The Scheme of Squares

Although the values given above are only hypothetical, a large number of organic and inorganic electrochemical processes are found to foUow a square scheme. 

At pH values above three but below 11, the overall reaction will be the one-proton one-electron reduction of species A to AH. Consequently, the voltammetric signal will be found to shift in potential with approximately 59 mV per pH (at 25°C), irrespective of whether electron transfer or proton transfer occurs first. 

Above pH 11, the reaction corresponds to the reduction of species A to A without proton transfer. Accordingly the peak position will not vary with pH. This result is shown within Fig. 4.8. 

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Of course, proton transfer can also occur between two reactants in the solution. As such, it is not an electrochemical reaction, unless it is combined with an electron exchange with the electrode. Such a combined electron-proton transfer can be represented by the scheme of squares shown in Fig. 2.8. Both electron and proton transfer... 

Studies in the area of electrochemical molecular recognition deal with bifunctional receptor molecules that contain not only binding sites but also one or more redox-active centres whose electron transfer reaction is coupled to the receptor s complexation. Such systems can be described by the scheme of squares as shown in Scheme 1. 

We have applied this technique to the study of the proton flux that takes place when a modified electrode, the thionine-coated electrode, is either oxidised or reduced. We were particularly interested in the question as to whether the proton and electron fluxes were in time with one another or not. Typical results for proton and electron fluxes for reduction and oxidation at a number of different values of pH are displayed in Fig. 7. At first sight, we were bewildered by the variety of behaviour. However, we can explain the different transients as follows. In Table 2, we set out the scheme of squares [18, 19] for the thionine/leucothionine system with a number of vital pKk values. Starting at pH 4 in the oxidation direction (LH + - Th+ + 2e + 3H+), we see that the proton flux is indeed larger than the electron flux and that both fluxes are in time with each other. In the opposite reduction direction, the electron flux is similar but the proton flux is smaller and delayed. The reason for this is that, to start with, protons are used up and the pH crosses the pKa at 5.5 (Th+ + 3H+ + 2e - LH +). However, for pH > 5.5, the reaction can utilise the H+ stored in the coat (Th+ + 2 LH2+ + 2 e - 3 LH2+). This means that bulk H+ is not consumed, leading to a smaller H+ transient. When the electron flux dies away, the pH drifts back to the equilibrium value of 4. As it does so, there is an H+ flux from the relaxation LH2+ + H+ - LH +. The explanation of the transients at pH 5 is similar. In the reduction direction, the H+ flux has almost completely collapsed. In this case, the pH crosses the pKa boundaries at 8.5 where there will be no H+ flux (Th+ + 2e -> L ). The relaxation flux after the electron flux has died away will also be small since the bulk concentration of H+ (pH = 5) is so small. At pH 6, the reduction transients are similar to those at pH 5. In the oxidation direction, the pH rapidly crosses the pKa = 5.5 boundary. Now the coat mops up the H+, releasing no H+ to the solution (3LH2+ - ... 

The scheme of squares is used to describe mechanistic pathways involving electron and proton transfers, and was first proposed by J. Jacq [/. Electroanal. Chem. 29 (1971) 149]. The scheme is based upon the assumption that the reactions occur in a stepwise manner. Figure 4.7 depicts a simple one-proton one-electron scheme. In the following, assume E = 0.0 V, E2 = -1-0.473 V, pKai = 3 and pKa2 = 11. 

Fig. 4.8 The variation of the midpoint potential for the scheme of squares, as outlined in Fig. 4.7.

Initially, it was thought that all two-electron reaction pathways were of this general sort, and a so-called scheme of squares was elaborated, which, for the system shown, would take the form ... 

Scheme 1 The scheme of one square for guest binding and electron transfer.

The last class of derivatives able to mimic the haemoprotein function that we will consider is constituted by the series of square-planar Rh(I) and Ir(I) complexes with the Shiff base o-Ph2C6H4CH=NR, Scheme 6.25... 

Convenient analysis of these complex electrode processes with separation of electron and proton charge transfer steps can be achieved by the use of a scheme of squares [13]. 

Also, in complex electrode reactions involving multistep proton and electron transfer steps, the electrochemical reaction order with respect to the H+ or HO may also vary with pH, indicating a change of mechanism with pH. In this respect, the use of schemes of squares outlined in Sect. 2.2 is very useful in the analysis of these complex kinetics [13]. 

The wave function of the two-shell configuration (17.42) corresponds to the representation of uncoupled quasispin momenta of individual shells. The eigenfunction of the square of the operator of total quasispin and its z-projection can be written as follows in the scheme of the vectorial coupling of momenta in quasispin space ... 

