Profiles, concentration-distance

Let us see now what happens in a similar linear scan voltammetric experiment, but utilizing a stirred solution. Under these conditions, the bulk concentration (C0(b, t)) is maintained at a distance S by the stilling. It is not influenced by the surface electron transfer reaction (as long as the ratio of electrode area to solution volume is small). The slope of the concentration-distance profile [(CQ(b, t) — Co(0, /))/r)] is thus determined solely by the change in the surface concentration (Co(0, /)). Hence, the decrease in Co(0, t) duiing the potential scan (around E°) results in a sharp rise in the current. When a potential more negative than E by 118 mV is reached, Co(0, t) approaches zero, and a limiting current (if) is achieved ... 

The reaction was followed by the local measurement of chloride ions, at a potentio-metric Ag/AgCl microelectrode probe, positioned in the aqueous receptor phase, as DCE droplets containing TPMCl were grown (feeder phase). The reaction was shown unambiguously to occur interfacially, and was first-order in TPMCl with a hydrolysis rate constant of 6.5 x 10 cms. A typical concentration-distance profile determined in these experiments is shown in Fig. 18. 

Figure 28 The biophysical model for passive diffusion and concurrent intracellular metabolism of a drug for a simple A-to-B reaction process. Concentration-distance profiles are depicted in the aqueous boundary layer and intracellular domain for the drug and metabolite. The bottom diagram depicts the direction of the fluxes of drug and metabolite viewed from the donor and receiver sides of the cell monolayer. Details of basic assumptions are found in the text.

Figure 2. Development of elemental concentration-distance profiles as a function of time for a mass-transport-limiting situation, (a) Diffusion to a large (r0 > <5) organism, (b) Diffusion to a small (ro

W. M. Reichert, J. T. Suci, J. T. Ives, and J. D. Andrade, Evanescent detection of adsorbed protein concentration-distance profiles Fit of simple models to variable-angle total internal reflection fluorescence data, Appl. Spectrosc. 41, 503-507 (1987). 

What do these perturbations in concentration represent They are quantitative measures of the accumulation or depletion of species in the interphase region. Unfortunately, only a Gibbs angel could directly provide the concentration-distance profile for the various cationic and anionic species in the double layer. At present, there are no techniques sensitive enough to experimentally determine the distance variation of the concentration changes in the various spedes in solution. One must settle for knowledge obtained by indirect argument and therefore of lesser certainty. 

Figure 2.8 Diffusional relaxation following momentary creation of a vacancy at an inert barrier. Semi-infinite conditions prevail. Density map and concentration-distance profiles are shown, (a) Initial condition (b) vacancy creation (c) vacancy extends farther into bulk of medium (d) relaxation begins (e) relaxation continues (f) relaxation continues (g) initial condition restored.

Figure 3.1 Concentration-distance profiles during diffusion-controlled reduction of O to R at a planar electrode. D0 = DR. (A) Initial conditions prior to potential step. Cq = 1, CR = 0 mM. (B) Profiles for O (dashed line) and R (solid line) at 1,4, and 10 ms after potential step to Es. R is soluble in solution. (C) Profiles for O and R at 1,4, and 10 ms after potential step to Es. R is soluble in the electrode. (D) Potential stepped from Es to Ef at t = 10 ms so that oxidation of R to O is now diffusion-controlled. Profiles are for 11, 15, and 20 ms after step to Es. R is soluble in the electrode.

The potential step Es is generally terminated by switching the potential to some final value Ef at which R is now oxidized back to O. If this final potential is sufficiently positive, the concentration of R at the electrode surface is made to be essentially zero. Consequently, accumulated R now diffuses to the electrode, where it is consumed by oxidation back to O, the original species in solution. This is illustrated in Figure 3. ID. Since the concentration of R is zero both at the surface and in the bulk of the electrode, R diffuses both toward and away from the interface under the influence of both downhill sides of its concentration-distance profile. For this reason, all of the R originally generated is not oxidized back to O unless the potential is maintained for a considerable length of time. 

