The Derivative of a Constant

The first term ( = 0) in the summation on the right-hand side vanishes because it is the derivative of a constant. The exponential on the left-hand side is the generating function g(, s), for which equation (D.l) may be used to give... 

When in some range qu(P/P0) becomes independent of P/P0 (i.e., a condition of measurement of the micropore volume) the differentiation of Equation 9.22 withdraws qfi(P/P()) from the consideration (the derivative of a constant parameter is zero). Thus, the result of the measurement (of course under the named assumption) does not depend on the absence or presence of micropoies at all. To the contrary, the BET method assumes that the entire adsorbed gas lays on a flat surface, including the portion adsorbed volumetrically (i.e., by different mechanism) in micropoies. 

The first formula means that the derivative of a constant is zero. Eq. (6.21) is also valid for fractional or negative values of n. Thus, we find... 

Intuitively, all vapor liquid equilibrium models obey this relationship in positive composition space, but in order to be sure that it is useful, and valid, to analyze negative compositions, the Gibbs Duhem relationship should be maintained for negative composition too. Since 7/= I for the ideal and constant relative volatility cases, this condition is always valid for the entire composition spectrum, since the derivative of a constant is zero. Fortunately, it can be shown that all other models where y, is a function of composition (NRTL, Wilson, and so on) obey this rule, except in areas of discontinuity. Therefore, it is safe and thermodynamically sound, to analyze and interpret these negative compositions as they still obey fundamental thermodynamic rules. 

When redefining variables in this way, one must be sure that the original defining equation is unchanged. Thus, since the derivative of a constant (r .) is always zero, then Eq. 1.31 for the new dependent variable 0 is easily seen to be unchanged... 

When the limits are not specified, the integral is an indefinite integral. The derivative of a constant C is zero. Therefore, when we take the derivative of a function /(x) = g x) + C, all the information about... 

The rate of a process is expressed by the derivative of a concentration (square brackets) with respect to time, d[ ]/dt. If the concentration of a reaction product is used, this quantity is positive if a reactant is used, it is negative and a minus sign must be included. Also, each derivative d[ ]/dt should be divided by the coefficient of that component in the chemical equation which describes the reaction so that a single rate is described, whichever component in the reaction is used to monitor it. A rate law describes the rate of a reaction as the product of a constant k, called the rate constant, and various concentrations, each raised to specific powers. The power of an individual concentration term in a rate law is called the order with respect to that component, and the sum of the exponents of all concentration terms gives the overall order of the reaction. Thus in the rate law Rate = k[X] [Y], the reaction is first order in X, second order in Y, and third order overall. 

The assumptions of transition state theory allow for the derivation of a kinetic rate constant from equilibrium properties of the system. That seems almost too good to be true. In fact, it sometimes is [8,18-21]. Violations of the assumptions of TST do occur. In those cases, a more detailed description of the system dynamics is necessary for the accurate estimate of the kinetic rate constant. Keck [22] first demonstrated how molecular dynamics could be combined with transition state theory to evaluate the reaction rate constant (see also Ref. 17). In this section, an attempt is made to explain the essence of these dynamic corrections to TST. 

The foundation of our approach is the analytic calculations of the perturbed wave-functions for a hydrogenic atom in the presence of a constant and uniform electric field. The resolution into parabolic coordinates is derived from the early quantum calculation of the Stark effect (29). Let us recall that for an atom, in a given Stark eigenstate, we have ... 

The theoretical approach involved the derivation of a kinetic model based upon the chiral reaction mechanism proposed by Halpem (3), Brown (4) and Landis (3, 5). Major and minor manifolds were included in this reaction model. The minor manifold produces the desired enantiomer while the major manifold produces the undesired enantiomer. Since the EP in our synthesis was over 99%, the major manifold was neglected to reduce the complexity of the kinetic model. In addition, we made three modifications to the original Halpem-Brown-Landis mechanism. First, precatalyst is used instead of active catalyst in om synthesis. The conversion of precatalyst to the active catalyst is assumed to be irreversible, and a complete conversion of precatalyst to active catalyst is assumed in the kinetic model. Second, the coordination step is considered to be irreversible because the ratio of the forward to the reverse reaction rate constant is high (3). Third, the product release step is assumed to be significantly faster than the solvent insertion step hence, the product release step is not considered in our model. With these modifications the product formation rate was predicted by using the Bodenstein approximation. Three possible cases for reaction rate control were derived and experimental data were used for verification of the model. 

Optimal sampling. As was pointed out earlier, the error in the derivative of A is proportional to ak j Jnk(N). The optimal sampling is therefore obtained when ak/ Jnk(N) is constant as a function of k. In regions where ak is large, additional sample points should be added to compensate. This is often a small effect but in some special cases is worth considering. In order to obtain the optimal sampling, the potential energy should be corrected as... 

We have derived previously [4, 5] that the following expression relates the noise on data to the noise of a constant multiple of that data ... 

