A1.6.3.2 SECOND-ORDER AMPLITUDE CLOCKING CHEMICAL REACTIONS [Pg.241]

The most important second-order forces are dispersion forces. London [3, 31, 32] showed that they are caused by a correlation of tlie electron distribution in one molecule with tliat in the other, and pointed out that the [Pg.191]

The details of the second-order energy depend on the fonn of exchange perturbation tiieory used. Most known results are numerical. However, there are some connnon features that can be described qualitatively. The short-range mduction and dispersion energies appear in a non-expanded fonn and the differences between these and their multipole expansion counterparts are called penetration tenns. [Pg.198]

Other examples of order-disorder second-order transitions are found in the alloys CuPd and Fe Al. Flowever, not all ordered alloys pass tlirough second-order transitions frequently the partially ordered structure changes to a disordered structure at a first-order transition. [Pg.632]

If the desorption rate is second-order, as is often the case for hydrogen on a metal surface, so that appears in Eq. XVIII-1, an equation analogous to Eq. XVIII-3 can be derived by the Redhead procedure. Derive this equation. In a particular case, H2 on Cu3Pt(III) surface, A was taken to be 1 x 10 cm /atom, the maximum desorption rate was at 225 K, 6 at the maximum was 0.5. Monolayer coverage was 4.2 x 10 atoms/cm, and = 5.5 K/sec. Calculate the desorption enthalpy (from Ref. 110). [Pg.739]

There are many other examples of second-order transitions involving critical phenomena. Only a few can be mentioned here. [Pg.656]

We start from the expression for the second-order wavefunction [Pg.249]

Figure Al.6.9. Feymnan diagram for the second-order process described in the text. |

Tavrin, V, Zhang, Y, Wolf, W, Braginski, AI A second-order SQUID gradiometer operating at 77 K, Supercond. Sci. Technol. 7 (1994) 265 [Pg.992]

In words, equation (Al.6.89) is saying that the second-order wavefunction is obtained by propagating the initial wavefunction on the ground-state surface until time t", at which time it is excited up to the excited state, upon which it evolves until it is returned to the ground state at time t, where it propagates until time t. NRT stands for non-resonant tenn it is obtained by and cOj -f-> -cOg, and its physical interpretation is [Pg.249]

The ordinary Debye-Huckel interionic attraction effects have been neglected and are of second-order importance. [Pg.179]

These systems are solved by a step-limited Newton-Raphson iteration, which, because of its second-order convergence characteristic, avoids the problem of "creeping" often encountered with first-order methods (Law and Bailey, 1967) [Pg.116]

From (1) it is clear that the phase contrast can be interpreted simply in terms of tbe variation (second order derivative) of the projected image density, and increases with improving resolution of the system, in agreement with the findings of [3]. [Pg.575]

A related phenomenon with electric dipoles is ferroelectricity where there is long-range ordermg (nonzero values of the polarization P even at zero electric field E) below a second-order transition at a kind of critical temperature. [Pg.635]

This expression may be interpreted in a very similar spirit to tliat given above for one-photon processes. Now there is a second interaction with the electric field and the subsequent evolution is taken to be on a third surface, with Hamiltonian H. In general, there is also a second-order interaction with the electric field through which returns a portion of the excited-state amplitude to surface a, with subsequent evolution on surface a. The Feymnan diagram for this second-order interaction is shown in figure Al.6.9. [Pg.242]

If the long-range mteraction between a pair of molecules is treated by quantum mechanical perturbation theory, then the electrostatic interactions considered in section Al.5.2.3 arise in first order, whereas induction and dispersion effects appear in second order. The multipole expansion of the induction energy in its fill generality [7, 28] is quite complex. Here we consider only explicit expressions for individual temis in the [Pg.190]

Its ratio to the first temi can be seen to be (5 J / 5 Ef) E HT. Since E is proportional to the number of particles in the system A and Ej, is proportional to the number of particles in the composite system N + N, the ratio of the second-order temi to tire first-order temi is proportional to N N + N. Since the reservoir is assumed to be much bigger than the system, (i.e. N) this ratio is negligible, and the truncation of the [Pg.397]

The initial classification of phase transitions made by Ehrenfest (1933) was extended and clarified by Pippard [1], who illustrated the distmctions with schematic heat capacity curves. Pippard distinguished different kinds of second- and third-order transitions and examples of some of his second-order transitions will appear in subsequent sections some of his types are unknown experimentally. Theoretical models exist for third-order transitions, but whether tiiese have ever been found is unclear. [Pg.613]

Here the coefficients G2, G, and so on, are frinctions ofp and T, presumably expandable in Taylor series around p p and T- T. However, it is frequently overlooked that the derivation is accompanied by the connnent that since. . . the second-order transition point must be some singular point of tlie themiodynamic potential, there is every reason to suppose that such an expansion camiot be carried out up to temis of arbitrary order , but that tliere are grounds to suppose that its singularity is of higher order than that of the temis of the expansion used . The theory developed below was based on this assumption. [Pg.643]

It was pointed out that a bimolecular reaction can be accelerated by a catalyst just from a concentration effect. As an illustrative calculation, assume that A and B react in the gas phase with 1 1 stoichiometry and according to a bimolecular rate law, with the second-order rate constant k equal to 10 1 mol" see" at 0°C. Now, assuming that an equimolar mixture of the gases is condensed to a liquid film on a catalyst surface and the rate constant in the condensed liquid solution is taken to be the same as for the gas phase reaction, calculate the ratio of half times for reaction in the gas phase and on the catalyst surface at 0°C. Assume further that the density of the liquid phase is 1000 times that of the gas phase. [Pg.740]

S. Chains in the S phase are also oriented normal to the surface, yet the unit cell is rectangular possibly because of restricted rotation. This structure is characterized as the smectic E or herringbone phase. Schofield and Rice [204] applied a lattice density functional theory to describe the second-order rotator (LS)-heiTingbone (S) phase transition. [Pg.134]

The measurement of a from the experimental slope of the Tafel equation may help to decide between rate-determining steps in an electrode process. Thus in the reduction water to evolve H2 gas, if the slow step is the reaction of with the metal M to form surface hydrogen atoms, M—H, a is expected to be about If, on the other hand, the slow step is the surface combination of two hydrogen atoms to form H2, a second-order process, then a should be 2 (see Ref. 150). [Pg.214]

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