Reaction ladder diagrams

Ladder diagrams are particularly useful for evaluating the reactivity of acids and bases. An acid and a base cannot coexist if their respective areas of predominance do not overlap. If we mix together solutions of NH3 and HE, the reaction... 

We can also construct ladder diagrams using cumulative formation constants in place of stepwise formation constants. The first three stepwise formation constants for the reaction of + with NH3... 

Ladder diagrams can also be used to evaluate equilibrium reactions in redox systems. Figure 6.9 shows a typical ladder diagram for two half-reactions in which the scale is the electrochemical potential, E. Areas of predominance are defined by the Nernst equation. Using the Fe +/Fe + half-reaction as an example, we write... 

Using standard-state potentials to construct a ladder diagram can present problems if solutes are not at their standard-state concentrations. Because the concentrations of the reduced and oxidized species are in a logarithmic term, deviations from standard-state concentrations can usually be ignored if the steps being compared are separated by at least 0.3 A trickier problem occurs when a half-reaction s potential is affected by the concentration of another species. For example, the potential for the following half-reaction... 

Ladder diagram showing the effect of a change in pH on the areas of predominance for the U02 +/U + half-reaction. 

Solubility of AgCI as a function of pCI. The dashed line shows the predicted SAgci, assuming that only reaction 8.1 and equation 8.2 affect the solubility of AgCI. The solid line is calculated using equation 8.7, and includes the effect of reactions 8.3-8.5. A ladder diagram for the AgCI complexation equilibria is superimposed on the pCI axis. 

The ladder diagram for this system is shown in Figure 11.24a. Initially the potential of the working electrode remains nearly constant at a level near the standard-state potential for the Fe UFe redox couple. As the concentration of Fe + decreases, however, the potential of the working electrode shifts toward more positive values until another oxidation reaction can provide the necessary current. Thus, in this case the potential eventually increases to a level at which the oxidation of H2O occurs. 

A "potential ladder" diagram models the potential difference. The rungs on the ladder correspond to the values of the reduction potentials. For a galvanic cell, the half-reaction at the cathode is always on the upper rung, and the subtraction... 

Kuo and Brown [3], where energies of the low-lying sd-like states of and were calculated from the underlying nucleon-nucleon interaction. The effective interaction used in this type of calculation is the reaction or G-matrix which sums the ladder diagrams including only particle-particle intermediate states. 

The simulated complex-plane impedance diagram is shown in Figure 4.27b. As can be seen in the figure, this ladder structure is characterized by two semicircles with two time constants, r, = RclCd] and r2 = R3C2, accounting for the two-step reaction. The element C2 symbolizes the adsorption capacitance, and r2 represents the relaxation of the adsorbing process. 

However, this problem is overcome by introducing the reaction matrix G, displayed by the summation of ladder type of diagrams in Fig. 1, which accounts for the effects of two-nucleon correlations. The G-matrix is given by the solution of the Bethe-Goldstone equation... 

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