Eliminating first-order terms

We now substitute eqns 2, 3, and 4 into eqn 1, eliminate second order terms in AC, multiply on the left by ct, and separate the resulting equation into two as follows. We find, first of all ... 

It is seen that k varies little with solvent changes, while larger variations are found with ki. The small variations in k may be due to cross-termination or in instances where radical traps were used, to further induced decomposition derived from these traps. Also the employment of low peroxide concentrations may not completely eliminate induced decomposition. With phenol as the solvent, a large deviation in from other values is observed. This value is suspect. It should be noted that low concentrations of peroxide were used with this solvent and it was assumed that the observed rate coefficient corresponded to k. The activation parameters in terms of induced and unimolecular decomposition are given in Table 71. Activation parameters for the first-order term are not significantly influenced by changes in solvent polarity. 

Clearly, these quantities depends on the choice of the correlation volume if is very large, then the local composition should approach the bulk composition hence, the PS should approach zero. Therefore, in order to eliminate the dependence on Vcor and obtain an intrinsic PS measure of A and B, the suggestion has been made (Ben-Naim 1988,1990b, 1992, 2006) of introducing the first-order term in the expansion of 5A,B(Vcor) in power series around KoJ. The coefficient of the first-order term in the expansion is (Ben-Naim 2006)... 

A glance at (2.50) shows that j(i ) is simply the second factor in (2.50), which includes the density expansion of g(i ). The elimination of the factor exp[--jfff/(7 )] appearing in (2.50) is often useful, since the remaining function y(R) becomes everywhere an analytic function of R even for hard spheres. We illustrate this point by considering only the first-order term in the expansion (2.50) for hard-sphere particles, where we have... 

The operator T, converts the dipole moment p,- at site / (composed of permanent and induced moments as given by (4.20) into the electric field produced by it at site j. Elimination of F, between (4.20) and (4.21) and retaining only first-order terms in E gives... 

The derivation proceeds as in ref. [Cl 85] except that the following transformation is introduced, which eliminates first derivative terms in the second-order equation ... 

On subsciCuLlng (12.49) into uhe dynamical equations we may expand each term in powers of the perturbations and retain only terms of the zeroth and first orders. The terms of order zero can then be eliminated by subtracting the steady state equations, and what remains is a set of linear partial differential equations in the perturbations. Thus equations (12.46) and (12.47) yield the following pair of linearized perturbation equations... 

Under steady-state conditions, variations with respect to time are eliminated and the steady-state model can now be formulated in terms of the one remaining independent variable, length or distance. In many cases, the model equations now result as simultaneous first-order differential equations, for which solution is straightforward. Simulation examples of this type are the steady-state tubular reactor models TUBE and TUBED, TUBTANK, ANHYD, BENZHYD and NITRO. 

A typical drug clearance curve is shown in Fig. 5.9. The curve in Fig. 5.9 is a first-order curve that is, the elimination rate is proportional to the amount of drug in the bloodstream. As the amount of drug in the blood is reduced, the elimination rate is also reduced. Another term often used is half-fife. This is the time taken to clear half (50%) of the remaining drug in the body. Mathematically, it is given by... 

Pseudo-first-order rate constants for carbonylation of [MeIr(CO)2l3]" were obtained from the exponential decay of its high frequency y(CO) band. In PhCl, the reaction rate was found to be independent of CO pressure above a threshold of ca. 3.5 bar. Variable temperature kinetic data (80-122 °C) gave activation parameters AH 152 (+6) kj mol and AS 82 (+17) J mol K The acceleration on addition of methanol is dramatic (e. g. by an estimated factor of 10 at 33 °C for 1% MeOH) and the activation parameters (AH 33 ( 2) kJ mol" and AS -197 (+8) J mol" K at 25% MeOH) are very different. Added iodide salts cause substantial inhibition and the results are interpreted in terms of the mechanism shown in Scheme 3.6 where the alcohol aids dissociation of iodide from [MeIr(CO)2l3] . This enables coordination of CO to give the tricarbonyl, [MeIr(CO)3l2] which undergoes more facile methyl migration (see below). The behavior of the model reaction closely resembles the kinetics of the catalytic carbonylation system. Similar promotion by methanol has also been observed by HP IR for carbonylation of [MeIr(CO)2Cl3] [99]. In the same study it was reported that [MeIr(CO)2Cl3]" reductively eliminates MeCl ca. 30 times slower than elimination of Mel from [MeIr(CO)2l3] (at 93-132 °C in PhCl). 

The pharmacokinetic term clearance (CT) best describes the efficiency of the elimination process. Clearance by an elimination organ (e.g., liver, kidney) is defined as the volume of blood, serum, or plasma that is totally cleared of drug per unit time. This term is additive the total body or systemic clearance of a drug is equal to the sum of the clearances by individual eliminating organs. Usually this is represented as the sum of renal and hepatic clearances CT = CT renal -I- CL hepatic. Clearance is constant and independent of serum concentration for drugs that are eliminated by first-order processes, and therefore may be considered proportionally constant between the rate of drug elimination and serum concentration. 

The hydration of an aldehyde RCHO can be expressed with two (pseudo-first-order rate laws consisting of an elimination (minus sign) and a production (plus sign) term ... 

