Experimental tests of the theory

Taken together, eqns 8.27 and 8.28 suggest that the expression for has the 

Marcus expression for the rate constant of electron transfer 

Many of the key features of Marcus theory have been tested by experiments, showing in particular the predicted dependence of on the standard reaction Gibbs energy and the edge-to-edge distance between electron donor and acceptor. 

It is difficult to measure the distance dependence of k t when the reactants are ions or molecules that are free to move in solution. In such cases, electron transfer occurs after a donor-acceptor complex forms and it is not possible to exert control over r, the edge-to-edge distance. The most meaningful experimental tests of the dependence of k on r are those in which the same donor and acceptor are positioned at a variety of distances, perhaps by covalent attachment to molecular hnkers. Under these conditions, the term becomes a constant and, after 

P 9 run when the intervening medium is a molecular hnk between donor and acceptor. Electron transfer between protein-bound cofactors can occur at distances of up to about 2.0 run, a long distance on a molecular scale, corresponding to about 20 carbon atoms, with the protein providing an intervening medium between donor and acceptor. 

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The principles of thin-layer anodic stripping voltammetry were discussed, and a model for the stripping stage was developed for anodization by a linear potential ramp [35]. The experimental test of the theories was carried out for cadmium and lead amalgams. 

Spencer, The Determination of Reaction Rates of Nonideal Explosives from Shaped Charge Penetration Data , Univ of Utah Inst for Study of Rate Processes, TechRept No XLVIl(1955) Contract N7-onr-45107 46) R.J. Eichelberger, JApplPhys 26, 392-402(1955) (Re-examination of the unsteady theory of jet formation by lined cavity charges) 47) Ibid, 27, 63-8 (1956) (Experimental test of the theory of penetration by metallic jets) 48) T.C. Poulter ... 

The very broad crossover has important implications for an experimental test of the theory. Indeed, to illustrate clearly the effects, in the figures of this chapter we have driven the parameters into extreme regions. No experiment, for instance, will cover six decades of overlap, nor will wre reach a range of z sufficient to cover both the 0- and the excluded volume limits. Any physical experiment or any simulation will see only some small section of the crossover function, the full function being tested only by combining results for several chemical systems. In particular we have no chance to see a double crossover as typically exhibited here, in a single experiment. 

Helfferich, F. (1962b). Ion exchange kinetics. III. Experimental test of the theory of particle-diffusion controlled ion exchange. J. Phys. Chem. 66, 39-44. 

Obtaining experimental tests of the theory of hydrogen-like ions at moderate Z is challenging. Except for hydrogen there appear to be no measurements more precise than current theory. In the next decade, small but significant improvements in precision can be expected for Z = 2, 7 and 14. For moderate-.Z helium-like ions, laser techniques probe relativistic QED effects at higher precision than... 

Observation of the mechanisms of lamella formation in single capillaries, etched media, and bead packs, followed by development of pore-level theory for the formation, flow, and disappearance of lamellae ("bubbles") and experimental tests of the theories (36-41). 

We discuss some of the properties of maps and the techniques for analyzing them in Sections 10.1-10.5. The emphasis is on period-doubling and chaos in the logistic map. Section 10.6 introduces the amazing idea of universality, and summarizes experimental tests of the theory. Section 10.7 is an attempt to convey the basic ideas of Feigenbaum s renormalization technique. 

A careful experimental test of the theory was carried out by Perrin (1910). Emulsions composed of droplets about 0.4 jum in diameter were observed under an optical microscope and the positions of the particles were noted at regular time intervals. The Stokes-Einstein equation was checked by writing it in the form... 

However, there are other types of experimental tests of the theory of polymers in solution these are experiments in which the chemically homogeneous solute is a mixture of two (or more) samples whose polymer chains have very different average sizes. However, we cannot test the theories of strongly polydisperse solutions as we did in the weakly polydisperse case i.e. by referring to a monodisperse solute of molecular mass equal to the average molecular mass of the sample. Polydispersion here becomes an essential parameter. 

VII. GENERAL PROPERTIES OF THE SIZE DISTRIBUTION AND EXPERIMENTAL TEST OF THE THEORY... 

For an experimental test of the theory for thermal conduction see Sect. 7.2. 

Experimental tests of the theory have shown that in some cases the correlation between theory and experiment is good 110), but sometimes deviations from dependence occur, and these can be both positive 110) and negative 107,111) the latter may be attributed to the non equilibrium conditions of the crystallization of loi blocks 107) and to the difference between the d/f values for copolymers and homopolymers 111). At any rate, the data on melting point depression make it eaqr to differentiate between random and block copolyn rs erf the same composition 111). 

A recent experimental test of the theory was performed using time-resolved fluorescence. Lissamine fluorophores were attached to gold nanoparticles via thio ether groups. Both the radiative and non-radiative decay chaimels were studied. The sizes of the Au particles were varied in several steps between a = 1 and a = 30 nm, whereas the distance between the fluorphore and the nanoparticle was kept constant at about d=r-a = 1 nm. The optical excitation source was a 120 fs laser pulse at a wavelength of 400 nm (3.10 eV), well away from the dipolar plasmon resonance at 520 run (2.4 eV) for a gold sphere in water. It was believed that the transition dipole of the fluorophore was parallel to the surface of the particle. 

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