Concentration interface

As discussed later, the reaction-enhancement factor ( ) will be large for all extremely fast pseudo-first-order reac tions and will be large tor extremely fast second-order irreversible reaction systems in which there is a sufficiently large excess of liquid-phase reagent. When the rate of an extremely fast second-order irreversible reaction system A -t-VB produc ts is limited by the availabihty of the liquid-phase reagent B, then the reac tion-enhancement factor may be estimated by the formula ( ) = 1 -t- B /VCj. In systems for which this formula is applicable, it can be shown that the interface concentration yj will be equal to zero whenever the ratio k yV/k B is less than or equal to unity. 

The relation between CAi[ and CAi2 is determined by the phase equilibrium relationship since the molecular layers on each side of the interface are assumed to be in equilibrium with one another. It may be noted that the ratio of the differences in concentrations is inversely proportional to the ratio of the mass transfer coefficients. If the bulk concentrations, CAt> and CA02 are fixed, the interface concentrations will adjust to values which satisfy equation 10.98. This means that, if the relative value of the coefficients changes, the interface concentrations will change too. In general, if the degree of turbulence of the fluid is increased, the effective film thicknesses will be reduced and the mass transfer coefficients will be correspondingly increased. 

The capacity of the film will be assumed to be small so that the hold-up of solute is negligible. If Henry s law is applicable, the interface concentration in the second (penetration) phase is given by ... 

In some cases, such as the evaporation of a liquid at an approximately constant temperature or the dissolving of a highly soluble gas in a liquid, the interface concentration may be either substantially constant or negligible in comparison with that of the bulk. In such cases, die integral on the left-hand side of equation 10.158 may be evaluated directly to... 

The boundary conditions require knowledge of the interface concentration of hydrogen ChjL to compute E (see below). For hydrogenations, the equilibrium concentration (ChjL= CfJ L) can be used, albeit with the assumption of no mass transfer resistance on the gas side. Otherwise, it must be determined using Eq. (4). The boundary conditions for the substrate S state that it is not transferred to the gas phase - that is, S is not vaporized. This assumption is most often... 

Figure 8.14 Two half-spaces in contact with each other at x = 0 have the concentration C0 for x<0 and 0 for x>0. Interface concentration is constant and equal to C0/2. The curves are labeled for different values of the parameter On.

The rate at which interface concentration builds up or goes down is shown for various a in Figure 8.19. Neglecting density change upon solidification and con-... 

Figure 6.3 Plots of concentration against distance from the electrode solution interface ( concentration profiles ) as a function of time during the chronoamperometry experiment for (a) the concentration of Tl (as reactant) remaining in solution (b) the concentration of Tl + (as product). Movement of the material through the solution is by diffusion, i.e. a convection-free situation.

The rate of mass transfer of a snbstance across a water-gas bonndary is controlled by the diffnsion film model as well. Gas transfer from a water sonrce is faster than from a solid sonrce, and the chemical does not nndergo a chemical reaction during the transfer process. Under these conditions, the interface concentration may be interpreted in terms of the Henry constant (K ), which indicates whether the controlling resistance is in the liqnid or the gas film. When 5, a water film is the controlling factor, while a gas film controls the behavior when K >500. 

For a diffusion couple, the definition of Amid requires some thinking because the mid-concentration of the whole diffusion couple is right at the interface, which does not move with time. This is because for a diffusion couple every side is diffusing to the other side. On the other hand, if a diffusion couple is viewed as two half-space diffusion problems with the interface concentration viewed as the fixed surface concentration, then. Amid equals 0.95387(Df), the same as the half-space diffusion problem. 

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. 

Figure 1-11 Concentration profile for (a) crystal growth controlled by interface reaction (the concentration profile is flat and does not change with time), (b) diffusive crystal growth with t2 = 4fi and = 4t2 (the profile is an error function and propagates according to (c) convective crystal growth (the profile is an exponential function and does not change with time), and (d) crystal growth controlled by both interface reaction and diffusion (both the interface concentration and the length of the profile vary).

