Half-reaction equilibrium

Online mass spectrometry data presented and discussed in the previous sections suggest that catalytic hypophosphite oxidation on nickel in D2O solutions proceeds via the coupling of anodic (19.11) and cathodic (19.12) half-reactions at the catalyst surface. The classical mixed-potential theory for simultaneously occurring electrochemical partial reactions [14] presupposes the catalyst surface to be equally accessible for both anodic (19.11) and cathodic (19.12) half-reactions. Equilibrium mixtures of H2, HD, and D2 should be formed in this case due to the statistical recombination of Hahalf-reactions (19.11) and (19.12) for example, the catalytic oxidation of hypophosphite on nickel in D20 solution under open-circuit conditions should result in the formation of gas containing equal amounts of hydrogen and deuterium (H/D=l) with the distribution H2 HD D2= 1 2 1 (the probability of HD molecule formation is twice as high as for either H2 or D2 formation [75]). Therefore, to get further mechanistic insight, the distribution of H2, HD, and D2 species in the evolved gas was compared to the equilibrium values at the respective deuterium content [54]. 

This half reaction equilibrium together with the electron s linear free energy dependence on electrode potential allows for determination of the proton-electron pair free energy without direct DFT consideration of a solvated proton ... 

A connnon approach has been to measure the equilibrium constant, K, for these reactions as a fiinction of temperature with the use of a variable temperature high pressure ion source (see section (Bl.7.2)1. The ion concentrations are approximated by their abundance in the mass spectrum, while the neutral concentrations are known from the sample mlet pressure. A van t Hoff plot of In K versus /T should yield a straight Ime with slope equal to the reaction enthalpy (figure B1.7.11). Combining the PA with a value for basicityG at one temperature yields a value for A.S for the half-reaction involving addition of a proton to a species. While quadnipoles have been tire instruments of choice for many of these studies, other mass spectrometers can act as suitable detectors [19, 20]. 

At equilibrium at 298 K the electrode potential of the half-reaction for copper, given approximately by... 

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... 

At the equivalence point, the moles of Fe + initially present and the moles of Ce + added are equal. Because the equilibrium constant for reaction 9.16 is large, the concentrations of Fe and Ce + are exceedingly small and difficult to calculate without resorting to a complex equilibrium problem. Consequently, we cannot calculate the potential at the equivalence point, E q, using just the Nernst equation for the analyte s half-reaction or the titrant s half-reaction. We can, however, calculate... 

Table 1.7 shows typical half reactions for the oxidation of a metal M in aqueous solutions with the formation of aquo cations, solid hydroxides or aquo anions. The equilibrium potential for each half reaction can be evaluated from the chemical potentials of the species involved see Appendix 20.2) and it should be noted that there is no difference thermodynamically between equations 2(a) and 2(b) nor between 3(a) and 3(b) when account is taken of the chemical potentials of the different species involved. 

The data given in Tables 1.9 and 1.10 have been based on the assumption that metal cations are the sole species formed, but at higher pH values oxides, hydrated oxides or hydroxides may be formed, and the relevant half reactions will be of the form shown in equations 2(a) and 2(b) (Table 1.7). In these circumstances the a + will be governed by the solubility product of the solid compound and the pH of the solution. At higher pH values the solid compound may become unstable with respect to metal anions (equations 3(a) and 3(b), Table 1.7), and metals like aluminium, zinc, tin and lead, which form amphoteric oxides, corrode in alkaline solutions. It is evident, therefore, that the equilibrium between a metal and an aqueous solution is far more complex than that illustrated in Tables 1.9 and 1.10. Nevertheless, as will be discussed subsequently, a similar thermodynamic approach is possible. 

Equations 20.176 and 20.179 emphasise the essentially thermodynamic nature of the standard equilibrium e.m.f. of a cell or the standard equilibrium potential of a half-reaction E, which may be evaluated directly from e.m.f. meeisurements of a reversible cell or indirectly from AG , which in turn must be evaluated from the enthalpy of the reaction and the entropies of the species involved (see equation 20.147). Thus for the equilibrium Cu -)-2e Cu, the standard electrode potential u2+/cu> hence can be determined by an e.m.f. method by harnessing the reaction... 

One of the most important characteristics of a cell is its voltage, which is a measure of reaction spontaneity. Cell voltages depend on the nature of the half-reactions occurring at the electrodes (Section 18.2) and on the concentrations of species involved (Section 18.4). From the voltage measured at standard concentrations, it is possible to calculate the standard free energy change and the equilibrium constant (Section 18.3) of the reaction involved. 

The value of this list is obvious. Any half-reaction can be combined with the reverse of another half-reaction (in the proportion for which electrons gained is equal to electrons lost) to give a possible chemical reaction. Our list permits us to predict whether equilibrium favors reactants or products. We would like to expand our list and to make it more quantitative. Electrochemical cells help us do this. 

Opposing reactions. The reversible interconversion of d.y-[Cr(cn)2(OH)2 + to trans is characterized by an equilibrium constant of 0.16 and a forward rate constant of 3.30 X 10 4 s"1.In an experiment starting with the pure cis isomer, how long does it take to form half the equilibrium concentration of trans Starting with pure trans, how long does it take for half the equilibrium concentration of trans to form ... 

One of the most useful applications of standard potentials is in the calculation of equilibrium constants from electrochemical data. The techniques that we develop here can be applied to any kind of reaction, including neutralization and precipitation reactions as well as redox reactions, provided that they can be expressed as the difference of two reduction half-reactions. 