Assuming that no significant in-situ degradation of PCBs occurs (k htm = photo = Kio = 0 ) three elimination pathways remain which, if described in terms of first-order reaction rates, can be directly compared with respect to their relative importance for the elimination of each PCB congener from the water column. As shown by the removal rates listed in Table 23.4, for both compounds the flux to the atmosphere is by far the most important process. Because of its larger Kd value, removal of the heptachlorobiphenyl to the sediments is predicted to be also of some importance. By the way, from this simple model we would expect to find the heptachlorobiphenyl relatively enriched in the sediments compared to the trichloro-biphenyl. We shall see later whether this is true. 

However, for the enumerations considered in the present paper no such complications arise and it is reasonable to rely on the results of extrapolations. The suggestion made by Flory and Fisk31 that the exact enumerations in the range n = 1 to 14 are dominated by immediate reversals is incorrect. The discussion of Section IV-G indicates that the nth step in the enumeration eliminates the effect of polygons of order n, and the contribution of the nth stage is of order l/nc. Thus the exact enumeration procedure is equivalent to assessing the asymptotic behavior of l/nc from the first 14 terms. 

G( ) is the transfer function relating 0O and 0X. It can be seen from equation 7.18 that the use of deviation variables is not only physically relevant but also eliminates the necessity of considering initial conditions. Equation 7.19 is typical of transfer functions of first order systems in that the numerator consists of a constant and the denominator a first order polynomial in the Laplace transform parameter s. The numerator represents the steady-state relationship between the input 0O and the output 0 of the system and is termed the system steady-state gain. In this case the steady-state gain is unity as, in the steady state, the input and output are the same both physically and dimensionally (equation 7.16h). Note that the constant term in the denominator of G( ) must be written as unity in order to identify the coefficient of s as the system time constant and the numerator as the system... 

Then the differences in rate caused by the electronic effect of the substituent are correlated by the Hammett equation log(kz/kH) = poz, where kz is the rate constant obtained for a compound with a particular meta or para substituent, ku is the rate constant for the unsubstituted phenyl group, and crz is the substituent constant for each substituent used. The proportionality constant p relates the substituent constant (electron donating or wididrawing) and the substituent s effect on rate. It gives information about the type and extent of charge development in the activated complex. It is determined by plotting log(kz/kQ) versus ov for a series of substituents. The slope of the linear plot is p and is termed the reaction constant. For example, the reaction shown above is an elimination reaction in which a proton and the nosy late group are eliminated and a C-N n bond is formed in their place. The reaction is second order overall, first order in substrate, and first order in base. The rate constants were measured for several substituted compounds ... 

The minus sign and prime on reabsorption indicate a different driving force concentration than for filtration and secretion and drug transport in the opposite direction. Elimination is the generic term given to the first-order rate constant K, or sometimes [h describing the parent drug lost by both metabolism km and excretion ke (Eq. 1.17) ... 

This equation assumes a one-compartment model and constant, first-order elimination. Based on Equation 7.17, the e kt term approaches 0 as t increases, and Cp approaches Rmi/kx]Vd. The value of 7 inf/A eiVd corresponds to Cpss (Equation 7.18). 

Valid values of F fall between 0 and 1. The term FD0/ Vd is related to the concentration of drug (same units as Cp) theoretically available to the bloodstream in the absence of elimination. Equation 7.21 assumes first-order elimination and a single compartment... 

Description of the Model. The Keys et al. (1999) model simulates six tissue compartments small intestine, blood, liver, testis, slowly perfused tissues, and poorly perfused tissues. Conversion of DEHP to MEHP in the small intestine is simulated with km and Vmax terms, whereas conversion in liver and blood are simulated with separate first order rate constants, kj and kb, respectively. Elimination of DEHP and MEHP is assumed to be entirely by metabolism of DEHP to MEHP, and MEHP to unspecified metabolites the latter transformation is represented in the model by km and Vmax terms. 

The mechanisms of the reductive eliminations in Scheme 5 were studied [49,83], and potential pathways for these reactions are shown in Scheme 6. The reductive eliminations from the monomeric diarylamido aryl complex 20 illustrate two important points in the elimination reactions. First, these reactions were first order, demonstrating that the actual C-N bond formation occurred from a monomeric complex. Second, the observed rate constant for the elimination reaction contained two terms (Eq. (49)). One of these terms was inverse first order in PPh3 concentration, and the other was zero order in PPh3. These results were consistent with two competing mechanisms, Path B and Path C in Scheme 6, occurring simultaneously. One of these mechanisms involves initial, reversible phosphine dissociation followed by C-N bond formation in the resulting 14-electron, three-coordinate intermediate. The second mechanism involves reductive elimination from a 16-electron four-coordinate intermediate, presumably after trans-to-cis isomerization. 

Kominar and coworkers115 used the toluene carrier technique for the pyrolysis of bromobenzene. The elimination process was found to be homogeneous and of first order. The Arrhenius expression for equation 26 is log kx (s-1) = (14.6 0.3) - [(317.000 5.4550)/19.157]. The result of this pyrolysis was discussed in terms of the following mechanism (equations 26-31) ... 

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