For the dissolution of a crystal into a melt, if one wants to predict the interface melt composition (that is, the composition of the melt that is saturated with the crystal), the dissolution rate, and the diffusion profiles of all major components, thermodynamic understanding coupled with the diffusion matrix approach is necessary (Liang, 1999). If the effective binary approach is used, it would be necessary to determine which is the principal equilibrium-determining component (such as MgO during forsterite dissolution in basaltic melt), estimate the concentration of the component at the interface melt, and then calculate the dissolution rate and diffusion profile. To estimate the interface concentration of the principal component from thermod5mamic equilibrium, because the concentration depends somewhat on the concentrations of other components, only... 

The results above have the following applications (i) estimation of diffusive crystal dissolution distance for given crystal and melt compositions, temperature, pressure, and duration if diffusivities are known and surface concentrations can be estimated and (ii) determination of diffusivity (EBDC) and interface-melt concentrations. Those diffusivities and interface concentrations can be applied to estimate crystal dissolution rates in nature. 

Solution The calculated concentration profiles at f= 100 years are plotted in Figure 4-32. The calculation of the interface concentrations is explained in the figure caption. Because diffusion in olivine is much more rapid, the profile is longer in olivine and deviation from the initial concentration is smaller. 

Figure 5-25 (a) Diffusion profile across a diffusion couple for a given cooling history. This profile is an error function even if temperature is variable as long as D is not composition dependent, (b) Diffusion profile across a miscibility gap for a given cooling history. Because the interface concentration changes with time, each half of the profile is not necessarily an error function. 

Next we turn to the inference of cooling history. The length of the concentration profile in each phase is a rough indication of (jDdf) = (Dot), where Do is calculated using Tq estimated from the thermometry calculation. If can be estimated, then x, Xc and cooling rate q may be estimated. However, because the interface concentration varies with time (due to the dependence of the equilibrium constants between the two phases, and a, on temperature), the concentration profile in each phase is not a simple error function, and often may not have an analytical solution. Suppose the surface concentration is a linear function of time, the diffusion profile would be an integrated error function i erfc[x/(4/Ddf) ] (Appendix A3.2.3b). Then the mid-concentration distance would occur at... 

For semi-infinite media, this interface concentration remains constant, but for a particle it changes with time towards. Equations (3-72) and (3-73) are compared with more complete solutions below. 

Equation (CCC) can be solved to obtain the rate, Rt, under certain boundary conditions. Take the case where the concentration in the bulk liquid is given by c, bulk at time / = 0 as well as at x = for times t > 0. It is assumed that there is a thin layer at the surface that contains the dissolved species in equilibrium with the gas immediately adjacent to the surface. This interface concentration is denoted as ci,inicrfacc- Under these conditions, Eq. (CCC) can be solved (see Danckwerts, 1970, pp. 31-33) to obtain the rate of transfer per unit surface area after exposure time t, as... 

The interface concentration is called 6 sat> even though an air-water mixture may not be saturated. 

All these mechanisms along with any reaction in the solid phase are considered to be processes in series (Smith, 1981). In three-phase systems, three interface concentrations, two in the gas-liquid interface CG i and CLi, and one in the liquid-solid interface Cs, have to be eliminated. If equilibrium exists at the bubble-liquid interface, CG>i and CLi are related by Henry s law ... 

In Table 4.4.1 we present some typical estimates for lim and the maximal interface concentration variation Ac for z = 2. 

Ac — the maximal interface concentration variation at the limiting current through an inhomogeneous membrane. 

Additional examples are presented in Figs. 4.4.4 and 4.4.5 where we depict the interface concentration profiles for 8 = 200 and 8 = 20 respectively, for different values of R at the limiting current. 

The immediate question is then how is this compatible with the arguments concerning sensitivity of the system to the value of concentration at the minimum and the expected related positive rectification To answer, we have to examine the detailed time evolution of the minimal electrolyte concentration Cm n(t) (the interface concentration in the electro-neutral picture) during one period. Bear in mind that, since at the plateau of the VC curve practically all of the system s resistance is concentrated at the location where the concentration is at its minimum, the electric current in the system is proportional to Cmjn(f) V(t). In Fig. 5.4.4 we present the calculated time plots of Cm[n = C(t, 1), V(t) during one period for / = 1, A = 10, Vcr= 15. 

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