Diffusion-limited condition

Takahashi et al. [220] first reported the formation of Bi-Te alloy films with varying chemical composition by means of cathodic electrodeposition from aqueous nitric acid solutions (pH 1.0-0.7) containing Bi(N03)3 and Te02. The electrodeposition took place on Ti sheets at room temperature under diffusion-limited conditions for both components. In a subsequent work [221], it was noted that the use of the Bi-EDTA complex in the electrolyte would improve the results, since Bi " is easily converted into the hydrolysis product, Bi(OH)3, a hydrous polymer, thus impairing the reproducibility of electrodeposition. The as-produced films were found to consist of mixtures of Te and several Bi-Te alloy compounds, such as Bi2Tc3, Bi2+xTe3 x, Bi Tee, and BiTe. Preparation of both n- and p-type Bi2Te3 was reported in this and related works [222]. 

The deposition takes place from HTeOs and cadmium-EDTA complex solutions at a potential whereat, whilst Te is deposited from HTeOs under a diffusion-limited condition, the Cd-EDTA complex ion is not reduced to metallic Cd. The first step is the dark deposition of one monolayer of elemental Te on the p-Si substrate (Fig. 4.11a, i). After completion of this step, as specified by measuring the charge passed, the electrode is illuminated by light with energy higher than the band gap energy of silicon for a limited time. Then conduction band electrons are... 

Another example is dendritic crystal growth under diffusion-limited conditions accompanied by potential or current oscillations. Wang et al. reported that electrodeposition of Cu and Zn in ultra-thin electrolyte showed electrochemical oscillation, giving beautiful nanostmctured filaments of the deposits [27,28]. Saliba et al. found a potential oscillation in the electrodeposition of Au at a liquid/air interface, in which the Au electrodeposition proceeds specifically along the liquid/air interface, producing thin films with concentric-circle patterns at the interface [29, 30]. Although only two-dimensional ordered structures are formed in these examples because of the quasi-two-dimensional field for electrodeposition, very recently, we found that... 

Under steady-state conditions, the internalisation flux equals the rate of supply by diffusive transport and chemical reactions. As was shown earlier (cf. equations (12) and (13)), the maximum flux (rate) of solute internalisation by a microscopic cell under diffusion-limited conditions can be given by ... 

In the case that the chemical reaction proceeds much faster than the diffusion of educts to the surface and into the pore system a starvation with regard to the mass transport of the educt is the result, diffusion through the surface layer and the pore system then become the rate limiting steps for the catalytic conversion. They generally lead to a different result in the activity compared to the catalytic materials measured under non-diffusion-limited conditions. Before solutions for overcoming this phenomenon are presented, two more additional terms shall be introduced the Thiele modulus and the effectiveness factor. 

Channel techniques employ rectangular ducts through which the electrolyte flows. The electrode is embedded into the wall [33]. Under suitable geometrical conditions [2] a parabolic velocity profile develops. Potential-controlled steady state (diffusion limiting conditions) and transient experiments are possible [34]. Similar to the Levich equation at the RDE, the diffusion limiting current is... 

A zero-order reaction thus becomes a half-order reaction, a first-order reaction remains first-order, whereas a second-order reaction would have an apparent order 3/2 for diffusion-limited conditions. 

A dislocation source that climbs rapidly enough so that ideal diffusion-limited conditions are achieved therefore operates with an efficiency of unity. On the other hand, slowly acting sources can have efficiencies approaching zero. Applications of these concepts to the source action of interfaces are discussed in Section 13.4.2. 

Under diffusion-limited conditions, the vacancies can be assumed to be maintained at equilibrium at the dislocations. The dislocations act as ideal sources (Section 11.4.1) and, therefore, at the dislocations = 0. When an atom is inserted at a dislocation of type 2 acting as a sink (Fig. 13.3), the dislocation will move forward along x by the distance y/2 Ll/b. The force on it acting in that direction is axyb/y/2, and the work performed by the stress is therefore ( /2 Q/b)(oxyb/ /2) = axy 2. The boundary value for the diffusion potential >A at the cores of these dislocations is, therefore,... 

Four basic situations can occur both 0i s small, both large, and two cases of one large and the other small. When 0X and 02 are both small, c = cf and c/3t a c 7) and the /3-phase layer should grow under diffusion-limited conditions. Putting these conditions into Eqs. 20.26 and 20.28 and summing them,... 

Spheres. Consider a 5-rich sphere of /3 phase of radius R = R(t) growing in an infinite a matrix under diffusion-limited conditions as shown in Fig. 20.6. This problem can be solved by using the scaling method with r) defined by rj = r/ ADat)1/2. The diffusion equation in the a phase in spherical coordinates in rt-space (see Eq. 5.14) becomes, after transformation into 77-space,... 

The growth of spherical precipitates under diffusion-limited conditions has been observed in a number of systems, such as Co-rich particles growing in Cu supersaturated with Co (see Chapter 23). In these systems, the particles are coherent with the matrix crystal and the interfaces possess high densities of coherency dislocations, which are essentially steps with small Burgers vectors, The interfaces therefore possess a high density of sites where atoms can be exchanged and the particles operate as highly efficient sources and sinks. 

