Equation direction

The measurement of pK for bases as weak as thiazoles can be undertaken in two ways by potentiometric titration and by absorption spectrophotometry. In the cases of thiazoles, the second method has been used (140, 148-150). A certain number of anomalies in the results obtained by potentiometry in aqueous medium using Henderson s classical equation directly have led to the development of an indirect method of treatment of the experimental results, while keeping the Henderson equation (144). 

In the special case when d = 1 and of = 1, that is, when if a fault occurs it will not be detected and that failure is certain, occurrence equates directly to the probability of a fault. The probability of a fault in turn depends on the capability of the process used for the component/characteristic. 

The differential rate equations of a complex reaetion, expressing rates as functions of concentrations, are usually simpler in form than are the corresponding integrated equations, whieh express concentrations as funetions of time moreover, it is always possible to write down the differential rate equations for a postulated kinetie seheme, whereas it may be difficult or impossible to integrate them. Of course, we usually measure concentration as a funetion of time. If, however, we can measure rates, we may use the differential equations directly. 

The quantitative description of enzyme kinetics has been developed in great detail by applying the steady-state approximation to all intermediate forms of the enzyme. Some of the kinetic schemes are extremely complex, and even with the aid of the steady-state treatment the algebraic manipulations are formidable. Kineticists have, therefore, developed ingenious schemes for writing down the steady-state rate equations directly from the kinetic scheme without carrying out the intermediate algebra." -" ... 

The investigation above is due initially to Gibbs (Scient. Papers, I., 43—46 100—134), although in many parts we have followed the exposition of P. Saurel Joum. Phys. diem., 1902, 6, 474—491). It is chiefly noteworthy on account of the ease with which it permits of the deduction, from purely thermodynamic considerations, of all the principal properties of the critical point, many of which were rediscovered by van der Waals on the basis of molecular hypotheses. A different treatment is given by Duhem (Traite de Mecanique chimique, II., 129—191), who makes use of the thermodynamic potential. Although this has been introduced in equation (11) a the condition for equilibrium, we could have deduced the second part of that equation directly from the properties of the tangent plane, as was done by Gibbs (cf. 53). 

To calculate the wavefunction for any particle we use Schrodinger s great contribution, the Schrodinger equation. Although we shall not use the equation directly (we shall need to know only the form of some of its solutions, not how those solutions are found), it is appropriate at least to see what it looks like. For a particle of mass m moving in a region where the potential energy is V(x) the equation is... 

Note that neither temperature nor a change in temperature equates directly with the heat flow, q. We can relate q to A T only if we know the identity and amount of the material that undergoes a change of state. Example provides some practice. 

Determination of Current Distributions Governed by Laplace s Equation Direct Methanol Fuel Cells From a Twentieth Century Electrochemist s Dream to a Twenty-First Century Emerging Technology West, A. C. Newman, J. Lamy, C. Ueger, J.-M. Srinivasan, S. 23... 

In principle, the task of solving a linear algebraic systems seems trivial, as with Gauss elimination a solution method exists which allows one to solve a problem of dimension N (i.e. N equations with N unknowns) at a cost of O(N ) elementary operations [85]. Such solution methods which, apart from roundoff errors and machine accuracy, produce an exact solution of an equation system after a predetermined number of operations, are called direct solvers. However, for problems related to the solution of partial differential equations, direct solvers are usually very inefficient Methods such as Gauss elimination do not exploit a special feature of the coefficient matrices of the corresponding linear systems, namely that most of the entries are zero. Such sparse matrices are characteristic of problems originating from the discretization of partial or ordinary differential equations. As an example, consider the discretization of the one-dimensional Poisson equation... 

Of course, this equation directly follows from Fig. 3.3. Multiplying Equation (3.34)... 

The last part of Eq. (1) is derived from the pH dependence of permeability, given a pH gradient between the two sides of the intestinal barrier, based on the well known Henderson-Hasselbalch equation. Direct measurement of in situ intestinal perfusion absorption rates confirmed this pH dependence [14]. 

