Replacement equations

Rather than solve a Schrodinger equation with the Nuclear Hamiltonian (above), a common approximation is to assume that atoms are heavy enough so that classical mechanics is a good enough approximation. Motion of the particles on the potential surface, according to the laws of classical mechanics, is then the subject of classical trajectory analysis or molecular dynamics. These come about by replacing Equation (7) on page 164 with its classical equivalent ... 

Similarly, when the pressure-volume relation is Pvk — constant, equation 4.30 replaces equation 6.26 ... 

These component balances are conceptually identical to a component balance written for a homogeneous system. Equation (1.6), but there is now a source term due to mass transfer across the interface. There are two equations (ODEs) and two primary unknowns, Og and a . The concentrations at the interface, a and a, are also unknown but can be found using the equilibrium relationship, Equation (11.4), and the equality of transfer rates. Equation (11.5). For membrane reactors. Equation (11.9) replaces Equation (11.4). Solution is possible whether or not Kjj is constant, but the case where it is constant allows a and a to be eliminated directly... 

Replacing equation 25 into 11 an expression for the efficiency is obtained. 

This formula replaces Equation 16.17 in the calculation procedure. 

If f(x) is not given by a formula, or the formula is so complicated that analytical derivatives cannot be formulated, you can replace Equation (5.5) with a finite difference approximation... 

A more satisfactory definition is that of a dimensionless specific speed N s. The above deficiency can be removed by replacing equation 4.20 by equation 4.22 ... 

In the case of a many-body problem, the Bohmian mechanics for an A-body dynamics follows from the one for a single system, but replacing Equation 8.25 by... 

The correct derivation of the remaining working equations involves replacing equation 13.1, which only takes into account the thermal volume increase, with equation 13.10, where both contributions are considered. The thermal volume change in equations 13.1 and 13.2 was renamed A v to distinguish it from the intrinsic (or chemical ) volume change Ac lcmv. 

If analytic equations for the heat capacities of reactants and products are unavailable, we still can carry out the integration required by Equation (4.78) by graphical or numerical methods. In essence, we replace Equation (4.79) by the expression... 

Because the electrolyte H2C03(a) does not directly influence the acidity of the aqueous phase, it is preferable to avoid the lack of precision inherent in the definition of the form of the aqueous CO2 by replacing equation 8.76 by... 

However, in Figure 5 and Figure 6 the intercept b is not included since the values of b cannot be correlated easily with the surface adsorption effect or any known effects of the experimental variables. Therefore, the following expression has been chosen for D to replace Equation 15 in order to use the generalized correlations and to examine their applicabilities and the predictability of the theory proposed in this study. 

The problems surrounding Hamilton s test (vide supra), as well as some misconceptions and misuses encountered in the literature, led Rogers to propose an alternative and more reliable method for determining the absolute structure58. A factor >7 is introduced which multiplies the imaginary component Aff of the anomalous scattering terms of the atomic scattering factors of all atoms (equation 11, which replaces equation 9, see Section 4.2.2,1.1), and which is treated as a variable in the least-squares refinement. 

For analytical studies involving the interplay of flow processes and chemical reactions, we require a set of chemically balanced equations that lead from the known reactants to the measured reaction products. For the complete oxidation of CH4, this last statement implies that we need a reaction scheme producing only H20 and C02. This requirement is incompatible with Equations 1-1 to 1-10. To achieve consistency, we first replace Equation I—1 by the balanced process... 

This equation is transcendental in the unknown /. To solve for / with its two dimensionless parameters e/D and Nre given, we replace equation (3.24) by the equivalent equation in standard form F(f) = 0, namely... 

In the present case, the extension of the scalar counting rules for n(S) to symmetry theorems is straightforwardly achieved by replacing n(S), n(v), n(e) n( > ) by the permutation representations F,T(S), r,T(v), r,T(e) and ra(vj) [13]. A permutation representation /j/v), of a set of objects x has character y(R) under operation (R) of the symmetry group of the undistorted framework, where x(R) is equal to the number of objects wnshifted under operation R. The subscript a is often dropped if there is no danger of confusion. With these replacements, equation (4) becomes... 

Since we replaced the classical partition functions in GR ° and G ° by quantum mechanical ones, we have included quantum effects on all degrees of freedom of the reactants and all but the missing degree of freedom at the transition state. One then includes quantum effects on the remaining degree of freedom by a transmission coefficient k, thereby replacing Equation (3.15) by... 

By first replacing equation 23 into 10, it can be readily shown, that the resulting integral can be expresed as a weighted sum of the moments of the particle size distribution... 

If lithium aluminum deuteride is used, then controlled introduction of deuterium is achieved84. Other solvents such as THF or DME have also been employed76. Under normal circumstances, fluorides are inert but vinyl fluorides may be selectively replaced (equation 8)85-87. Other fluorides may also been reduced if cerium salts or other Lewis acids are added88. 

To analyze this system, we assume that the enzymes obey the flux expression of Equation (4.10). Using this Michaelis-Menten expression, we replace Equation (5.5) with the following... 

The activities (or concentrations) of a species in all of its ionization states and complexed forms are generally summed to obtain the total activity (or concentration) of that species. The number of relations and equilibrium constants needed to describe a reaction such as ATP formation is then reduced to one that is, a separate equilibrium constant is not needed for every possible combination of ionization states and complexed forms of all of the reactants and the products. Using this convention, we can replace Equation 6.10 and many others like it, which also describe ATP formation, by the following general reaction for the phosphorylation of ADP ... 

When the TTS principle is carried out, the non-isothermic crystallization kinetics can be described by replacing argument t in the dependence of non-crystallinity degree of time during isothermic crystallization. The replacement equation is... 

For a number of purposes it is convenient to replace equation (6) by an alternative expression. If equation (A-5) is applied to a closed system consisting of a unit mass of the reacting fluid, then we obtain... 

The restriction to one-step reactions may be removed by replacing equation (19) by equation (21) ZND detonations involving chain reactions can be analyzed more easily than the deflagrations with chain reactions discussed in Section 5.4. 

If Pi,i/Pi,e( d fhen equation (11) merely provides a small correction to equation (6). However, Pi,f/Pi,e(7]) may approach unity for some burning solids. In such cases, the gasification process, equation (1), no longer governs the overall burning rate some other process becomes rate controlling. In extreme cases it is sometimes reasonable and convenient to replace equation (11) by the simpler requirement of surface equilibrium, namely,... 

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