Diffusion layer thickness calculations

Natural convection depends strongly on cell geometry. No convection can arise in capillaries or in the thin liquid layers found in narrow gaps between electrodes. The rates of natural convective flows and the associated diffusion-layer thicknesses depend on numerous factors and cannot be calculated in a general form. Very rough estimates show that the diffusion-layer thickness under a variety of conditions may be between 100 and 500 pm. 

Early investigators assumed that this so-called diffusion layer was stagnant (Nernst-Whitman model), and that the concentration profile of the reacting ion was linear, with the film thickness <5N chosen to give the actual concentration gradient at the electrode. In reality, however, the thin diffusion layer is not stagnant, and the fictitious t5N is always smaller than the real mass-transfer boundary-layer thickness (Fig. 2). However, since the actual concentration profile tapers off gradually to the bulk value of the concentration, the well-defined Nernst diffusion layer thickness has retained a certain convenience in practical calculations. 

The thin-layer thickness calculated in this manner may well have some serious errors associated with it due to sample purity, errors in the weighing out of the solution, the diffusion of species near the thin layer into it within the timescale of the experiment, etc. 

Example 6.2. Calculate the diffusion limiting current density for the deposition of a metal ion at a cathode in a quiescent (unstirred) solution assuming a diffusion layer thickness 8 of 0.05 cm. The concentration of ions in the bulk (cj,) is 10 moEL (10 moEcm ), the same as in Example 6.1. The diffusion coefficient D of in the unstirred solution is 2 X lO cm /s. Using Eq. (6.83), we calculate that the limiting diffusion current density for this case is... 

Determine the diffusion layer thickness S as a function of time t for constant-current deposition of a metal ion at a cathode in a quiescent (unstirred) solution at 25°C. Use time t values of 20, 40, 80, 100, 200, and 300 s. Plot the function 8 = f(t) using the 8 values calculated. 

At the other boundary, bulk concentration of A must be maintained at some finite distance from the electrode, while the concentration of B will be zero at the same point. This distance may be regarded as the diffusion layer thickness. In terms of the simulation, the establishment of the semi-infinite boundary condition requires the determination of the number of volume elements making up the diffusion layer. This will be a function of the number of time iterations that have taken place up to that point in the simulation. At any time in the physical experiment, the diffusion layer thickness is given by 6(Dt)1/2. This rule of thumb may be combined with Equation 20.7 to calculate Jd, the number of volume elements in the diffusion layer ... 

The best approach is normally an in situ determination based on voltammetry or charging curves, usually within the hydrogen adsorption region [96]. It is of course necessary to know the actual value of 0H for absolute determinations, but the method is practicable on a relative basis. The method becomes absolute only in a few cases, in particular for Pt electrodes [97] for which the catalytic activity per metal atom, which is the parameter really needed to evaluate electrocatalytic effects, can be calculated [98]. Sometimes, results are reported relative to the surface area measured on the basis of the limiting current for a redox reaction [99], but what is obtained is only the macroscopic surface in which asperities of a height higher than the diffusion layer thickness can only be accounted for. 

The concentration profiles are very sensitive to the kinetics of the electrode reaction. In this context, the determination of the diffusion layer thickness is of great importance in the study of non-reversible charge transfer processes. This magnitude can be defined as the thickness of the region adjacent to the electrode surface where the concentration of electro-active species differs from its bulk value, and it can be accurately calculated from the concentration profiles. In the previous chapter, the extensively used concept of Nemst diffusion layer (8), defined as the distance at which the linear concentration profile (obtained from the straight line tangent to the concentration profile curve at the electrode surface) takes its bulk value, has been explained. In this chapter, we will refer to it as linear diffusion layer since the term Nemst can be misunderstood when non-reversible processes... 

The variation of 7fphe with the potential for different values of the electrode radius obtained in CV with spherical electrodes, including the limiting case of a planar electrode (rs —> oo),canbeseeninFig.5.10a.Fromthesecurves, whichhave been calculated using Eq. (5.71) for a sweep rate v = 0.1 V s, it can be observed that the diffusion layer thickness decreases with the electrode radius and the constant value <5 = rs is reached when rs < 5 pm (i.e., a truly stationary I-E response is obtained as can be seen in Fig. 5.10b). 

Fig. 5.10 Nemst diffusion layer thickness 5 le obtained in LSV (a) and Cyclic Voltammograms (b) corresponding to a spherical electrode. These curves have been calculated from Eq. (5.71)-(5.72) and (5.23) for A = 10 5mV and v = lOOrnVs-1. The values of the electrode radii appear on the curves. Reproduced with permission [29]...

A drug powder has uniform particle sizes of 500 pm diameter and 75 mg weight. If the solubility of the drug in water = 3 mg/mL, the density of the drug = 1.0 g/mL, and the diffusion coefficient of the drug in water = 5 x 10 6 cm2/sec, calculate the diffusion layer thickness when it takes 0.3 hours for the complete dissolution. 