It is important to mention that when one considers the derivative of a quantity at constant external potential, it means that the changes in the quantity are analyzed for a fixed position of the nuclei. These types of changes are known as vertical differences, precisely because the nuclei positions are not allowed to relax to a new position associated with a lower total energy. 

The Fukui function is primarily associated with the response of the density function of a system to a change in number of electrons (N) under the constraint of a constant external potential [v(r)]. To probe the more global reactivity, indicators in the grand canonical ensemble are often obtained by replacing derivatives with respect to N, by derivatives with respect to the chemical potential /x. As a consequence, in the grand canonical ensemble, the local softness sir) replaces the Fukui function/(r). Both quantities are thus mutually related and can be written as follows ... 

In Eq. (5.26), Tt is the interfacial pressure of the aqueous-organic system, equal to (Yo - Y) e to the difference between the interfacial tensions without the extractant (Yo) and the extractant at concentration c (y)], c is the bulk organic concentration of the extractant, and is the number of adsorbed molecules of the extractant at the interface. The shape of a typical n vs. In c curve is shown in Fig. 5.4 rii can be evaluated from the value of the slopes of the curve at each c. However, great care must be exercised when evaluating interfacial concentrations from the slopes of the curves because Eq. (5.26) is only an ideal law, and many systems do not conform to this ideal behavior, even when the solutions are very dilute. Here, the proportionality constant between dHld In c and is different from kT. Nevertheless, Eq. (5.26) can still be used to derive information on the bulk organic concentration necessary to achieve an interface completely saturated with extractant molecules (i.e., a constant interfacial concentration). According to Eq. (5.26), the occurrence of a constant interfacial concentration is indicated by a constant slope in a 11 vs. In c plot. Therefore, the value of c at which the plot n vs. In c becomes rectilinear can be taken as the bulk concentration of the extractant required to fully saturate the interface. 

One can easily note that Eq. (4.138) is similar to the solution given by Eq. (4.126), which is derived from the assumption of a constant friction and complete neglect of the Poisson expansion. The solution for Zma, which is the shortest bond length required to maintain a stable debonding process, is obtained from Eq. (4.137)... 

Since all electrophoretic mobility values are proportional to the reciprocal viscosity of the buffer, as derived in Chapter 1, the experimental mobility values n must be normalized to the same buffer viscosity to eliminate all other influences on the experimental data besides the association equilibrium. Some commercial capillary zone electrophoresis (CZE) instruments allow the application of a constant pressure to the capillary. With such an instrument the viscosity of the buffer can be determined by injecting a neutral marker into the buffer and then calculating the viscosity from the time that the marker needs to travel through the capillary at a set pressure. During this experiment the high voltage is switched off. 

The best correlations were obtained with the values of a- constants generally applied to correlating chemical reactions which involve the phenolic oxygen.89 It is typical of the adverse influence of substituents that Eqs. (1) and (2) are significantly different in p values. The two nitro derivatives in both do not lie on the correlation lines. It was found that nitro-substituted aryl esters preferentially undergo alcoholysis and reduction.70... 

Box and McKellar (1978) derived the sum rule (4.81) under the assumption of a constant refractive index and within the framework of the anomalous diffraction approximation of van de Hulst (1957, Chap. 11). 

The relationship between the mean current and the potential (Fig. 7.99) will now be derived. Suppose that at the drop/solution interface, an electronation reaction A + ne —> D is driven by the imposition of a constant potential, E. The reaction results in the depletion of A in the interfacial region, and therefore in the diffusion of A toward the drop/solution interface. Let it be assumed that the species D produced by the... 

To see why this is a general recipe for locating turning points in the locus we can argue as follows. At any point along the locus, the function F has the same constant value (zero). Thus the derivative of F along the stationary-state curve, like the derivative of any constant, must be zero dF = 0. We... 

However this would necessitate the determination of the derivative of a with respect to Z. This quantity cannot be measured with enough accuracy, especially when the association constant KA is small (this is the case in the high dielectric constant values of the solvents studied D 30). We have made two conductance runs, in a 59.02% water-THF mixture and in a 81.90% water-AC... 

A/iAg as a function of time with a single and spatially fixed sensor at , or one can determine D with several sensors as a function of the coordinate if at a given time [K.D. Becker, et al. (1983)]. An interesting result of such a determination of D is its dependence on non-stoichiometry. Since >Ag = DAg d (pAg/R T)/d In 3, and >Ag is constant in structurally or heavily Frenkel disordered material (<5 1), DAg(S) directly reflects the (normalized) thermodynamic factor, d(pAg/R T)/ In 3, as a function of composition, that is, the non-stoichiometry 3. From Section 2.3 we know that the thermodynamic factor of compounds is given as the derivative of a point defect titration curve in which nAg is plotted as a function of In 3. At S = 0, the thermodynamic factor has a maximum. For 0-Ag2S at T = 176 °C, one sees from the quoted diffusion measurements that at stoichiometric composition (3 = 0), the thermodynamic factor may be as large as to 102-103. 

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