Understanding the shape of the chronoamperogram requires consideration of concentration-distance profiles for a potential-step excitation in conjunction with Faraday s law. Faraday s law is so fundamental to dynamic electrochemical experiments that it cannot be emphasized too much. It is important to keep in mind that the charge Q passed across the interface is related to the amount of material that has been converted, and the current i is related to the instantaneous rate at which this conversion occurs. Current is physically defined as the rate of charge flow therefore,... 

The product D0 (dCo/dx)x=0 t is the flux or the number of moles of O diffusing per unit time to unit area of the electrode in units of mol/(cm2 s). (The reader should perform a dimensional analysis on the equations to justify the units used.) Since (3Co/3x)x=01 is the slope of the concentration-distance profile for species O at the electrode surface at time t, the expected behavior of the current during the chronoamperometry experiment can be determined from the behavior of the slope of the profiles shown in Figure 3. IB. Examination of the profiles for O at x = 0 reveals a decrease in the slope with time, which means a decrease in current. In fact, the current decays smoothly from an expected value of oo at t = 0 and approaches zero with increasing time as described by the Cottrell equation for a planar electrode,... 

In this case, the electrogenerated R will encounter Z as it diffuses away from the electrode and will react to form P. This diffusional mixing of two reactants (R and Z in this example) is the basis for chronoabsorptometry (and other electrochemical techniques) as a kinetic method. The homogeneous chemical reaction of R will perturb its accumulation by a magnitude that is proportional both to k and to the concentration of Z. The influence of this perturbation is apparent from the concentration-distance profiles for R, which are illustrated for a rate constant of 107 L/(mol s) in Figure 3.9. 

Figure 3.9 Concentration-distance profiles for EC mechanism with k = 107 L/ (mol s). See Figure 3.1 legend for conditions. [Adapted from R. F. Broman, W. R. Heineman, and T. Kuwana, Faraday Discuss. Chem. Soc. 56 16 (1974).]...

Figure 3.18 Stationary-electrode voltammogram for 1 mM O in supporting electrolyte. Electrode reaction is O + e R Eqr = 0 V vs. SCE. Dashed line, SEV for supporting electrolyte without O. Scan rate = -0.2 V/s. Concentration-distance profiles during the potential scan are shown to the left in each block. (A) Start, (B) 3.0 s, (C) 3.15 s, (D) 5.0 s.

The physical situation in the solution adjacent to the electrode during the potential scan is illustrated by the concentration-distance profiles included in Figure 3.18 for selected potentials [38]. The C-x profiles in Figure 3.18A are for O and R when Ej is imposed. Note that the application of Ej does not measurably alter the concentration of O at the electrode surface as compared to the solution bulk. As the potential is scanned positively, the concentration of O at the electrode surface decreases in order to establish a Cr/Cq ratio that satisfies the Nernst equation for the applied potential at any particular instant. This is illustrated by profiles in B-D. Note that the profile in B (for which the concentration of O at the electrode surface equals the concentration of R) corresponds to an Eappljed that is the formal electrode potential (vs. SCE) of the couple. 

Figure 3.19 Voltammograms and concentration-distance profiles for (a) fast, and (b) slow scan rate. Simulation by DigiSim. ...

Figure 3.22 Cyclic voltammogram of 1 mM O in supporting electrolyte. Scan initiated at 0.6 V vs. SCE in negative direction at 200 mV s1. Concentration-distance profiles a-h keyed to voltammogram. Eq R = 0 V vs. SCE. Simulation by DigiSim. ...

Figure 3.27 Polarogram of 1.0 mM Pb2+, 1.0 mM Cd2+, and 0.1 M KC1. 02 removed. Concentration-distance profiles during potential scan shown below polarogram.

Figure 3.37 illustrates the Nernst diffusion layer in terms of concentration-distance profiles for a solution containing species O. As pointed out previously, the concentration of redox species in equilibrium at the electrode-solution interface is determined by the Nernst equation. Figure 3.37A illustrates the concentration-distance profile for O under the condition that its surface concentration has not been perturbed. Either the cell is at open circuit, or a potential has been applied that is sufficiently positive of Eq R not to alter measurably the surface concentrations of the 0,R couple. 

