Cross relation

The enthalpy is the same as the heat exchanged with the surroundings. The Gibbs free energy can be related to enthalpy 

Hess s law referring to the heat evolved in a chemical reaction is conveniently formulated in terms of enthalpy. 

Maxwell first noted the cross relations based on a property of the total differentials of the state functions. The cross differentiations of a total differential of the state function are equal to each other. Table 1.14 summarizes the total differentials and the corresponding Maxwell relations. The Maxwell relations may be used to construct important thermodynamic equations of states. 

The cross relations can be seen in a reversible change of a rectangular rubber sheet subjected to two perpendicular forces Fx and Fy under isothermal conditions. If the extent of stretching in both directions of x andy are Ax and Ay, we have 

The ordinary elastic moduli in the x andy directions are denoted by M] andM22 

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A powerful application of outer-sphere electron transfer theory relates the ET rate between D and A to the rates of self exchange for the individual species. Self-exchange rates correspond to electron transfer in D/D (/cjj) and A/A (/c22)- These rates are related through the cross-relation to the D/A electron transfer reaction by the expression... 

The cross relation has proven valuable to estimate ET rates of interest from data tliat might be more readily available for individual reaction partners. Simple application of tire cross-relation is, of course, limited if tire electronic coupling interactions associated with tire self exchange processes are drastically different from tliose for tire cross reaction. This is a particular concern in protein/protein ET reactions where tire coupling may vary drastically as a function of docking geometry. 

These equations do not necessarily show the actual charges the important point is that all three are single-electron events. The asterisks can be thought of as an isotopic label, but need not be anything that concrete, since certain line-broadening techniques (Section 11.5) provide EE rate constants without them. The Marcus cross relation is an expression for kA% as a function of kAA, bb> and A, the equilibrium constant for Eq. (10-67). It reads,... 

Data are given in Table 10-7 to illustrate certain facets of the Marcus cross relation. They refer to six reactions in which the cage complex Mn(sar)3+ is reduced or Mn(sar)2+ oxidized.34 These data were used to calculate the EE rate constant for this pair. The results of the calculation, also tabulated, show that there is a reasonably self-consistent value of fcEE for Mn(sar)3+/Mn(sar)2+ lying in the range 3-51 L mol-1 s-1. When values34 for an additional 13 reactions were included the authors found an average value of kEE = 17 L mol 1 s l. 

Marcus theory. Consider that the reorganization energy for the ET reaction, AAb, can be approximated as the mean of the reorganization energies for the EE reactions Aab = (Aaa + ABb)/2. Show that substitution of this expression into Eq. (10-63) gives the usual form of the Marcus cross relation. 

Marcus cross relation, 243-246 Marcus theory, 239-248 Mean lifetime (see Lifetime)... 

Once we have obtained these barriers, we are now able to go back, insert these values into the cross-relation, and test the... 

X 10 M at 1.0 M ionic strength. A simple outer-sphere mechanism can be ruled out since k is more than two orders of magnitude greater than predicted by the Marcus cross-relation, and an oxygen atom transfer mechanism has been proposed. ... 

A model has been considered for Sn2 reactions, based on two interacting states. Relevant bond energies, standard electrode potentials, solvent contribntions (nonequi-librinm polarization), and steric effects are included. Applications of the theory are made to the cross-relation between rate constants of cross- and identity reactions, experimental entropies and energies of activation, the relative rates of Sn2 and ET reactions, and the possible expediting of an outer sphere ET reaction by an incipient SN2-type interaction (Marcus, 1997). 

Figure 3.5 Total control fields avoided crossing related correction fields and associated population dynamics in the model five-level system, (a) Two Lorentzian pulses centered at r j and (see Figure 3.4). (b) Four Lorentzian pulses centered at (see Figure 3.4). The total...

Hydrogen atom abstraction by non-radical, metal- species, once considered unlikely, has been shown in recent years to operate in a large number of reactions (113-115) with the kinetics responding to the thermodynamic driving force and intrinsic barriers as predicted by the Marcus cross relation (116). 

The Maxwell-Heaviside theory of electrodynamics has no explanation for the Sagnac effect [4] because its phase is invariant under 7 as argued already, and because the equations are invariant to rotation in the vacuum. The d Alembert wave equation of U(l) electrodynamics is also 7 -invariant. One of the most telling pieces of evidence against the validity of the U(l) electrodynamics was given experimentally by Pegram [54] who discovered a little known [4] cross-relation between magnetic and electric fields in the vacuum that is denied by Lorentz transformation. 

From Eqn. (14.3), one can easily derive the cross relations between extensive and intensive state variables as, for example,... 

For convenience we will make a simple demonstration of how to transform a 2x2 matrix problem to complex symmetric form. In so doing we will also recognise the appearence of a Jordan block off the real axis as an immediate consequence of the generalisation. The example referred to is treated in some detail in Ref. [15], where in addition to the presence of complex eigenvalues one also demonstrates the crossing relations on and off the real axis. The Hamiltonian... 

M. M. Cross, Rheology of Non-Newtonian Fluids a New Flow Equation for Pseudoplastic Systems, J. Colloids Sci., 20, 417 137 (1965) also M. M. Cross, Relation Between Viscoe-lasiticity and Shear-thinning Behaviour in Liquids, Rheological Acta, 18, 609-614 (1979). 

There were many predictions arising from the theory and its extension to electrochemical and other systems [11, 23, 24]. One such prediction, the cross-relation , was based on the relation between the A for reactions between two different redox systems, A 2, to the A s of the self-exchange reactions, An and A12, for each of the two systems (A,2 = l/2(An + A22)). The result for k12, the rate constant for the cross-reaction, is... 

The cross-relation (equation (1.6)), has also been applied successfully to transfers of CH3 [31] and to transfers of H" [32-33], while equation (1.4) has been used to treat proton transfers [34] and proton bound dimers [34d]. As already noted, the intersecting parabolas of Fig. 1.3 would not be applicable and so some other treatment was needed to understand the... 

It was recently shown (Ratner and Levine, 1980) that the Marcus cross-relation (62) can be derived rigorously for the case that / = 1 by a thermodynamic treatment without postulating any microscopic model of the activation process. The only assumptions made were (1) the activation process for each species is independent of its reaction partner, and (2) the activated states of the participating species (A, [A-], B and [B ]+) are the same for the self-exchange reactions and for the cross reaction. Note that the following assumptions need not be made (3) applicability of the Franck-Condon principle, (4) validity of the transition-state theory, (5) parabolic potential energy curves, (6) solvent as a dielectric continuum and (7) electron transfer is... 

Even in the domain of inorganic redox chemistry relatively little use has been made of the full potential of the Marcus theory, i.e. calculation of A, and A0 according to (48) and (52) and subsequent use of (54) and (13) to obtain the rate constant (for examples, see Table 5). Instead the majority of published studies are confined to tests of the Marcus cross-relations, as given in (62)-(65) (see e.g. Pennington, 1978), or what amounts to the same type of test, analysis of log k vs. AG° relationships. The hesitation to try calculations of A is no doubt due to the inadequacy of the simple collision model of Fig. 4, which is difficult to apply even to species of approximately spherical shape. 

Calculation of rate constants (k]2) for organic electron transfer processes, using the Marcus cross relations (62) and (63)... 

As for estimates of individual rate constants via the cross relations, this procedure seems to work well for organic electron-transfer processes, and the few existing limitations are of the same kind as those encountered for inorganic redox processes. 

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