Identical equation

The present derivation can easily be generalized to systems with an arbitrary number of internal degrees of freedom, and it leads to coupled channel equations identical with equation (A3.11.63). where the coupling temis (A3.11.62) are expressed as matrix elements of the interaction potential using states which depend on these internal degrees of... 

An equation identical to (13-129) defines the transformed mole fraction of component i in the vapor phase, Yj, where the terms in x are replaced by terms in y... 

Since Sg satisfies a system of equations identical with that satisfied by X, except that Rg replaces H, a standard method for improving the result of a direct method is to do the replacement and solve as before for the correction. This obtained correction will not, of course, be the true Sg) but only an approximation Sg, and it will have been obtained as a result of a set of operations that is equivalent to the formation of CaRg, in accordance with Eq. (2-3). The Ca = C is not known explicitly, but is defined implicitly (see the methods of factorization below). 

Not only is this equation identical to the form in Eq. (9-16), but we also can interpret the analysis as equivalent to a problem where we want to find K such that the steady state error e(t) approaches zero as quickly as possible. 

Batch partial melting will hereafter be understood as equilibrium melting, which is in contrast to fractional melting discussed in Section 9.3.3. The foundation of this model is remarkably simple and was first laid down by Schilling and Winchester (1967). A number of more or less complex modifications enabling useful information to be extracted from the data were later introduced by Gast (1968), Shaw (1970) and Albarede (1983). Bulk equilibrium crystallization of a liquid batch can be handled with equations identical to those for batch-melting. 

This is not the first time that the kinetics of bulk polymerizations has been analysed critically. Szwarc (1978) has made the same objection to the identification of the rate constant for the chemically initiated bulk polymerization of tetrahydrofuran as a second-order rate constant, k, and he related the correct, unimolecular, rate constant to the reported by an equation identical to (3.2). Strangely, this fundamental revaluation of kinetic data was dismissed in three lines in a major review (Penczek et al. 1980). Evidently, it is likely to be relevant to all rate constants for cationic bulk polymerizations, e.g., those of trioxan, lactams, epoxides, etc. Because of its general importance I will refer to this insight as Szwarc s correction and to (3.2) as Szwarc s equation . 

An equation, identical in form, is obtained for the rate of formation of the trans isomer. 

Following the same derivation as for case 4 of section 5.8 and referring again to Fig. (5.8) (here, however, the concentrations [A] and [A] are at equi-librinm only at the end of the extraction process), we obtain for the interfacial concentrations, equations identical with Eqs. (5.71) and (5.72). By snbstituting them into Eq. (5.92), keeping into acconnt that at the steady state... 

When the rate constants are large numbers (fast reactions), both the numerators and the denominators of Eq. (5.94) can be divided by k-. Since kjk. = A da and ( i) can be neglected relative to the other terms in the sum (A da A + A -I- A ), an equation identical with Eq. (5.73) is obtained for which only diffusion controls the partition rate. At the other extreme, when interfacial film diffusion is very fast, and the reactions are very slow, we can set 1 > QcA + k-i A ) and the rate equation becomes equal to... 

Putting the energy equation into dimensionless form yields an equation identical to Eq. (1-40) with... 

The first attempt to account for surface contamination in creeping flow of bubbles and drops was made by Frumkin and Levich (FI, L3) who assumed that the contaminant was soluble in the continuous phase and distributed over the interface. The form of the concentration distribution was controlled by one of three rate limiting steps (a) adsorption-desorption kinetics, (b) diffusion in the continuous phase, (c) surface diffusion in the interface. In all cases the terminal velocity was given by an equation identical to Eq. (3-20) where C, now called the retardation coefficient , is different for the three cases. The analysis has been extended by others (D6, D7, N2). 

If each normal mode is equally weighted in the total response, equations identical in form to (2.18) and (2.19) also hold for the cubic array since the connection between x and v obtains regardless of dimensionality. Here, g(v) — g3 (v) may be approximated by the familiar Debye equation ... 

This whole process can be repeated, reversing the roles of A and B, so that a large known initial concentration [A]e, is used. As a result, a set of equations identical to Equations 15-13 and 15-14 would be obtained having B in place of A, and b in place of a. The same is true for Figure 15-5. 

