First-term approximation

The first term of this series is the rate of rate-limiting step calculated at the equilibrium of the rest of reactions. This first term approximation is widely applied in heterogeneous catalysis. We have derived the following explicit formula (in the assumptions of the Basic case-, see Lazman and Yablonskii, 1988). 

Figures 10-13 compare the exact dependencies of the (feasible) reaction rate and their first term approximation (i.e. R — —(fco)/(fci)) as well as approximation corresponding to tn — 3 ...

Mathematically, any function can be estimated by a series of approximations referred to a Taylor series expansion. Each approximation or term of the Taylor series is based on a corresponding derivative. For a bond, duration is the first-term approximation of the price change and is related to the first derivative of the bond s price with respect to a change in the required yield. The convexity measure is the second approximation and related to the second derivative of the bond s price. 

Fig. 6.5 The form of the resistive force // kT/I) vs. the fractional extension r/(nl) for a randomly advancing chain (eqs. (6.48) and (6.49) the dashed line shows the first-term approximation of eq. (6.49)) (from Treloar (1975) courtesy of Clarendon Press).

Where R is the rate matrix that describes the NOE interactions across the system, and are the two NOE mixing and A(0) is the initial magnetization. To simplify this equation, a Taylor series expansion of the exponential can be made. Usually, only the first few terms in the expansion are kept for the approximation. The first term approximation is equivalent to the two-spin approximation (5,14). At realistic mixing times (50 ms or more), the Taylor series approximation also yields systematic error in determining the inter-proton distances (9). Figure 1 shows comparison of volumes simulated from the two-term Taylor series approximation and an exact rate-matrix calculation for the Dickerson dodecamer... 

Now in the present case i/a 1, so it follows from equation (A.1.7) that G/a 1, provided f differs significantly from zero. Thus che first term on the right hand side of (A.L.8) is a close approximation to the familiar Poisoiille flux. The second term, on the other hand, represents thermal transpiration. In particular, setting N 0, we find... 

Those involving series truncation. The quantity In (1 - X2) can be represented by the infinite series - [x2 + (1/2) x + (1/3) x - - ]. Truncating this series after the first term is a valid approximation for dilute solutions and also simplifies the form of the equation. It is an optional step, however, and can be avoided or mitigated by simply retaining more terms in the series. 

The first term on the right side of Eq. (5-179) is so nearly dominant for most furnaces that consideration of the main features of chamber performance is clarified by ignoring the loss terms and Lr or by assuming that they and have a constant mean value. The relation of a modified chamber efficiency T g(1 o) lo modified firing density D/(l — and to the normahzed sink temperature T = T-[/Tp is shown in Fig. 5-23, which is based on Eq. (5-178), with the radiative and convective transfer terms (GSi)/ja(TG — T ) -i- hiAijTc Ti) replaced by a combined radiation/conduction term (GS,) ,a(T - T ). where (GS])/ = (GS])/ + /jiA]/4oTgi Tg is adequately approximated by the arithmetic mean of Tg and T. ... 

Sethna [1981] considered two limiting cases. The calculation of action in the fast flip approximation (a>j CO ) proceeds by utilizing the expansion exp ( — cu,-1t ) 1 — cu t. After substituting the first term, i.e. the unity, in (5.72) we get precisely the quantity which yields the Franck-Condon factor in the rate constant. The next term cancels the adiabatic renormalization and changes KM)... 

Equation 6-107 gives the total energy loss in fixed beds as the sum of viseous energy loss (the first term on the right side of the equation) and the kinetie or turbulent energy loss (the seeond term on the right side of the equation). For gas systems, approximately 80% of the energy loss depends on turbulenee and ean be represented by the seeond term of Equation 6-107. In liquid systems, the viseous term is the major faetor. 

In the linear or first-order approximation, it is postulated that these activity coefficient terms are directly proportional, as in Eq. (8-92) ... 

