The second-order term in equation

After the application of Green s theorem to the second order term in Equation (2.81) we get the weak form of the residual statement as... 

It was suggested48 that the second-order term in equation (35) corresponds to a mechanism in which the complex tra .y-(PtL2MeX) is converted to a six-coordinate platinum(IV) complex by attack of H+ and a solvent molecule, S. The six-coordinate complex then slowly eliminates methane to yield tra s-(PtL2XS) and finally this latter complex presumably is converted to fra/ j-(PtL2XCl) in a rapid substitution reaction, viz. 

Values of 2bs and k3bs are given in Table 20, and it may be seen that, whereas only the second-order term in equation (40) is of importance for substitution of tetramethyltin and tetraethyltin, yet the third-order term is of as equal importance as the second-order term for the substitution of tetraisopropyltin. The value of k2bs obtained in the latter case (0.35) is naturally rather different to the original value of 1.61 l.mole-1.sec-1 given in Table 19. However, the values of k2bs given in Table 19 for the /i-tetraalkyls are quite valid. 

The brominolysis of a number of unsymmetrical tetraalkyltins was also studied35 and second-order rate coefficients for the cleavage of given alkyl-tin bonds determined by combination of rate studies with product analyses. Details are given in Table 21 and it should be noted that for all of the unsymmetrical tetraal-kyls listed in this table only the second-order term in equation (40) is of importance35. It is convenient now to discuss the two terms in equation (40) separately. 

Gielen and co-workers35 suggested that the second-order term in equation (40) corresponds to a mechanism in which a five-co-ordinate tin complex (XI) is formed as an intermediate. It seems evident that (XI) can be present in but very... 

The second approximation is to neglect the second-order term in Equation A.95. This gives... 

Experimental results confirm this linear relationship. An example is depicted in Figure 1. The second order term in Equation (18), we approximate by a second order polynomial in blend composition W. It follows with respect to Equation (21) and formulated in concentratiorrs c... 

The second order terms in equation A3 are reduced to first order by the introduction of a new dependent variable... 

In the modified Ridd mechanism for region B the deprotonation of the A-nitroso-anilinium ion Ar —NH2NO in Scheme 3-23 is rapid, and therefore does not influence the overall rate. However, the second-order term in the rate equation for region C (Scheme 3-25) is consistent with a mechanism in which the deprotonation of the A-nitrosoanilinium ion (Scheme 3-24) and of the C-nitrosoanilinium dication (Scheme 3-22) belongs to the rate-determining part of the reaction. 

For turbulent flow in single-phase systems, the predicted temperature profile is not changed significantly if the Peclet number is assumed to be infinite. Therefore, in turbulent two-phase systems the second-order terms in Eqs. (9) probably do not have a significant effect on the resulting temperature profiles. In view of the uncertainties in the present state of the art for determining the holdups and the heat-transfer coefficients, the inclusion of these second-order terms is probably not justified, and the resulting first-order equations should adequately model the process. 

Second-order correction can be implemented in a similar way. Let us illustrate a simple method for the exponential law. Retaining the second-order term of Equations (17) and (19), we obtain ... 

The third-order term in equation (40), p. 157, was suggested to correspond to a mechanism in which the intermediate (XI) is now attacked by a second bromine molecule, viz. 

As written, equation 22 depicts a simplified picture in which a single field, E(co,t) acts on the material. The general picture of second-order NLO involves the interaction of two distinct waves with electric fields Ei and E2 with the electrons of the NLO material. Suppose for example that we use two laser beams with different frequencies. The second-order term of equation 23 with two interacting waves of amplitudes Ei and E2 at an arbitrary point in space becomes ... 

In conformity with the significance of the terms employed by investigators of anisotropy (Tsvetkov et al. 1964), the effects associated with the first-order terms in equation (10.6) may be called the effects of intrinsic anisotropy, while the second-order effects may be referred to as the effects of mutual interaction. In the second approximation, the principal axes of the relative permittivity tensor do not coincide, generally speaking, with the principal axes of the orientation tensor. It is readily seen that interesting situations may arise when Aa < 0 in this case, the coefficients of the first- and second-order terms have different signs. 

