Derivation of expression for

Once the piston-driven flow field is known, the flame-driven flow field is found by fitting in a steady flame front, with the condition that the medium behind it is quiescent. This may be accomplished by employing the jump conditions which relate the gas-dynamic states on either side of a flame front. The condition that the reaction products behind the flame are at rest enables the derivation of expressions for the density ratio, pressure ratio, and heat addition... 

Winefordner JD, Parsons ML, Mansfield JM, McCarthy WJ (1967) Derivation of expressions for calculation of limiting detectable atomic concentration in atomic fluorescence flame spectrometry. Anal Chem 39 436... 

Details concerning derivation of expressions for the four parameters mentioned, based on the model shown in Table 7.1, are found in Vollertsen and Hvitved-Jacobsen (1999). The final expressions are as follows ... 

Derivation of expressions for the potential, A V, and steady state currents,... 

Winefordner, J. D., M. L. Parsons, J. M. Mansfield, and W. J. McCarthy Derivation of Expressions for Calculation of the Limiting Detectable Atomic Concentration in Atomic Fluorescence Flame Spectrometry. Anal. Chem. 39, 436 (1967). 

In general thermoeconomic optimization requires the derivation of expressions for entropy production, via nonequilibrium thermodynamics, due to each independent extensive property transport. 

The prediction of r and as a function of the natures of surfactant, cosurfactant, and hydrophobic phase requires the derivation of expressions for y and C that should account for, among other things, their dependence on the curvature. Such expressions are not yet available. 

The concept of an effectiveness factor ij for the description of the interaction between diffusion and chemical reaction in solid catalyst particles is well understood. Therefore, the derivation of expressions for the effectiveness factor are not elaborated. The following two points should be appreciated ... 

Adsorption isotherm equations can in principle be derived by first formulating the chemical potential of the adsorbate p° in terms of a model, then equating p to p. Although it is not impossible to derive expressions for p by thermodynamic means, statistical approaches are more appropriate because in this way the molecular picture can be made explicit. Moreover, adsorbates are not macroscopic systems, which is a prerequisite for applying thermodynamics, and statistical thermodynamics lends itself very well to the derivation of expressions for the surface pressure. Another approach is based on kinetic considerations expressions for the rates of adsorption and desorption are formulated at equilibrium the two are equal. 

Thus far we have focused on the formal development of quantum-classical dynamics and the derivation of expressions for transport coefficients which utilize this dynamics. We now turn to a discussion of how quantum-classical Liouville dynamics can be simulated for arbitrary many-body systems. 

Levich. V. G. (1962) Physicochemical Hydrodynamics (English translation), Prentice-Hall. Englewood Cliffs, NJ, pp. 80-85. This reference contains much information on diffusion in aqueous solutions. Included are derivations of expressions for mass transfer coeflicienis for different flow regimes with thin concentration boundar layers. 

The derivation of expressions for the multiple kinetic isotope effects of the triple hydrogen transfer case is analogous to the HH-transfer but more tedious. Therefore, the reader is referrred to refs. [25] and [26]. The main results are included in Table 6.2. As in the case of the HH-transfer, the kinetic isotope effects derived for the stepwise transfers are valid in the presence of turmeling and are independent of the tunneling model used. By contrast, the kinetic isotope effects of the single barrier reaction are affected by tunneling. 

In the previous Subsection, the derivation of expressions for the calculation of the electric field gradient at the quasi-relativistic level of theory has been outlined. Similar expressions must be used in order to obtain accurate values for other first-order electrical properties at quasi-relativistic level of theory. The expressions obtained in the present derivation show that at the quasi-relativistic level of theory, first-order properties must not be calculated as pure expectation values of the nonrelativistic property operator, but other operators also appear in the expressions. This is the so called picture-change effect previously discussed in several articles [71-76]. 

Derivation of Expression for Overall F-Factor for Any Chemical Reaction... 

B7. Derivation of Expressions for the Speed of Sound for Ideal and Real Gases... 

Equation (73) simplified by the application of Eq. (52) may be used for fast derivation of expressions for spreading pressures of single-component adsorbates in homogeneous approximation. Introducing the potential as an integration variable, we obtain... 

In order to incorporate the effects of nuclear motion we have to go back to the Hamiltonian, Eq. (2.1), which includes the kinetic energy operators for the nuclei. The corresponding eigenfunctions are the so-called vibronic wavefunctions ) with energy E J and are characterized by the electronic, k, and vibrational, v, quantum numbers, where v stands throughout the chapter collectively for the vibrational quantum numbers of all vibrational modes of the molecule. The proper approach for the treatment of the nuclear motion effects would be to use these unperturbed vibronic wavefunctions R/f ) instead of the unperturbed electronic wavefunctions k ) hi the derivation of expression for... 

The derivation of expressions for the relaxation amplitudes is fairly straightforward in the case of single-step reac-tions. The calculations are more complex for multistep reactions but well within the reach of present computers, as demonstrated in a recent study of micellar equilibria. 

Tr is renamed in Ahf in A/ica and in CacJ. Proceeding in an analogous manner in the derivation of expressions for the other size-dependent phase transitions, one obtains for Tmcn the equation... 

UnUke Flory s combinatorial approach, a Markovian analysis such as tot proposed by Macosko and Miller leads to easy derivations of expressions for M and for nonlinear polymers. Before generalizing to a reaction implying multivalent molecules having valence v, the case of a step-growth polymerization involving X4 tetravalent molecules and Ny Y- -Y molecules wiU be considered. 

Because r and AG appear at the maximum on the free energy-versus-radius curve of Figure 10.2b, derivation of expressions for these two parameters is a simple matter. For r, we differentiate the AG equation (Equation 10.1) with respect to r, set the resulting expression equal to zero, and then solve for r (= r ). That is. 

The derivation of expressions for branch point diffusivity that apply to general polymer branched architectures, and... 

Derivations of expressions for a are limited to crystallization in a uniaxial temperature gradient [10,11] and require additional simplifying assumptions, like linear dependence of G on a spatial coordinate and... 

The derivations of expressions for the self-diffusion coefficient and conductivity are simple in the case of a single crystal which contains a small concentration of diffusant dispersed in a regular lattice of mostly vacant sites separated by identical potential barriers. The frequency of jumps per unit volume if,) is given by the following expression involving the concentration, the barrier height EJ and the thermal energy kT... 

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