The first term is referred to as the diamagnetic contribution, while the latter is the paramagnetic part of the magnetizability. Each of the two components depend on the selected gauge origin however, for exact wave functions these cancel exactly. For approximate wave functions this is not guaranteed, and as a result the total property may depend on where the origin for the vector potential (eq. (10.61)) has been chosen. 

If we are comparing reactions which have approximatively the same steric requirements, the first term is roughly constant. If the species are very polar the second term will dominate, and the reaction is charge controlled. This means for example that an electrophihc attack is likely to occur at the most negative atom, or in a more general sense, along a path where the electrostatic potential is most negative. If the molecules are non-polar, the third term in (15.1) will dominate, and the reaction is orbital controlled. 

This is the Verlet algorithm for solving Newton s equation numerically. Notice that the term involving the change in acceleration (b) disappears, i.e. the equation is correct to third order in At. At the initial point the previous positions are not available, but may be estimated from a first-order approximation of eq. (16.29). 

Mean Field Approximation as a first order approximation, we will ignore all correlations between values at different sites and parameterize configurations purely in terms of the average density at time t p. The time evolution of p under an arbitrary rule [Pg.73]

This expression is exact within our original approximation, where we have neglected relativistic effects of the electrons and the zero-point motions of the nuclei. The physical interpretation is simple the first term represents the repulsive Coulomb potential between the nuclei, the second the kinetic energy of the electronic cloud, the third the attractive Coulomb potential between the electrons and the nuclei, and the last term the repulsive Coulomb potential between the electrons. 

In the ordinary Hartree-Fock scheme, the total wave function is approximated by a single Slater determinant and, if the system possesses certain symmetry properties, they may impose rather severe restrictions on the occupied spin orbitals see, e.g., Eq. 11.61. These restrictions may be removed and the total energy correspondingly decreased, if instead we approximate the total wave function by means of the first term in the symmetry adapted set, i.e., by the projection of a single determinant. Since in both cases,... 

This equation is the first term of an infinite series which appears in the rigorous solution of the quasi-diffusion. This equation describes the regular process of quasi-diffusion. For the low values of the Fourier number (irregular quasi-diffusion) it is necessary to use Eq. (5.1) or Boyd-Barrer approximation [105, 106] for the first term in Eq. (5.1)... 

With x = 1, Eq. (27) gives directly the relationship being sought. With x = 2 and x = 3, the required expression for (nBt/nam) is inserted from Eq. (26), since the value of em is surely such as to permit both the neglect of E,(e0) and the approximation by the first term of the expansion (25). Finally one gets... 

The first order approximation may be found by assuming p and A to be small, but not zero. If Eq. (1-86) is multiplied by p, all coefficients on the left side, being proportional to some power of i, give terms... 

It is important to note that in all these methods, the first term in the series solution constitutes the so-called approximation of zero order. This is generally the solution of a simple linear problem e.g., the harmonic oscillator the second term appears as the first approximation, and so on. The amount of labor increases very rapidly with the order of approximation, but the additional information obtained from approximations of higher orders (beginning with the second) does not increase our knowledge from the qualitative point of view. It merely adds small quantitative corrections to the first approximation, and in most applied problems, these corrections are scarcely worth the considerable complication in calculations. For that reason the first approximation is generally sufficient in exploring a new problem, or in investigating the qualitative aspect of a phenomenon. 

Suppose at(w,0) kl > only the hP1 state of the system is present initially. The first term on the right of Eq. (7-73) can then contribute nothing to any state l that differs from k it will induce no transitions from the initial state. Retaining it alone constitutes the adiabatic approximation. The second term contributes to at(w,t) provided (il)uc is finite. It is the first diabatic term in the expansion. 

The new pathway, too, is a chain reaction Note that the first term of Eq. (8-31) does not give a meaningful transition state composition. Since the scheme in Eqs. (8-20M8-23) seems valid for the Cu2+-free reaction, we can seek to modify it to accommodate the new result. This approach is surely more logical than inventing an entirely new sequence. To arrive at the needed modification, we simply replace Eq. (8-23) by a new termination step, Eq. (8-30). With that, and the steady-state approximation, the rate law is... 

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