The boundary conditions are defined in the same way as with the flow analysis network. The nodes whose control volumes are empty or partially filled are assigned a zero pressure, and the gate nodes are either assigned an injection pressure or an injection volume flow rate. Just as is the case with flow analysis network, a mass balance about each nodal control volume will lead to a linear set of algebraic equations, identical to the set finite element formulation of Poisson s or Laplace s equation. The mass balance (volume balance for incompressible fluids) is given by... 

Reactions 1 and 2 in the diagram shown in Figure 6.13 can be easily reduced to gross equations, identical to conjugated oxidation reactions [115] ... 

From the century s beginning, through its midpoint, the electrochemistry of electrodes was based upon the treatment given by Nemst (Section 7.2.36). This had been derived first, for an interface between a metal and its ions in solution, but the treatment had spread (Planck and Henderson, 1890-1907) to the potential difference between two liquids containing different concentrations of electrolytes. The first of these two treatments yields an equation (Nemst equation) identical in form to the... 

Burg135 derived a means of estimating a power spectrum from the limited number of autocorrelation coefficients of a short time series, without adding extra information. It can be shown that this approach leads to equations identical to an autoregressive LP method, with the power spectrum P(f) given by... 

If the 10 point groups allowed are arranged in nonredundant patterns allowed by the five 2D Bravais lattices, 17 unique two-dimensional space groups, called plane groups, are obtained (Fedorov, 1891a). Surface structures are usually referred to the underlying bulk crystal structure. For example, translation between lattice points on the crystal lattice plane beneath and parallel to the surface (termed the substrate) can be described by an equation identical to Eq 1.10 ... 

The structure of the system in Eq. (2.11) is formally very simple, although apart from the kinematic reversibility of the individual particle motions, which is a consequence of the time invariance of the quasistatic Stokes and continuity equations (Slattery, 1964), very little else can be said explicitly. Equation (2.11) would appear to pose a fruitful future study within the more general framework of dynamical systems (Collet and Eckmann, 1980) whose temporal evolution is governed by a system of equations identical in structure... 

This equation identical to equation 13.12 for slip casting, where a is now the resistance of the frit. Using this equation, a plot of the consolidated layer thickness, L, versus gives a straight line where F is the slope, from which the permeability of the cake of consolidated ceramic powder, o, and frit, a , can be determined. 

The distribution method for studying hydrolysis can be applied to salts of a weak acid, provided a suitable solvent for the acid is available the hydrolysis constant is given by an equation identical with (38), except that m now represents the concentration of free acid in the non-aqueous liquid. The same principle can be applied to the investigation of salts of tw o-sided weakness provided a solvent can be found which dissolves cither the weak acid or the weak base, hut not both. 

Olofsson (1951) also neglected this factor in obtaining an equation for the amounts of sulfate and chloride bound by wool from a mixture of HCl and H2SO4. In subsequent papers (Olofsson, 1952, 1954, 1956) he derived an equation, identical with Eq. (21), based on the alternative assumption that each sulfate ion occupies two positively charged sites. The predictions of the two models were compared with experimental data, but no firm conclusion could be drawn. In part this may have been due to inaccuracies in the determination of the amounts of the various ions bound by the wool. 

For a classical molecular weak base, such as ammonia, the reaction NH3 -i-HjO NH4 -1- OH leads to equations identical to (3-27) or (3-28) with KJK replaced by K, the classical ionization constant of a weak base. 

In the SAE calculations we solve an equation identical to Eqs. (6) and (10) except that the 1/r term is replaced by the proper radial potential corresponding to the particular /-component of the wave function. This does not alter the structure of the coupled equations, and so the propagation is essentially the same as for a hydrogen atom. Note that if we wish to consider the excitation of one of the p-electrons with quantum number m different from zero, the aj coefficients defined in Eqs. (10-12) become... 

The n-component monomolecular system may be treated in exactly the same manner except that an n-dimensional composition space is used. Although n-dimensional spaces with n > 3 cannot be simply put into pictures, a geometrical language still aids our ability to solve problems using the concepts, language, and techniques of two and three dimensional systems. The set of Eqs. (5) reduces to a single equation identical to Eq. (11) except that a is now the column matrix or vector in n-dimensional space given by... 

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