Figure 3.37 Concentration-distance profiles for voltammetry in stirred solution. Do = Dr.

Stirred-solution voltammetry utilizes current-voltage relationships that are obtained at a stationary electrode immersed in a stirred solution. In order to understand this aspect of electrochemistry, it is extremely useful to consider a typical current-voltage curve (voltammogram) in terms of the concept of concentration-distance profiles presented in the preceding section. The discussion will consider the potential, rather than the current, as the controlled variable. 

The current resulting from an applied potential is determined by the slope of the concentration-distance profile of the reactant at the electrode surface as... 

Figure 3.38 illustrates a typical current-voltage curve that would be obtained for a solution containing equal concentrations of O and R. The shape of this curve can be understood by considering the slopes of the concentration-distance profiles that are depicted for several representative potentials. During oxidation the current is determined by the slope of the profile for R, whereas the profile... 

Figure 3.38 Voltammogram with representative concentration-distance profiles for a solution containing equal concentrations of 0 (solid line) and R (dotted line). D0 = DR.

As was the case for techniques based on potential excitation, current-excitation methods are best understood by studying the time-dependent concentration changes in solution caused by the excitation signal applied to the electrode. Concentration-distance profiles for the case of species O being reduced to R by a current-step excitation signal (application of constant current to the cell) are shown in Figure 4.2. Consider first the profiles in Figure 4.2A for the reactant, O. An important concept is the relationship between the applied current and the slope of the profile at the electrode surface as expressed by... 

If D0 = Dr, the concentration-distance profiles of O and R will have slopes of the same magnitude but opposite sign as those shown in Figure 4.2B. At i, all of the reactant O at the electrode surface is converted to product R. 

The effect of reversing the polarity of the current excitation signal on the concentration-distance profiles is shown in Figure 4.2C. The original reactant O is regenerated at a rate now determined by the flux of R back to the electrode. When the surface concentration of R becomes zero, a reverse transition time ir occurs. At this time, another species must undergo oxidation to enable the cell to accept the applied current. 

The concentration-distance profiles predicted by Equation 1.48 and resulting in the current in Equationl.49 are given in Fig. 1.13. A complete expression for the current that results from semi-infinite diffusion is obtained as follows, which in fact is also called the Cotrell equation for a planar electrode ... 

Figure 5 Concentration-distance profiles for reactants generated in double potential step experiments involving initial oxidation of reagent A followed by reduction of A. (From Ref. 36.)...

In systems where ECL arises upon application of a single potential and reaction of a coreactant, as outlined in Eqs. (5) through (9), concentration distance profiles differ and depend on many more factors. In cases where a positive potential is applied and both the ECL chromophore and coreactant are oxidized at the electrode surface, concentrations of the two initial radical ion species will decay with increasing distance from the electrode, as will the concentration of the strong reducing agent formed upon decomposition of the coreactant. The zone where luminescence arises depends on relative rate constants for... 

In general, the methods are difficult to interpret quantitatively in terms of aerosol properties because of ambiguities in the size distribution-concentration-distance profiles and variations in chemical properties contributing to the index of refraction. Nevertheless, remote sensing continues to be important for the surveillance of aerosol behavior in planetary atmospheres. 

Since a permeability coefficient of 2 x 10 cm/sec was used in our theoretical calculation for case B, this is equivalent to a single porosity value for all calculations with Equation 6. For any given set of ethanol concentration at the boundary, the appropriate ethanol concentration distance profile was calculated from the experimentally determined profile for the pure ethanol/water system and taken as that of the system. For any given set of ethanol concentrations at the boundaries the appropriate portion of the experimentally determined ethanol concentration distance profile, was calculated from the experimentally determined profile for the pure ethanol/water system. For any given ethanol concentration distance profile, the corresponding solubility profile of 3-estradiol was obtained by interpolation of experimental solubility data (Figure 2). 

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