Laplace transformation equilibrium

In the context of chemical kinetics, the eigenvalue technique and the method of Laplace transforms have similar capabilities, and a choice between them is largely dependent upon the amount of algebraic labor required to reach the final result. Carpenter discusses matrix operations that can reduce the manipulations required to proceed from the eigenvalues to the concentration-time functions. When dealing with complex reactions that include irreversible steps by the eigenvalue method, the system should be treated as an equilibrium system, and then the desired special case derived from the general result. For such problems the Laplace transform method is more efficient. 

Inputs + Sources = Outputs + Sinks + Accumulation Formulation of differential equations in general is described in Chapter 1. Usually the ODE is of the first or second order and is readily solvable directly or by aid of the Laplace Transform. For example, for the special case of initial equilibrium or dead state (All derivatives zero at time zero), the preceding equation has the transform... 

Having obtained two simultaneous equations for the singlet and doublet correlation functions, X and, these have to be solved. Furthermore, Kapral has pointed out that these correlations do not contain any spatial dependence at equilibrium because the direct and indirect correlations of position in an equilibrium fluid (static structures) have not been included into the psuedo-Liouville collision operators, T, [285]. Ignoring this point, Kapral then transformed the equation for the singlet density, by means of a Laplace transformation, which removes the time derivative from the equation. Using z as the Laplace transform parameter to avoid confusion with S as the solvent index, gives... 

There are three points to emphasize. First, the expressions for the concentration or concentration gradient distribution for non-sector-shaped centerpieces can be applied to other methods for obtaining MWD s, such as the Fourier convolution theorem method (JO, 15, 16), or to more recent methods developed by Gehatia and Wiff (38-40). The second point is that the method for the nonideal correction is general. Since these corrections are applied to the basic sedimentation equilibrium equation, the treatment is universal. The corrected sedimentation equilibrium equation (see Equation 78 or 83) forms the basis for any treatment of MWD s. Third, the Laplace transform method described here and elsewhere (11, 12) is not restricted to the three examples presented here. For those cases where the plots of F(n, u) vs. u will not fit the three cases described in Table I, it should still be possible to obtain an analytical expression for F(n, u) which is different from those in Table I. This expression for F (n, u) could then be used to obtain an equation in s using procedures described in the text (see Equations 39 and 44). Equation 39 would then be used to obtain the desired Laplace transform. 

Here, (Q2) is the fluctuation of the Brownian oscillator coordinate at equilibrium, the classical and quantum statistical expressions of which are given in Table L.l Now, take the Laplace transform of both sides of Eq. (L.3),... 

Here we show how the modified Kubo formula (187) for p(co) leads to a relation between the (Laplace transformed) mean-square displacement and the z-dependent mobility (z denotes the Laplace variable). This out-of-equilibrium generalized Stokes-Einstein relation makes explicit use of the function (go) involved in the modified Kubo formula (187), a quantity which is not identical to the effective temperature 7,eff(co) however re T (co) can be deduced from this using the identity (189). Interestingly, this way of obtaining the effective temperature is completely general (i.e., it is not restricted to large times and small frequencies). It is therefore well adapted to the analysis of the experimental results [12]. 

The main results of this sectionjire the out-of-equilibrium generalized Stokes-Einstein relation (203) between Ax2(z) and p(z), together with the formula (206) linking Teff(ffi) and the quantity, denoted as ( ), involved in the Stokes-Einstein relation. One thus has at hand an efficient way of deducing the effective temperature from the experimental results [12]. Indeed, the present method, which avoids completely the use of correlation functions and makes use only of one-time quantities (via their Laplace transforms), is particularly well-suited to the interpretation of numerical data. 

When the environment of the particle is itself out-of-equilibrium, as is the case for a particle evolving in an aging medium, we showed how the study of both the mobility and the diffusion of the particle allows one to obtain the effective temperature of the medium. We derived an out-of-equilibrium generalized Stokes-Einstein relation finking the Laplace transform of the mean-square displacement and the z-dependent mobility. This relation provides an efficient way of deducing the effective temperature from the experimental results. 

We notice that the correlation function defined by Eq. (147) is stationary. Thus, it fits the Onsager principle [101], which establishes that the regression to equilibrium of an infinitely aged system is described by the unperturbed correlation function. The authors of Ref. 102 have successfully addressed this issue, using the following arguments. According to an earlier work [96] the GME of infinite age has the same time convoluted structure as Eq. (59), with the memory kernel T(t) replaced by (1>,XJ (f). They proved that the Laplace transform of Too is... 

In addition to the enhanced diffusivity effect, another issue needs to be taken into account when considering stationary-phase mass transfer in CEC with porous particles. The velocity difference between the pore and interstitial space may be small in CEC. Under such conditions the rate of mass transfer between the interstitial and pore space cannot be very important for the total separation efficiency, as the driving mechanism for peak broadening, i.e., the difference in mobile-phase velocity within and outside the particles, is absent. This effect on the plate height contribution II, s has been termed the equilibrium effect [35], How to account for this effect in the plate height equation is still open to debate. Using a modified mass balance equation and Laplace transformation, we first arrived at the following expression for Hc,s, which accounts for both the effective diffusivity and the equilibrium effect [18] ... 