This is the pressure equation for a one-component system of spherical particles obeying the pairwise additivity for total potential energy. Note that the first term is the ideal gas pressure. The second is due to the effect of the inter-molecular forces on the pressure. Note that in general, g(R) is a function of the density hence, the second term in (3.37) is not the second-order term in the density expansion of the pressure. 

The last four terms depend only on the reference density, po, and represent the repulsive energy contribution, Flcp, discussed above. Thus, we just have to deal with the second-order terms. The second-order term in the charge density fluctuations dp(r), that is, the second term in Equation 5.51, is approximated by writing Ap as a superposition of atomic contributions, Ap0 = Apv. This approach decays quickly with the increasing distance from the corresponding center. To simplify the second term further, Elstner applied a monopole approximation ... 

For consistent expansion, is necessary the second order term in space. Momentum equations of f are expanded directly an addition the next form... 

At r = 300 K, fcr 3 X lO J, which is an order of magnitude less that the dispersion contribution. The actual difference between the two terms (dispersion and electrostatic) will be reduced by mathematical cancellations in the second (dispersion) term in Equation (4.47), but only rarely will the electrostatic contribution constitute the dominant factor in the total interaction. The presence of imaginary frequencies in the second term may cause some problems in terms of physical concepts of the processes involved however, their use is actually a result of mathematical manipulations (i.e., tricks) that disappear as one works through the complete calculation. 

Although the possibility of the order-disorder transition was recognized in most of the block copolymer theories, it is Leibler who has expressedly addressed this problem. He derived the free energy of a block copolymer system in a series expanded in powers of the order parameter j denoting the deviation of the local density from the mean. The coefficients of this expansion up to the fourth ordef term were evaluated by a method which is a generalization of the random phase approximation method described above (Equation (16) was, in fact, derived as the second order term in the... 

Before discussing the discrepancies between Eqs. (7.104) and (5.21) it would be in order to comment further on the nature of Eq. (7.104). This equation in its general form is the so-called telegraphist s equation. The unique feature of this equation is that it describes physical phenomena which exhibit both wavelike characteristics and residual disturbance effects. The wavelike behavior is given by the second-order term in <, and the residual disturbance effect, by the first-order term in t. Wave effects propagate with a finite velocity, and time-dependent neutron phenomena should properly include such a term since time-wise variations in neutron population cannot be sensed at a given distance from the source in less time than it takes a neutron of speed v to reach that point. However, after the passage of this wave, the disturbance persists and this effect is described by the first-order term. 

The first integral on the right-hand side of this equation vanishes since H t) = 0. If the time t is such that a t) differs slightly from fr(0), then only the second-order term in the expansion is kept. Next fr(0) is replaced in the bracket by d(0) d(oo), where fr(oo), the equilibrium density matrix, is given by... 

There are two parts to the expression for the second-order term in the energy of equation 7.8, Ep (q ). I i d H/dq Pj) is the classical force constant which describes motion of the nuclei in the electronic state i), namely, the frozen electronic charge distribution of the undistorted, q = 0, structure. The second term, Em- — °) is always negative (i.e., stabilizing) since repre-... 

Only the second-order terms are included here - the first-order terms vanish for closed-shell systems, and the higher order terms are exceedingly small (because of the smallness of the magnetic perturbations) and can also be neglected. Of the three second-order terms in equation (11), the first represents the direct interaction of the molecular system with the external field and is represented by the magnetizability tensor at zero field ... 

The traditional explanation for the occurrence of the g = 4.3 line goes back to the classic paper by Castner et al (1960), where only second order terms in equation (22) are considered. Castner et al (1960) considered the limiting case of extreme rhombicity as governed by the condition D = 0, E 0 and possible symmetry arrangements which could lead to its fulfillment. It can easily be shown by a permutation of axes that an exactly equivalent condition is E/D < 1/3. From many points of view it is useful to maintain the convention, E/D < 1/3, which covers all possible situa-... 

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