Solution. Yes. When D varies with concentration we have shown in Section 4.2.2 that the diffusion equation can be scaled (transformed) from zt-space to 77-space by using the variable rj = x/ /4Di (see Eq. 4.19). Also, under diffusion-limited conditions where fixed boundary conditions apply at the interfaces, the boundary conditions can also be transformed to 77-space, as we have also seen. Therefore, when D varies with concentration, the entire layer-growth boundary-value problem can be transformed into 77-space. Since the fixed boundary conditions at the interfaces require constant values of 77 at the interfaces, they will move parabolically. 

Thus (L / Dma)1/2 is the appropriate factor used to convert an ambiguous material flux to a meaningful current term. In the simulations reported next, the dimensionless current parameter is given by Z(K). If diffusion-limited conditions are obtained, the relative concentration of electroactive species in the first element is zero. Thus the form most frequently used is... 

It is now possible to write a computer program that will simulate the currenttime behavior that occurs following a single step to diffusion-limiting conditions. The solution to this problem (the Cottrell equation) is well known and there is no need to perform this simulation to obtain any new electrochemical information. However, it is instructive to see how the principles developed earlier may be used to write a program and how the results of that simulation may be verified by comparing them to known results. Any additional programming will be left as an exercise to the reader. 

A different set of boundary conditions is obtained during the second halfcycle if the potential of the electrode is stepped to a value such that both B and A are converted to a third oxidation state (D) of the initial substance under diffusion-limited conditions. (A situation like this is encountered in the study of electrogenerated chemiluminescence when an aromatic hydrocarbon is first reduced to its anion radical, and then both the parent and the anion radical are oxidized to the cation radical at the same electrode.) In this case the electrode boundary conditions are... 

In the event that the electrode potential is stepped to a level that does not bring about diffusion-limiting conditions, the relative concentration of A and B at the electrode surface must be calculated. If the reaction... 

No special treatment like Equation 20.54 is necessary in the calculation of the surface condition, even if D is assumed electroactive, if diffusion-limited conditions are obtained since fA(l) = 0 under these circumstances, no conversion of B to D is possible in the first volume element. 

The potential of the Pt ring electrode most suitable for detection of X2 was determined as the potential where reduction of X2 occurs under diffusion-limited conditions but oxygen and Zn2+ were not reduced at all. The potentials thus determined were 0 V for I , 0.05 V for Br2, and 0.2 V for Cl2 when 0.5 M K2S04 was used as the base electrolyte. In cases where the base electrolyte was more acidic than 0.5 M K SO., these potentials were a little changed depending on the pH value of electrolytes. 

This section deals with the solution corresponding to an EC mechanism (see reaction scheme 4.IVc) in Reverse Pulse Voltammetry technique under conditions of kinetic steady state (i.e., the perturbation of the chemical equilibrium is independent of time see Sect. 3.4.3). In this technique, the product is electrogenerated under diffusion-limited conditions in the first period (0 < t < ) and then exam-... 

The influence of the rate constants (k + 2) of an EC mechanism with an equilibrium constant K = 1 /Keq = 0.1 is shown in Fig. 4.27. As expected, when the second potential is set, like the first one, under diffusion-limited conditions for... 

The activities of the various catalysts were also compared using in situ IR under diffusion limiting conditions,21 by following the intensity of the carbonyl stretch-vibration of octan-2-one (1718 cm 1) in time (figure 5). Under the diffusion limiting conditions, the same trends as above were observed. PIPO was the most active catalyst and all the other heterogeneous systems had comparable activities (table 3). 

One of the most significant points that we must consider in scientific studies, not limited to studies on photocatalysis, is distinction between evidence and consistency, as least as far as the author thinks. In other words, it is necessary to recognize every fact to be a necessary condition but not a sufficient condition in a strict scientific sense. For example, the fact that a reaction rate obeys the first-order rate law giving a linear relation in a plot of data as in Fig. 6 is only a necessary condition for a monomolecular reaction in homogeneous phase and also a necessary condition for heterogeneous photocatalytic reaction in diffusion-limited conditions or that in surface-reaction limited conditions with a Henry-type adsorption or a Langmuir-type adsorption in the lower-concentration region. 

For irreversible n order reactions and assuming a constant D, the Thiele modulus at strongly diffusion limited conditions, Eqn. (8.75) becomes ... 

The nature of intermediates formed in diffusion flames is similar to the premixed ones, albeit differences in the contacting pattern. In Fig. 11, the species concentration profiles in a laminar ethylene diffusion flame front are presented. The fuel and oxygen diffuse toward each other undergoing virtual annihilation within the flame zone concomitant with the establishment of a peak temperature of about 1600°C. Because premixed systems provide a better control of combustor temperature, and many practical combustion devices operate under diffusion limited conditions, considerable effort has been expended to ensure the rapid mixing of fuel and oxygen in combustion chambers and approach premixed conditions. 

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