Diffusion, parabolic law

The parabolic diffusion law or equation can be used to determine whether diffusion-controlled phenomena are rate-limiting. This equation... 

The parabolic diffusion law (Chapter 2) has successfully described rate data for 2,4,5,-T and parathion (Weber and Gould, 1966) as well as other... 

A number of workers (Wollast, 1967 Huang and Kiang, 1972 Luce et al., 1972) have observed that feldspar weathering conforms to the parabolic diffusion law (Chapter 2). An example of this is shown in Fig. 7.2. Much research effort has gone into explaining why parabolic kinetics could be operational for feldspar dissolution. Several explanations have been given, and these are discussed below. 

Oxidation - Oxidation of a ceramic Is taken to be a surface effect governed by the parabolic diffusion law and a thermal activation process of constant activation energy, with an Arrhenius type of temperature dependence. By the assuming the Griffith flaw size to be proportional to the surface oxide thickness, we can structure the... 

Fig. 26. Sorption velocities. NHg on natrolite, showing tendency to autocatalytic sorption rate curves. (NH3 sorbed before commencing expt. = 12 11 c.c. at N.T.P. Inset shows NH3 on heulandite, sorption rate following the parabolic diffusion law.)...

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 ... 

Dilute binary alloys of nickel with elements such as aluminium, beryllium and manganese which form more stable sulphides than does nickel, are more resistant to attack by sulphur than nickel itself. Pfeiffer measured the rate of attack in sulphur vapour (13 Pa) at 620°C. Values around 0- 15gm s were reported for Ni and Ni-0-5Fe, compared with about 0-07-0-1 gm s for dilute alloys with 0-05% Be, 0-5% Al or 1-5% Mn. In such alloys a parabolic rate law is obeyed the rate-determining factor is most probably the diffusion of nickel ions, which is impeded by the formation of very thin surface layers of the more stable sulphides of the solute elements. Iron additions have little effect on the resistance to attack of nickel as both metals have similar affinities for sulphur. Alloying with other elements, of which silver is an example, produced decreased resistance to sulphur attack. In the case of dilute chromium additions Mrowec reported that at low levels (<2%) rates of attack were increased, whereas at a level of 4% a reduction in the parabolic rate constant was observed. The increased rates were attributed to Wagner doping effects, while the reduction was believed to result from the... 

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. 

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... 

We have reviewed today s knowledge of the mechanisms for growth of electrolyte crystals from aqueous solution Convection, diffusion, and adsorption ( ) mechanisms leading to linear rate laws, as well as the surface spiral mechanism (parabolic rate law) and surface nucleation (exponential rate law). All of these mechanisms may be of geochemical importance in different situations. 

It should be emphasized that n.. and JPS, and therefore c and T, refer to the condition at the pore tip. The dissolution valence and the temperature can be assumed to be independent of pore depth. This is not the case for the HF concentration c. Because convection is negligible in macropores, the mass transport in the pore occurs only by diffusion. A linear decrease in HF concentration with depth and a parabolic growth law for the pores according to Pick s first law is therefore expected, as shown in Fig. 9.18 a. The concentration at the pore tip can be calculated from the concentration in the bulk of the electrolyte c, the pore length l, the diffusion coefficient DHf (Section 1.4) and the flow of HF molecules FHf. which is proportional to the current density at the pore tip ... 

If the rate is controlled by diffusive mass transfer (Figure 1-1 lb) and if other conditions are kept constant, then (i) the growth (or dissolution) distance is proportional to the square root of time, referred to as the parabolic growth law (an application of the famous square root law for diffusion), (ii) the concentration in the melt is not uniform, (iii) the concentration profile propagates into the melt according to square root of time, and (iv) the interface concentration is near saturation. For the rate to be controlled by diffusion in the fluid, it cannot be stirred. 

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... 

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)]. 

Differential Rate Laws 5 Mechanistic Rate Laws 6 Apparent Rate Laws 11 Transport with Apparent Rate Law 11 Transport with Mechanistic Rate Laws 12 Equations to Describe Kinetics of Reactions on Soil Constituents 12 Introduction 12 First-Order Reactions 12 Other Reaction-Order Equations 17 Two-Constant Rate Equation 21 Elovich Equation 22 Parabolic Diffusion Equation 26 Power-Function Equation 28 Comparison of Kinetic Equations 28 Temperature Effects on Rates of Reaction 31 Arrhenius and van t Hoff Equations 31 Specific Studies 32 Transition-State Theory 33 Theory 33... 

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... 

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. 

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