Very recently, Belokon and North have extended the use of square planar metal-salen complexes as asymmetric phase-transfer catalysts to the Darzens condensation. These authors first studied the uncatalyzed addition of amides 43a-c to aldehydes under heterogeneous (solid base in organic solvent) reaction conditions, as shown in Scheme 8.19 [47]. It was found that the relative configuration of the epoxyamides 44a,b could be controlled by choice of the appropriate leaving group within substrate 43a-c, base and solvent. Thus, the use of chloro-amide 43a with sodium hydroxide in DCM gave predominantly or exclusively the trans-epoxide 44a this was consistent with the reaction proceeding via a thermodynamically controlled aldol condensation... 

Scheme 2. Calculation of the reactivity number (localization energy) for the 1-position of naphthalene according to the PMO and PMO-F method. (The denominator follows from the normalization condition, i.e. the normalized NBMO coefficients are inversely proportional to the root of the sum of squares of the unnormalized coefficients)...

Prior to about 1955 much of the nuclear information was obtained from application of atomic physics. The nuclear spin, nuclear magnetic and electric moments and changes in mean-squared charge radii are derived from measurement of the atomic hyperfine structure (hfs) and Isotope Shift (IS) and are obtained in a nuclear model independent way. With the development of the tunable dye laser and its use with the online isotope separator this field has been rejuvenated. The scheme of collinear laser/fast-beam spectroscopy [KAU76] promised to be useful for a wide variety of elements, thus UNISOR began in 1980 to develop this type of facility. The present paper describes some of the first results from the UNISOR laser facility. 

It was further developed the following year (22), and was based primarily on the scheme of Priest (12) with an idea from Gardon (9d). The latter suggested that the rate of capture of oligomeric radicals in solution by pre-existing particles, R, should be proportional to the collision cross-section, or tfie square of the radius of the particles, r. This has been called the "collision theory" of radical capture. In 1975 Fitch and Shih measured capture rates in MMA seeded polymerizations and came to the conclusion that R was proportional to the first power of the radius, as would e predicted by Fick s theory of diffusion (23). In his book, K. J. Barrett also pointed out that diffusion must govern the motions of these species in condensed media (10). 

Given that there are a limited number of possible intermediates, we can draw up a scheme of squares of the form shown above [3], where the x symbolises a Pt—C bond, and which summarises the following reactions ... 

The mononuclear precursor [Mn°(phen)2Cl2] has two cis arranged labile Cl ligands and, as such, is an appropriate building block for the preparation of squares or TBPs (Scheme 1). Nevertheless, the combination of this precursor with the mononuclear complex traK.y-[M° bpb)(CN)2] (M = Cr, Fe) resulted in the formation of the dinuclear compounds [Mn phen)2Cl] [M °(bpb)(CN)2] (71) (Fig. 7). Crystallization of the latter species from polar solvents used in the reaction is driven by electroneutrality of these complexes. 

Two-dimensional TLC is performed by spotting the sample in one corner of a square thin-layer plate and developing in the usual manner with the first eluent. The chromatographic plate is then removed from the developing chamber and the solvent is allowed to evaporate from the layer. Then, the plate is placed in the second eluent so that development can take place in a second direction which is perpendicular to that of the first direction of development. In 2-D TLC, usually, the layer is of continuous composition, but two different eluents must be employed to obtain a better separation of a mixture. The success of the separation will depend on the ability to modify the selectivity of the second eluent compared to the selectivity of the first eluent. Fig. 1 shows the scheme of spot distribution on a 2-D TLC plate, following two developments for a theoretical case. In 2-D TLC, any spot can be identified by a pair of x, and y, coordinates or Rfn and Rfi2, respectively, where x, divided by Zf,i is equal to Rf n for the first eluent and yi/Z 2 is equal Z fi2 for the second eluent. The final position of the spot can be determined by the coordinates (xi,yi), in which Rn2-o can be expressed as (Rf,ii,Rf,i2)-A very good method for selection of the appropriate mobile phase for 2-D TLC separations is with the use of the Prisma system. 

Another conceptually different approach is cross-validation. In Equation (2.19), X is regarded as a model for X, and as such the model should be able to predict the values of X. This can be checked by performing a cross-validation scheme in which parts of X are left out of the calculations and kept apart, the model is built and used to predict the left out entries. The sum of squared differences between the predicted and the real entries serves as a measure of discrepancy. All data in X are left out once, and the squared differences are summed in a so called PRESS statistics (PRediction Error Sum of Squares). The model that gives the lowest PRESS is selected and the pseudo-rank of X is defined as the number of components in that model. 

Figure 7. a) The tl-orbital scheme of square-pyramidal bispidine-copper(II)-coligand complexes... 

Partitioning of the variances of a linear regression follows the scheme given in Table 6.1 and Figure 6.1 by the appropriate sums of squares the total variance of the y values, SS, adds up from the sum of squares of the mean, SSj, and the sum of squares corrected for the mean, SS . 

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