From the experimental standpoint, the use of a.c. techniques offers many advantages. Sensitivity is much higher than in d.c. measurements, since phase-sensitive detection can be used and very small probe signals can be employed ( 5mV). The technique is therefore a truly equilibrium one, unlike cyclic voltammetry. An alternative approach to the commonly used sinusoidal signal superimposed on the selected d.c. potential is to use a potential step and to employ Laplace transform methods. Instrumentally, this is rather more demanding and the advantages are not clear [51]. Fourier transform methods have also been considered and their use will have advantages in terms of the time-scale for an experiment, especially at very low frequencies. 

The matrix equation for the Laplace transform fllxl (k, z) of the equilibrium time correlation functions flxl(k, t), where... 

In order to draw the main ideas of the GCM approach let us recall a general representation for the Laplace transform of an equilibrium time correlation function = (A t) A k(0)), derived by Mori [39],... 

We compare in Figure 6.20 two profiles that were calculated as numerical solutions of the equilibrium-dispersive model, using a linear isotherm. The first profile (solid line) is calculated with a single-site isotherm q = 26.4C) and an infinitely fast A/D kinetics (but a finite axial dispersion coefficient). The second profile (dashed line) uses a two-site isotherm model q — 24C - - 2.4C), which is identical to the single-site isotherm, and assumes infinitely fast A/D kinetics on the ordinary sites but slow A/D kinetics on the active sites. In both cases, the inverse Laplace transform of the general rate model given by Lenhoff [38] (Eqs. 6.65a to h) is used for the simulation. In the case of a surface with two t5q>es of adsorption sites, Eq. 6.65a is modified to take into accoimt the kinetics of adsorption-desorption on these two site types. 

We use the same symbol as in Section IX.D, but this should not lead to confusion.) Now, however, since F(12)=F(12, t=0) is arbitrary rather than an equilibrium distribution function, it is no longer true that F(12) = 0. Thus using projection operator methods on the Laplace transform of (11.1), we find... 

The quantities qi, qi and have the dimensions cm-1. They are inverse screening lengths. Later we shall see that qo also plays a role in the equilibrium structure of the ionic solution. Equation (9.2.15) can be solved by means of Laplace transformation with respect to time. When the solutions are multiplied by <5c (q, 0) and Sc (q, 0) and ensemble-averaged we obtain... 

Of possible computational interest is the fact that in Eq. (247) Qi/ T) may be regarded as the Laplace transform of j S,/ (E), and tables of such transforms and their inverses are available. If we now further assume an equilibrium distribution in internal states of the reactants, we can write for the partition function, ignoring degeneracies for simplicity. ... 

Schmidt number see equation (9-125) number of adsorption transfer units = kadsL/v) number of equilibrium stages reaction order Laplace transform parameter... 

In Eq. [7], the frequency-dependent friction is the Laplace transform of the time-dependent friction The presence of the Laplace transform means that the time-dependence of the friction must be known in order to determine the Laplace transform. This friction can be readily determined from molecular dynamics simulations in the approximation where the motion along the reaction coordinate is fixed at x = 0. (A discussion of some subtle, but important, aspects of this approximation is given by Carter et al. ) In that case, the random force R(t) can be calculated from equilibrium dynamics in the presence of this one constraint. From R(t), the time-dependent friction (t) can be calculated and the implicit Eq. [7] solved. The result gives the Grote—Hynes value of the transmission coefficient for that system. 

Laplace transform of C axial dispersion coefficient, m /s Henry s law solubility constant, Cl/Cq. adsorption equilibrium constant radial distance in the catalyst particle, radius of the catalyst particle Laplace transform variable t ime, s. 

Because the equilibrium ensemble is time translationally invariant, C is a function of t — t. Since we will be more interested in the Fourier and Laplace transforms of (19), we introduce the definitions... 

Solve by the method of Laplace transforms the population balance equation given in Section 3.3.6.2 under equilibrium conditions. And show that the solution is an exponential distribution. 

It is crucial to have good equilibrium pair distribution functions up to concentrations at least as high as IM, in order to be able to calculate transport properties up to these concentrations. It is the case when the Laplace transforms of the MSA distribution functions [27] are used in place of the Debye-Hiickel distribution functions. 

In other words, a system of spins fliat have perturbed from their equilibrium state return to the equilibrium condition according to an exponential law characterized by time constant T. We shall use this fact shortly as a rationale for invoking the Laplace transform and associated transfer functions. 

We have four partial differential equations and two equilibrium equations. Laplace transform is applied to solve four variables Ce(t), C e(t), Cm(r,t) and Ci(r,t). 

The problem for the diffusion-controlled adsorption of a mixture of surfactants at a small deviation from equilibrium can be solved analytically by using the Laplace transform the resulting long expressions can be found in Ref. 73. The results are compared with experimental data for different alkanoates at the air-liquid interface [73]. Simple adsorption isotherms for mixtures of surfactants are reported in Refs. 31 and 44 surface equations of state applicable to mixtures of surfactant molecules of different size can be found in Res. 27 and 74. 

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