

The dissolution distance is proportional to the square root of time (parabolic reaction law), and the dissolution rate is inversely proportional to the square root... 

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 ... 

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... 

If the growth rate is controlled by both interface reaction and diffusion (Figure 1-1 Id), then (i) the concentration profile is not flat, (ii) the interface concentration changes with time toward the saturation concentration, (iii) the diffusion profile propagates into the melt, and (iv) the growth rate is not constant, nor does it obey the parabolic growth law. 

The right hand side is the result of integration. As long as local equilibrium prevails, the average value, LA, of the transport coefficient, taken across the reaction layer, is determined by the thermodynamic parameters at the interfaces A/AB and AB/B, and thus is independent of the reaction layer thickness A . If one inserts Eqn. (1.27) into Eqn. (1.26), a parabolic rate law is found... 

The increase A will occur at interface A/AB if LA/LR< 1, and it will occur at AB/B if La >Lr (Fig. 1-5). We conclude that parabolic rate laws in heterogeneous solid state reactions are the result of two conditions, the prevalence of a linear geometry and of local equilibrium which includes the phase boundaries. 

For reactions of the type A + B = AB (or a+P = y), the situation is different. If one has a linear reaction geometry and the y product forms at different times and locations on the A/B interface, the patches of y eventually merge by fast lateral (interface) transport. Eventually, a full y layer is formed between a and / . At first, this layer has a non-uniform thickness (Fig. 6-4). In Chapter 11 we will show, however, that the uneven a/y and y/p interfaces are morphologically stable and become smooth during further growth. This leads to constant boundary conditions for the y formation after some time of reaction and eventually results in a parabolic rate law, as will be discussed later. 

These assumptions, however, oversimplify the problem. The parent (A,B)0 phase between the surface and the reaction front coexists with the precipitated (A, B)304 particles. These particles are thus located within the oxygen potential gradient. They vary in composition as a function of ( ) since they coexist with (A,B)0 (AT0<1 see Fig. 9-3). In the Af region, the point defect thermodynamics therefore become very complex [F. Schneider, H. Schmalzried (1990)]. Furthermore, Dv is not constant since it is the chemical diffusion coefficient and as such it contains the thermodynamic factor /v = (0/iV/01ncv). In most cases, one cannot quantify these considerations because the point defect thermodynamics are not available. A parabolic rate law for the internal oxidation processes of oxide solid solutions is expected, however, if the boundary conditions at the surface (reaction front ( F) become time-independent. This expectation is often verified by experimental observations [K. Ostyn, et al. (1984) H. Schmalzried, M. Backhaus-Ricoult (1993)]. 

Since this driving force is proportional to A "1, it again leads to a parabolic rate law. The AB formation rate is always decreased compared to a stress-free reaction as long as the layer adheres and does not form cracks. However, if the evolving stress energy contained in the A substrate is also taken into account, the overall stress energy depends on the thickness of the reaction layer, which invalidates the parabolic growth and slows down the reaction rate. In principle, this can stop the reaction before A or B are consumed. 

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... 

Depending on the relative rates of diffusion of these species, the reaction can take place at either the AO/AB2O4 or the B2O3/AB2O4 interface. When diffusion is slower than the rate of reaction, the thickness of the product layer follows a parabolic growth law like that observed in Figure 5.20 for NLA1204. 

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... 

Fig. 2.21. Normalized conductance of Zr/Co bilayer diffusion couples vs. square roots of time for three different temperatures. Note the deviations at short reaction times from a parabolic time law [2.84]...

Alkali oxides dissolved in molten alkali metals are able to react with solid metals in several ways. The simple exchange of oxygen between the liquid and solid metals sometimes oxidizes the solid metal under formation of a stable surface oxide, which has a protective character and reduces the reaction rate. The kinetics of this type of reaction follow a parabolic rate law. Diffusion of oxygen or metal ions through the slowly growing oxide layer is the rate determining step. An example of this type of... 

In both of these equations, x is the film thickness, t is the time of the oxidation, and k and are experimentally determined constants. The constant fep is called the parabolic rate constant. A linear rate is usually found when the film is porous or cracked. The parabolic equation is found when the film forms a coherent, impenetrable layer. As the rate of film growth, dx/dt, diminishes with time for the parabolic rate law, this equation is associated with protective kinetics. The parabolic rate law arises when the reaction is controlled by diffusion. The species with the lowest diffusion coefficient plays the most important role in this case. 

Many solid-state reactions give a parabolic rate law for the growth of an internal phase, and such a parabolic rate law is taken as evidence that the reaction is diffusion-controlled. The units of k are the same as the units of the diffusion coefficient, m s Generally, one can write ... 

In the oxidation, sulfidation, and so on of metals and alloys, solid reaction products are growing as film, scale, crystals, or in other morphologies, on the metal phase. A frequent case is the formation of a dense scale, separating the metal and gas phase. In this case, generally, a parabolic rate law is observed for the increase of film thickness x ... 

The parabolic rate law is based on the assumption that cation or anion diffusion through the oxide layer is the rate-Hmiting step. In this case, Fick s law of diffusion serves as the starting point for any reaction rate. According to Fick s law, the flux of the diffusing... 

---