Quantitative measurements of electrokinetic phenomena permit the calculation of the zeta potential by use of the appropriate equations. However, in the deduction of the equations approximations are made this is because in the interfacial region physical properties such as concentration, viscosity, conductivity, and dielectric constant differ from their values in bulk solution, which is not taken into account. Corrections to compensate for these approximations have been introduced, as well as consideration of non-spherical particles and particles of dimensions comparable to the diffuse layer thickness. This should be consulted in the specialized literature. 

Before we can consider the response of our model organism to lead in sea water we must define k (equation 14) and b (equation 15) and estimate the range of diffusion layer thicknesses ( S ) that are characteristic of phytoplankton cells. Since we have no direct experimental evidence on which to base an estimate of k we will treat it as a variable in the calculations. 

For a particular diffusion layer thickness S the thermodynamic availability, as measured with an ISE or calculated from a speciation model, and the electrochemical availability, as measured by ASV, represent limiting cases of a continuum of trace metal availability. The nature of this continuum is most simply defined by considering the flux of the free metal ion across the diffusion layer to a surface which senses the metal availability. The ratio of the observed flux (J) to the limiting flux (J] ) is unity for ASV measurements under current limiting conditions and zero for ISE measurements. 

Equation 9D is strictly applicable only in unstirred solutions, but for short transients the situation is better. As long as the value of 8, as calculated from Eq. 9D for a purely diffusion-controlled process, is small compared to the diffusion layer thickness set up by stirring,... 

Another study carried out by these authors [93] modeled the collapsing motion of a single bubble near an electrode surface, and equations for the motion of a spherical gas bubble were obtained. The jet speed and water hammer pressure during jet flow (liquid jet) were calculated, and when the jet speed was 120 m/s, the water hammer pressure was approximately 200 MPa upon the electrode surface. This pressure played an important part in the fineness of the crystal deposits. Mass transfer during the electrode reaction was by turbulent diffusion. The diffusion layer thickness was reduced to approximately 1/10th its size in the presence of the ultrasonic field. The baths contained the ions Cl-, SO -, and Zn2+. The ultrasonic frequency employed in the experiments was 40 kHz and it was seen that ultrasound considerably increased the deposition rate and current efficiency, as well as the smoothness and hardness of the deposit. Microscopy studies showed that the... 

Figure 2. Calculated pH at cathode vs. electrolyte HCO3 concentration with the diffusion layer thickness 5 = 0.01 cm and 0.001 cm. Two different current densities are assumed. The pH of the bulk solution is also indicated. Reproduced from Ref. 21, Copyright (2005) with kind permission of Springer Science and Business Media.

The determination of the real surface area of the electrocatalysts is an important factor for the calculation of the important parameters in the electrochemical reactors. It has been noticed that the real surface area determined by the electrochemical methods depends on the method used and on the experimental conditions. The STM and similar techniques are quite expensive for this single purpose. It is possible to determine the real surface area by means of different electrochemical methods in the aqueous and non-aqueous solutions in the presence of a non-adsorbing electrolyte. The values of the roughness factor using the methods based on the Gouy-Chapman theory are dependent on the diffuse layer thickness via the electrolyte concentration or the solvent dielectric constant. In general, the methods for the determination of the real area are based on either the mass transfer processes under diffusion control, or the adsorption processes at the surface or the measurements of the differential capacitance in the double layer region [56],... 

E3.10. Calculate the limiting current density for oxygen reduction in an alkahne solution if the oxygen concentration is 0.4 mol/m, the diffusion coefficient, D02, is equal to 5.0 x 10 vc /s, and the diffusion layer thickness is 195.3pm. 

Fig. 2 Plot of the average diffusion layer thickness versus ultrasound intensity calculated from the limiting currents observed for the reduction of cobalticenium cations in acetonitrile at various ultrasound intensities and in cells A and B .

In process engineering this parameter is called the mass transfer constant. It Is denoted by some authors by (a notation which has the advantage of underlining the parallel shown with the reaction rate constants denoted by k). To be precise, this quantity is not based on the diffusion layer thickness as defined in this document, but rather the value calculated from the interfaclal slope of the concentration profile (see section 4.3.1.4). For example. In the case of an experiment involving forced convection, one should use the thickness of the Nernst layer In order to define the mass transport rate constant. 

The thicknesses of the diffusion layers were calculated from the values and independently measured k values they are also shown in Table 5.1. These data, by order of magnimdes, correspond to the above estimations. 

Research into this area is dominated by microelectrodes. At short times, the diffusion layer thickness is much smaller than the microelectrode radius and the dominant mass transport mechanism is planar diffusion. Under these conditions, the classical theories, e.g., that of Nicholson and Shain, can be used to extract kinetic parameters from the scan rate dependence of the separation between the anodic and cathodic peak potentials. Using this approach, the standard heterogeneous electron transfer rate constant, k°, may be determined from the published working curves relating AEp to a kinetic parameter The variation of AEp with o is determined and, from this, T is calculated. k° is then determined by the following equation ... 

In Eqn (2.82), Cm is normally treated as the solubility of the oxidant in the ionomer membrane. The diffusion layer thickness 5o can be obtained using a rotating disk electrode technique, which will be given in a very detailed discussion in Chapter 5. The equivalent thickness of the ionomer membrane can be calculated according to the amount of ionomer applied in the electrode layer using the following equation ... 

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