The fact that we can calculate E° from standard potentials allows us to calculate equilibrium constants for any reaction that can be expressed as two half-reactions. The reaction does not need to be spontaneous nor does it have to be a redox reaction. Toolbox 12.3 summarizes the steps and Example 12.8 shows the steps in action. 

In addition to defined standard conditions and a reference potential, tabulated half-reactions have a defined reference direction. As the double arrow in the previous equation indicates, E ° values for half-reactions refer to electrode equilibria. Just as the value of an equilibrium constant depends on the direction in which the equilibrium reaction is written, the values of S ° depend on whether electrons are reactants or products. For half-reactions, the conventional reference direction is reduction, with electrons always appearing as reactants. Thus, each tabulated E ° value for a half-reaction is a standard reduction potential. 

This is a quantitative calculation, so it is appropriate to use the seven-step problem-solving strategy. We are asked to determine an equilibrium constant from standard reduction potentials. Visualizing the problem involves breaking the redox reaction into its two half-reactions ... 

Analytical methods based upon oxidation/reduction reactions include oxidation/reduction titrimetry, potentiometry, coulometry, electrogravimetry and voltammetry. Faradaic oxidation/reduction equilibria are conveniently studied by measuring the potentials of electrochemical cells in which the two half-reactions making up the equilibrium are participants. Electrochemical cells, which are galvanic or electrolytic, reversible or irreversible, consist of two conductors called electrodes, each of which is immersed in an electrolyte solution. In most of the cells, the two electrodes are different and must be separated (by a salt bridge) to avoid direct reaction between the reactants. 

Of course free electrons do not exist in a regular laboratory, so Equation 13.6 is a theoretical half reaction of the full equilibrium ... 

Fig. 2. Current-potential curves in Evans diagram [29] format for reduction of Cu2+ ions and oxidation of H2CO. and are the equilibrium, or open circuit, potentials for the Cu2+ reduction and H2CO oxidation reactions, respectively. Assuming negligible interfering reactions, the vertical dashed lines indicate the exchange current densities for the two half reactions, and the deposition current for the complete electroless solution. Adapted from ref. 23.

X is half committed to reaction and half in equilibrium with R. 

Fig. 22.6. Redox potentials (mV) of various half-cell reactions during mixing of fluid from a subsea hydrothermal vent with seawater, as a function of the temperature of the mixture. Since the model is calculated assuming 02(aq) and H2(aq) remain in equilibrium, the potential for electron acceptance by dioxygen is the same as that for donation by dihydrogen. Dotted line shows currently recognized upper temperature limit (121 °C) for microbial life in hydrothermal systems. A redox reaction is favored thermodynamically when the redox potential for the electron-donating half-cell reaction falls below that of the accepting half-reaction.

Metal indicator electrodes develop a potential which is usually determined by the equilibrium position of a redox half-reaction at the electrode surface. These are further classified into the following three types, namely ... 

The curves relative to the half-reactions intersect at the point corresponding to the formation of the so-called activated complex. The height of the energy barrier of the two redox processes (oxidation, /z0x reduction, hRed) is inversely proportional to the respective reaction rates. Since in this case h0x = hRed, it is immediately apparent that these conditions identify the equilibrium conditions. 

Table 7.1 Standard Electrode Potentials and Equilibrium Constants for Some Reduction Half-Reactions. ...

The Pourbaix diagram for the O2-H2O couple is presented in Figure 7.7, along with the (,J. -pH conditions characteristic of various natural and polluted waters. The equations for the boundary lines are calculated as follows. The redox half reaction that defines the upper boundary is given by Eq. 7.3T Its equilibrium constant is... 

A further informative example is the organic matter—CO2 couple, which is the principal reductant in natural systems. Consider a solution in equilibrium with atmospheric CO2 at neutral pH and containing 10p,M CH2O , where CH2O represents average organic C in natural systems, whose composition is similar stoichiometrically to that of carbohydrates. The half reaction is... 

Table 4.2 Equilibrium constants of reduction half-reactions at pH 7 and 25 °C...

Table 7.7 Equilibrium constants of reduction half-reactions of trace elements in submerged soils compared with those for Fe and Mn...

A mathematical equation indicating how the equilibrium constant of an enzyme-catalyzed reaction (or half-reaction in the case of so-called ping pong reaction mechanisms) is related to the various kinetic parameters for the reaction mechanism. In the Briggs-Haldane steady-state treatment of a Uni Uni reaction mechanism, the Haldane relation can be written as follows ... 

REDOX HALF-REACTIONS. Electron transfer reactions involve oxidation (or loss of electrons) of one component and reduction (or gain of electrons) by a second component. Therefore, a complete redox reaction can be treated as the sum of two half-reactions such that the stoichiometry and electric charge is balanced across a chemical equilibrium. For each such half-reaction, there is an associated standard potential E°. The hydrogen ion-hydrogen gas couple is ... 

Another important characteristic of ping pong reaction is the presence of exchange reactions in the absence of other substrates. See Isotope Exchange at Equilibrium Multisite Ping Pong Bi Bi Mechanism Half-Reaction Substrate Synergism... 

The half reaction for iron dissolution proceeds until equilibrium is reached. Further corrosion of iron requires that the single potential is raised to some nonequi-... 

The reaction produces hydroxyl ions which react directly with the Fe ions to produce an oxide precipitate. The combined anodic and cathodic reactions form the corrosion cell, the electrochemical potential of which lies between the single potential of the two half reactions. This mixed potential is termed the corrosion potential, corr> and for corrosion to proceed beyond the equilibrium state, the corrosion potential must be more positive than the equilibrium single potential of iron. For iron in water at pH 7 and with [Fe j = 10" M, for example, the potential of the anodic reaction is. 

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