Laplace transform relation

After instantaneous excitation of A, the normalized decay of A is given by R(t) = NA(t)/NA0). According to (3.10), its Laplace transformation, related to the lifetime xA, defines the relative quantum yield of luminescence ... 

Using again the Laplace transform relation between density and Slater sum, one readily obtains... 

C inverse LAPLACE transform relation for the ARRHENIUS equation. 

C generate rate constant using Inverse LAPLACE transform relation ... 

Elimination of Ci and C3 from these equations will result in the desired relation between inlet Cj and outlet Co concentrations, although not in an exphcit form except for zero or first-order reactions. Alternatively, the Laplace transform could be found, inverted and used to evaluate segregated or max mixed conversions that are defined later. Inversion of a transform hke that of Fig. 23-8 is facilitated after replacing the exponential by some ratio of polynomials, a Pade approximation, as explained in books on hnear control theory. Numerical inversion is always possible. 

This equation cannot be integrated directly since the temperature 9 is expressed as a function of two independent variables, distance jc and time t. The method of solution involves transforming the equation so that the Laplace transform of 6 with respect to time is used in place of 9. The equation then involves only the Laplace transform 0 and the distance jc. The Laplace transform of 9 is defined by the relation ... 

Thus, a single-valued connection is established between the kernel of this equation R(t) (a memory function ) and KM(t). Their Laplace transforms R and Km are related by... 

Without resorting to the impact approximation, perturbation theory is able to describe in the lowest order in both the dynamics of free rotation and its distortion produced by collisions. An additional advantage of the integral version of the theory is the simplicity of the relation following from Eq. (2.24) for the Laplace transforms of orientational and angular momentum correlation functions [107] ... 

This relation can be employed to obtain other Laplace transforms. For example, with the use of Eqs. (49) and (50),... 

The combination of Eqs. (28) and (22) gives the Laplace transform of the impulse response H(p) which allows us to solve Eq. (21). By the inverse transformation, the relation which gives the output of the linear system g(t) (the thermogram) to any input/(0 (the thermal phenomenon under investigation) is obtained. This general equation for the heat transfer in a heat-flow calorimeter may be written (40, 46) ... 

Using deterministic kinetics, one can force-fit the time evolution of one species—for example, eh but then those of other yields (e.g., OH) will be inconsistent. Stochastic kinetics can predict the evolutions of radicals correctly and relate these to scavenging yields via Laplace transforms. 

The statistical partition functions are seen to be related by Laplace transformation in the same way that thermodynamic potentials are related by Legendre transformation. It is conjectured that the Laplace transformation of the statistical partition functions reduces asymptotically to the Legendre transformation of MP in the limit of infinitely large systems. 

This involves obtaining the mean-residence time, 0, and the variance, (t, of the distribution represented by equation 19.4-14. Since, in general, these are related to the first and second moments, respectively, of the distribution, it is convenient to connect the determination of moments in the time domain to that in the Laplace domain. By definition of a Laplace transform,... 

A particular vessel behavior sometimes can be modelled as a series or parallel arrangement of simpler elements, for example, some combination of a PFR and a CSTR. Such elements can be combined mathematically through their transfer functions which relate the Laplace transforms of input and output signals. In the simplest case the transfer function is obtained by transforming the linear differential equation of the process. The transfer function relation is... 

The mathematical basis for the exponential series method is Eq. (5.3), the use of which has recently been criticized by Phillips and Lyke.(19) Based on their analysis of the one-sided Laplace transform of model excited-state distribution functions, it is concluded that a small, finite series of decay constants cannot be used to represent a continuous distribution. Livesey and Brouchon(20) described a method of analysis using pulse fluorometry which determines a distribution using a maximum entropy method. Similarly to Phillips and Lyke, they viewed the determination of the distribution function as a problem related to the inversion of the Laplace transform of the distribution function convoluted with the excitation pulse. Since Laplace transform inversion is very sensitive to errors in experimental data,(21) physically and nonphysically realistic distributions can result from the same data. The latter technique provides for the exclusion of nonrealistic trial solutions and the determination of a physically realistic solution. These authors noted that this technique should be easily extendable to data from phase-modulation fluorometry. 

