First-order theory

Because much experimental work has been stimulated by the quasi-chemical theory, it is important to gain proper perspective by first describing the features of this theory.12 The term, quasichemical will be used to include the Bragg-Williams approximation as the zeroth-order theory, the Bethe or Guggenheim pair-distribution approximations as the first-order theory, and the subsequent elaborations by Yang,69 Li,28 or McGlashan31 as theories of higher order. 

Shown in Figure 8 is essentially the same plot as Figure 7, but one of the expansions is made using the first order theory. This figure shows that although the first order theory is not adequate for an accurate description of the x2 surface even for a 5% variation around the true solution, the first order theory nevertheless results in a solution which is closer to the true solution compared with its starting point and that the method of iteration may be applied to improve the accuracy of the solution. 

In geometrical optics the numerical aperture of a lens is given by first-order theory and spherical aberrations are given by third-order theory (Jenkins and White 1976 Hecht 2002). In first-order theory the approximation is made... 

The term in curly brackets describes the deviation from the first-order theory. For paraxial rays h = 0 and that term vanishes this is the first-order result. But for other values of h the rays do not cross the axis at the same point as the paraxial rays, and this causes aberration. When parallel rays are incident from infinity as illustrated in Fig. 2.2, 1 /s y = 0. If the aberration is not too big, then in the term in the curly brackets the approximation Si q can be made. With n = nyjni, (2.4) then reduces to... 

It was assumed, as in the first-order theory for the Gouy interference phenomenon (6), that as a first approximation only rays passing through equal values of the refractive index gradient on either side of the boundary center would arrive together at the slit image plane. Thus pairs of positions with equal dn/dx values (see upper part of Figure 1) were... 

There are two areas in which it seems that substantial advances could be made even on the basis of first-order theory. These are the field of spontaneous reactions where isotope effects are sometimes large and where the existence of many closely related systems makes it likely that a useful framework of generalizations could be found. The second field is that of strictly non-aqueous solvent systems where a comparison of solvent isotope efFects with those in aqueous solution is likely to throw light on essential differences in chemistry. 

Fig. 13. Splittings for Ve > Vls. The spin-orbit splittings are drawn to an arbitrary scale and represent first-order theory only. The numbers indicate total degeneracy, (a) d1 in an octahedral interstice, (b) d9 in a tetrahedral interstice, (c) d6 in an octahedral interstice, (d) d7 in an octahedral interstice.

The first-order theory presented above is related to the treatment of Baur and Nosanow/ who used a similar but perhaps physically less realistic model to derive the same results [that is, Eqs. (7), (9), and (10)].t... 

The data plots of Fig. 15b (silica) are differentiated for the use of methyl- -butyl ether (MTBE) or acetonitrile (ACN) as localizing solvent C in the mobile phase. It is seen that for some solute pairs (Fig. 15a and c) the open squares (MTBE) fall on a different curve than the closed squares (ACN). This implies that the constant in Eq. (31a) is solvent-specific, rather than being constant for all solvents (as first-order theory would predict). A similar behavior is observed for alumina as well. Figure 15a plots data for 18 different polar solvents B or C, and some scatter of these plots of log a versus m is observed here, as in Fig. 15b for silica. The variation of Q with the localizing solvent C used for the mobile phase has been shown (18) to correlate with the relative basicity of the solvent, or its placement in the solvent classification scheme of Refs. 40 and 41. Thus, for relatively less basic solvents (groups VI or VII in Refs. 40 and 4/),... 

In the next few pages we shall discuss the question of local interfacial structures bounding idealised aggregates, tiled by blocks of fixed dimensions. The model represents one extreme idealisation of the molecular constituents that form the aggregate, most applicable to small surfactant molecules. At the other extreme, the block dimensions are not set a priori, they must be determined as a function of the temperature, concentration, etc. This case will be dealt with later. The welding of two concepts, a fluid-like mixture of hydrocarbons, with that of an idealised block is at first sight contradictory. However it can be shown to be consistent in a first order theory [2]. 

F30. Friedenwald, J. S., and Maengwyn-Davies, G. D., Elementary kinetic theory of enzymatic activity first order theory. In The Mechanism of Enzyme Action (W. D. McElroy and B. Glass, eds.), p. 163. Johns Hopkins Press, Baltimore, Maryland, 1954. 

Etzler (1983) has proposed a first-order theory of vicinal water based on the percolation theory treatment of bulk water developed by Stanley and Teixeira (1980a,b). Etzler assumes two distinct types of H-bond connectivity, namely, a perfect four-connected set of water molecules and the remainder, with three, two or one (or no) H-bonds. Based on the idea that two distinct populations of water molecule environments exist, Etzler calculated the increase in the amount of ice-like (four-coordinated) local environments and proposed that the tendency to create such more nearly tetrahedral arrangements is somehow induced by proximity to the surface. The vicinal water thus represents an enhancement of the ice-like structure. Etzler then calculated the density of the water in silica pores. This was... 

In order to gain insight into this matter, calculations of the vibration-rotation parameters [5, 6] were performed for several linear XYZ and YXY molecules using model anharmonic force fields and conventional first-order theory [41, 42, 55, 56]. With this procedure,... 

Transition properties first-order theory of oscillator strengths (FOTOS)... 

There is a special trap that lurks in analysis. Many spectra appear to be clearly first order, with insignificant mixing of states. These spectra invite analysis by first-order theory. It is, however, well known that this may be deceptive.6 7 Figure 2 illustrates this point. Displayed is the calculated spectrum for the XCH2CH2Y molecule with the parameters Ax 5.0, /AX = 7.0,/AA = -10.0,/xx -16.0, an entirely reasonable set of parameters for such a molecule. The spectrum, however, clearly invites the interpretation Ax = Ax = 6.0 Hz. In those cases where first-order analysis is applied, it is appropriate to ask, Would exact analysis yield a different answer, or at least much larger error limits "... 

If the transition considered is the HOMO LUMO transition of an alternant hydrocarbon, then first-order theory predicts that inductive perturbation will have no effect at all, because for = fo as a consequence of the pairing theorem. Small red shifts are in fact observed that can be attributed to hyper conjugation with the pseudo-7t MO of the saturated alkyl chain.290 On the other hand, alkyl substitution gives rise to large shifts in the absorption spectra of radical ions of alternant hydrocarbons whose charge distribution is equal to the square of the coefficients of the MO from which an electron was removed (radical cations) or to which an electron was added (radical anions), and these shifts are accurately predicted by HMO theory.291... 

In a first order theory, the polarization P should be proportional to the distortion ... 

Meyer s idea of flexoelectricity has been generalized to include a contribution due to the gradient of the orientational order parameter. The polarization in this case arises not from the curvature distortion of the director but from the spatial variation of the degree of orientational order of the molecules. In a simple first order theory, one may take P oc V5, where s is the order parameter as defined in 2.3.1. This effect has been termed as order electricity . 

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