Ouchiyama and Tanaka (Ol) did not solve this equation directly, but from the inspection of published data they assumed that the granule size is roughly uniformly distributed and self-preserving ... 

Heller equations, direct molecular dynamics, Gaussian wavepackets and multiple spawning, 399-402 Hellmann-Feynman theorem ... 

Nuclear motion Schrodinger equation direct molecular dynamics, 363-373 vibronic coupling, adiabatic effects, 382-384 electronic states ... 

The results shown in Figs. 6 and 7 can be combined to compute the mean mixture fraction ( ) and its variance (c2) from Eqs. (34) and (35), respectively. Example plots are shown in Liu and Fox (2006) and, as expected, they agree with the solution found by solving the moment transport equations directly (Eqs. 28 and 29). 

Comparison of these equations directly yields the fundamental theorem... 

PE Poisson equation (direct solution in physical space)... 

Another method for determining the activation energy involves using a modification of the Arrhenius equation. If we try to use the Arrhenius equation directly, we have one equation with two unknowns (the frequency factor and the activation energy). The rate constant and the temperature are experimental values, while R is a constant. One way to prevent this difficulty is to perform the experiment twice. We determine experimental values of the rate constant at two different temperatures. We then assume that the frequency factor is the same at these two temperatures. We now have a new equation derived from the Arrhenius equation that allows us to calculate the activation energy. This equation is ... 

Electrochemical experiments fall into two broad categories. Some experiments are concerned with standard cell voltages, while other experiments use the Nernst equation directly or indirectly. Experiment 21 in the Experimental chapter uses these concepts. 

Recently, the Pitzer equation has been applied to model weak electrolyte systems by Beutier and Renon ( ) and Edwards, et al. (10). Beutier and Renon used a simplified Pitzer equation for the ion-ion interaction contribution, applied Debye-McAulay s electrostatic theory (Harned and Owen, (14)) for the ion-molecule interaction contribution, and adoptee) Margules type terms for molecule-molecule interactions between the same molecular solutes. Edwards, et al. applied the Pitzer equation directly, without defining any new terms, for all interactions (ion-ion, ion-molecule, and molecule-molecule) while neglecting all ternary parameters. Bromley s (1) ideas on additivity of interaction parameters of individual ions and correlation between individual ion and partial molar entropy of ions at infinite dilution were adopted in both studies. In addition, they both neglected contributions from interactions among ions of the same sign. 

Many interesting phenomena can arise in nonlinear periodic structures that possess the Kerr nonlinearity. For analytic description of such effects, the slowly varying amplitude (or envelope) approximation is usually applied. Alternatively, in order to avoid any approximation, we can use various numerical methods that solve Maxwell s equations or the wave equation directly. Examples of these rigorous methods that were applied to the modelling of nonlinear periodical structures are the finite-difference time-domain method, transmission-line modelling and the finite-element frequency-domain method." ... 

D. A. Mazziotti, Anti-Hermitian contracted Schrodinger equation direct determination of the two-electron reduced density matrices of many-electron molecules. 97, 143002 (2006). 

Figure 2.18 Balance of energies at the TPL gives the Young equation directly.

It remains to calculate the coefficients that define the interpolating spline. One can obviously solve the 4(n-l) constraint equations directly, but there exists a much more efficient algorithm. Let and mi+1 denote the... 

Direct scale-up may be used to obtain a relief system size that is less conservative than the DIERS equation. Direct scale-up and its many conditions of applicability are detailed in A5.12. A direct scale-up test is only applicable if the test reactor empties totally by two-phase relief161, and the applicability of the method can therefore only be assessed after the scale-up test has been performed. Direct scale-up may not be feasible if the reacting system contains solids with a particle size similar to or larger than the diameter of the small-scale relief system. 

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