A similar approach, also based on the Kubo-Tomita theory (103), has been proposed in a series of papers by Sharp and co-workers (109-114), summarized nicely in a recent review (14). Briefly, Sharp also expressed the PRE in terms of a power density function (or spectral density) of the dipolar interaction taken at the nuclear Larmor frequency. The power density was related to the Fourier-Laplace transform of the time correlation functions (14) ... 

As shown by van der Laan (V4) and Aris (A7), if Laplace transforms are used to solve Eq. (16), the mean and variance may be easily found from the relations... 

The quantitative treatment for i as a function of a varying T f was first solved analytically by Sevdk in 1948. The solution involves Laplace transformation and the error function complement expressions applied in Vol. I, Section (4.2.11). It is better to quote here the rather simpler equations that can be found if one takes the entire surface as available for the exchange of electrons, i.e., the easy case of 0 = 0. Then (Gileadi, 1993),22 with this assumption, the peak potential is related to the rate constant (Ay) for the interfacial reaction, to the Tafel constant b, and to the sweep rate s, by the equation ... 

The solution of eqn. (44) for a coulomb potential with boundary conditions (45) and (46) for either initial conditions (48) or (49) has only been developed in recent years. Hong and Noolandi [72] showed that the solution of the Debye—Smoluchowski equation is related to the Mathieu equation. Many of the details of their analysis are discussed in the Appendix A, Sect. 4, and the Appendix eqn. (A.21) is the Green s function (fundamental solution), which is the probability that a reactant B is at r given that it was initially at r0. This equation is developed as the Laplace transform. To obtain the density of interest p(r, ), with either condition, the Green s function has to be averaged over the initial distribution, as in eqn. (A.12), and the Laplace transform inverted. Alternatively, the density p(r, ) can be found from the inverse Laplace transform of the linear combination of independent solutions (A.17) which satisfy the boundary and initial conditions. This is shown in Fig. 10. For a Boltzmann initial condition, Hong and Noolandi [72] found... 

Comparing eqns. (170) and (171) shows that the density distribution for the steady-state formation, recombination and scavenging, pss(r cs r0), is closely related to the Laplace transformed (time-dependent) density distribution for recombination and escape. The partially reflecting boundary condition [eqn. (46)] with p replaced by p or pss... 

To show how the lifetime distribution of eqn. (346) is directly related to the diffusion equation analysis, first substitute eqn. (343) into eqn. (346) and take the Laplace Transform... 

After substitution of these expressions into the Laplace transformed linear rate equation (198a), the following relation... 

The theory of kinematic waves, initiated by Lighthill Whitham, is taken up for the case when the concentration k and flow q are related by a series of linear equations. If the initial disturbance is hump-like it is shown that the resulting kinematic wave can be usually described by the growth of its mean and variance, the former moving with the kinematic wave velocity and the latter increasing proportionally to the distance travelled. Conditions for these moments to be calculated from the Laplace transform of the solution, without the need of inversion, are obtained and it is shown that for a large class of waves, the ultimate wave form is Gaussian. The power of the method is shown in the analysis of a kinematic temperature wave, where the Laplace transform of the solution cannot be inverted. 

Note that K(t) is a memory function. Here (S) and K(S) are the Laplace transforms of < >( ) and K(t), respectively. We can now show that the kernel K(t) is related to the autocorrelation function of the random force according to the equation... 

From these relations we see that the width and shift of the power spectrum and consequently the spectroscopic lines are related through the Kronig-Kramers dispersion relations. Exactly the same arguments apply to the Laplace transform of the time-correlation function, H(/co). The real and imaginary parts, C H(co) and C"//(/(0), are related by Kramers-Kronig dispersion relation. 

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